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The search for tensor interactions in the beta-decay of polarized ⁸⁰Rb Pitcairn, Robert Andrew 2007

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The Search for Tensor Interactions in the Beta-decay of Polarized 8 0 Rb by Robert Andrew Pitcairn B.Sc, The University of Manitoba, 2004 A THESIS SUBMITTED IN PARTIAL F U L F I L M E N T O F T H E REQUIREMENTS FOR T H E D E G R E E OF Master of Science in The Faculty of Graduate Studies (Physics) The University Of British Columbia September 20, 2007 © Robert Andrew Pitcairn, 2006 Abstract The spin polarized nuclear recoil asymmetry from /? emission is nearly zero in the standard model. This observable is sensitive to tensor interactions which exist in certain standard model extensions. The nuclear recoil energy is very small (a few eV) and therefore requires a sensitively controlled environment for measurement, the TRINAT atom trap provides such an envi-ronment. Rudidium-80 is an unstable isotope which /^ -decays (positron emission or electron cap-ture) with a half life of 30s. It is produced in large quantities at the ISAC facility located in TRIUMF. This isotope offers favorable nuclear and atomic properties for measurement in the TRINAT apparatus. Rubidium-80 is trapped in a vacuum by lasers combined with a magnetic field, and polarized with another laser. When the trapped Rubidium decays the sudden change of nuclear charge typically ejects a few low energy atomic electrons leaving a positive ion. An electric field accelerates the ions towards a position sensitive microchannel plate and the electrons to another microchannel plate. The direction of polarization is parallel to the plates surface, so an asymmetry manifests itself as a difference in the distribution of ion impacts when the polarization is inverted. The ion time of flight is used to discriminate between positron emission and electron capture events. This discrimination is required since electron capture has a large asymmetry in the standard model which would overwhelm the desired observable. There is a small polarization asymmetry expected to occur even within the standard model due to "recoil order" corrections to the V - A theory. Since these "recoil order" correc-tions have yet to be theoretically calculated, they are left as fit parameters. Unfortunately the data is statistically insufficient to fit both the "recoil order" corrections and the tensor couplings simultaneously. However, if the "recoil order" corrections are fixed to a crude theo-retical estimate a fit for tensor couplings sets limits consistent with zero and complementary to other experimental results, namely nuclear recoils in He-6 and positron-polarization from C-10. Theoretical limits based on neutrino mass and naturalness arguments remain more restrictive at this time. ii Table of Contents Abstract ii Table of Contents iii List of Tables v i List of Figures y i i i Acknowledgments xi Dedication x i i 1 Introduction 1 1.1 Introduction 1 1.2 History of Weak Interactions 1 1.3 Motivation 4 2 Nuclear Theory 6 2.1 /3-decay Overview 6 2.2 Parity Violation 8 2.2.1 Parity Violation in /3-decay 8 2.2.2 Parity Violation in Electron Capture 10 2.3 Proton and Neutron structure effects 10 2.4 Nuclear /3-decay beyond the Standard Model 14 2.5 Recoil Asymmetry 15 2.6 Sources of Tensor Interactions 16 2.6.1 Tensor Interactions at the quark level . 16 2.6.2 Supersymmetry 17 2.6.3 Technicolor 19 2.6.4 Current Tensor Limits 19 2.7 Nuclear /3-decay rate including all model extensions 20 iii Table of Contents 2.8 Complete Recoil Momentum Distribution 21 3 Atomic Physics 27 3.1 Magneto-Optical Traps (MOTs) 27 3.1.1 1-D M O T 27 3.2 Optical Pumping 30 3.3 Shakeoff Electrons 31 4 Experimental 32 4.1 Overview 32 4.2 8 0 R b 33 4.3 TRINAT Apparatus 33 4.3.1 Optical Pumping Details 36 4.3.2 Electric Field 37 4.4 TRINAT Experiment Timing 39 4.5 Detectors 40 4.5.1 M C P i 40 4.5.2 Bias Tee 42 4.5.3 /3-detectors 42 4.6 Main Data 43 4.6.1 M C P distribution 44 4.6.2 Time of flight [TOFe] (e_detector stop) 46 4.6.3 Photoion Time of Flight (photodiode stop) 47 4.6.4 Optical Pumping Excited state population [OPex] 47 4.7 Data Cuts . 50 4.7.1 M C P distribution (with cuts) 50 4.7.2 Asymmetries 53 4.7.3 TOFe with cuts 55 4.7.4 Photodiode T O F with cuts 55 4.8 Auxiliary Data Sets 56 4.8.1 CaF T O F (/?-detector stops) 56 4.8.2 M C P T O F (ion M C P stops) 58 4.9 Connection between observables and 8, Pr 59 5 Analysis 62 5.1 Fitting Photoion T O F 62 5.2 Polarization 64 Table of Contents 5.3 TOFe fit 65 5.3.1 TOFe background subtraction 65 5.3.2 Position Distribution fit 69 5.3.3 TOFe fit algorithm 69 5.3.4 TOFe systematics 73 5.4 Asymmetry analysis 74 5.4.1 Asymmetry Fitting Algorithm 75 5.5 Asymmetry systematics 79 5.6 Potential Sensitivity 80 6 Improvements 82 6.1 Polarization 82 6.2 Gamma detection 82 6.3 Charge State Separation 83 6.4 M C P Upgrade 83 Bibliography 84 A Decay correlation functions 87 B Fermi function coefficients 92 B.0.1 Fermi function 92 C Recoil Order Corrections to the Momentum Distribution 97 D Simulated Asymmetry Distributions 106 E Electric field Generation and Systematics 113 E . l Electric Field 113 E.1.1 Field Source • 113 E . 1.2 Field corrections 115 F Germanium 118 F. 0.3 Germanium 118 G Glossary of common abbreviations 120 v List of Tables 2.1 Allowed approx. selection rules 7 2.2 1 s t forbidden approx. selection rules 8 2.3 SM /3-decay coefficients 9 2.4 Non-relativistic reductions of hadronic currents 15 4.1 Potentials of Hoops creating the electric field in the TRINAT apparatus. . . 38 4.2 Charge state TOFe timing cuts 54 5.1 Fit parameters for Photoion T O F analysis 63 5.2 Division of charge-states in CaF T O F spectrum in relative time units. . . . 67 5.3 Division of charge-states in M C P T O F spectrum in relative time units. . . . 67 5.4 Division of charge-states for background spectrum to be subtracted from TOFe. 68 5.5 Time of flight fit parameters and statistical uncertainties 72 5.6 Time of flight fit parameters and statistical uncertainties for fit with trap width minus one sigma 73 5.7 Time of flight fit parameters and statistical uncertainties for fit with trap width plus one sigma 73 5.8 Hvalues for two runs 74 5.9 Final values and systematics 74 5.10 Real time cuts used to excluded electron capture events 75 5.11 Final values and uncertainties 77 5.12 Potential statistical limits on tensor coefficients 80 B . l Coefficients making a_i term 94 B.2 Coefficients making ao term 94 B.3 Coefficients making a\ term 95 B.4 Coefficients making a<i term 95 B.5 Coefficients making term 95 B.6 Coefficients making term 95 vi List of Tables B.7 Coefficients making as term 96 B.8 Coefficients appearing in Taylor expansion of Fermi function 96 E . l Potentials of Hoops creating the electric field in the TRINAT apparatus. . . 114 E.2 Coefficients of the polynomial used for modeling the electric field.. 115 List of Figures 1,1 TRINAT detection setup 5 2.1 Tensor Exclusion plot 20 2.2 N(6,\Pr\ distribution 22 2.3 C(6,\Pr\ distribution 23 2.4 L(6, \Pr| distribution 24 2.5 Momentum distribution for ground state decay of 8 0 R b . 25 3.1 M O T restoring force 28 3.2 ID M O T Force 29 3.3 Optical Pumping schematic . 30 4.1 8 0 R b decay scheme 34 4.2 8 0 R b round state and D2 energy levels '. 35 4.3 Birds eye view of TRINAT apparatus 36 4.4 Cross section of the mount assembly holding field hoops in place 38 4.5 Experiment timing 39 4.6 Schematic view of an M C P Z stack. . 41 4.7 Theoretical efficiency curve for electron detection in an M C P 42 4.8 Timing signal from the electron M C P 43 4.9 Raw M C P singles 44 4.10 Raw M C P singles (Projected on Polarization axis) 45 4.11 Time of flight from e~ detector stops 46 4.12 TOFe background spectrum 47 4.13 Photodiode T O F spectrum (photodiode stops) 48 4.14 OP photoions . 49 4.15 M C P singles with "GoodMCP" condition 50 4.16 M C P singles with "cgplus","krlnec" and "GoodMCP" conditions 51 4.17 M C P singles (with cuts). 52 viii List of Figures 4.18 Trap distribution during optical pumping 53 4.19 Asymmetry for charge state one. 54 4.20 Asymmetry for charge state two 54 4.21 Asymmetry for charge state three 55 4.22 This plot shows how TOFe spectrum changes with cuts 56 4.23 Photodiode T O F with cuts 57 4.24 CaF T O F (Calcium Fluoride stops) 58 4:25 M C P T O F (Position sensitive M C P stops) 59 4.26 Definition of Coordinate system 60 5.1 Photoion T O F fit . . . . . . . . . . 63 5.2 Shakeoff electron detection efficiency for various charge states 68 5.3 Fit to photoion position data 69 5.4 Fit to TOFe spectrum. 72 5.5 Position spectra for recoils of charge states 1 to 3 76 5.6 This shows the charge-state 1 fit (dashed line) and data 78 5.7 This shows the charge-state 2 fit (dashed line) and data 78 5.8 This shows the charge-state 3 fit (dashed line) and data 79 5.9 Limits on tensor interactions from this experiment 81 B. l (3+ energy spectrum with and without Fermi function 94 C. l bgsM spectrum 98 C.2 bgsc spectrum 99 C.3 bgsr, spectrum 100 C.4 dgsN spectrum 101 C.5 dgsc spectrum 102 C.6 dgsL spectrum 103 C.7 ipgsc spectrum 104 C. 8 4>gsc spectrum 105 D. l Asymmetry for three different values of bgs 106 D.2 Asymmetry for three different values of bex : . 107 D.3 Asymmetry for three different values of dgs. . . 108 D.4 Asymmetry for three different values of dex 109 D.5 Asymmetry for three different values of fex. . 110 D.6 Asymmetry for three different values of ip I l l List of Figures D. 7 Asymmetry for three different values of <p 112 E. l Cross section of the mount assembly holding field hoops in place 114 E.2 Electric field from Simlon calculation 115 E.3 TOF for various initial velocities . 116 E. 4 Systematic differences in TOF translate into systematics in position 117 F. l - Gamma ray spectrum from 80Rb decay. • • • • 119 x Acknowledgments I have been very fortunate to work with such great people throughout the course of my masters. I'm not very good at writing this sort of thing but I'll do my best, sorry if its not all that poetic, but its all completely honest. I would like to thank John Behr my supervisor for always being available to help me with any issues I may have both big and small. John provided with an excellent second opinion on a variety of analysis techniques I attempted, and always steered me on the right track. Matt Pearson has provided invaluable insights into atomic/laser physics, as well as helping with shifts for my experiment, thank you. I would like to thank Daniel Roberge for helping with his portion of the analysis, but more importantly for giving me a friendly face to talk to on a daily basis for two years. Even if that friendly face didn't make it to work in the morning! Connie Hoehr helped me with handling the electric field in this experiment. Thank you very much. I have also received assistance from co-op students along the way: Andre Gaudin,Baborra Dej. Thank you both very much. I would like to thank both Tom Mattison and Janis McKenna for giving my thesis a thorough reading and helping me significantly elevate its quality. I would also like to thank my girl friend Melissa Iverson. Although she did nothing to help with my experiment she has been very supportive during the actual writing my of thesis, thanks hon! x i Dedication This is dedicated to my parents...all four of them. I wouldn't be able to do this without their love and support. xii Chapter 1 Introduction 1.1 Introduction How does one search when one doesn't know what one is searching for? This is the problem currently faced by the particle physics community. The Standard Model (SM) is quite pos-sibly the greatest intellectual triumph of the 20th, century, endowing us with fundamental insight into the subatomic world. The SM has given us theoretical predictions with un-precedented precision (eg, the anomalous magnetic moment of the muon (g-2)) and at the same time provided excellent overall agreement with world data. Despite this remarkable success, it is well known that the SM doesn't answer all questions (What is the origin of Dark Energy? What causes the observed mass hierarchy? Why are there three generations of particles?). The SM is thus not likely a final theory. The question is then "where do we look for a new theory?". Ironically, here the strength of the SM turns out to be its weakness. We are unable to find any shortcomings in SM predictions, so we don't have any insight where to search for a new theory. Theorists have proceeded by proposing extensions to the SM that are still consistent with SM data: SO(10), E(8) and MSSM. However, a theory is only as good as its agreement with experiment, so the onus is now on experimentalists to test these theories and provide new insights as to how we can extend our beloved SM. This thesis will discuss an experiment aimed at constraining non-SM contributions to weak interactions. When searching for SM extensions there are generally two approaches that are taken: smashing particles together with tremendous force with the aim of directly creating new particles ("energy frontier approach") or performing experiments at lower energies and inferring particle existences through interferences and loop effects ("low energy approach"). The experiment described in this thesis takes the low energy approach to particle detection. 1.2 History of Weak Interactions The history of weak interactions dates back to 1900, when Becquerel identified that the /3-rays from certain radioactive decays are actually just electrons. The first successful weak 1 Chapter 1. Introduction interaction theory came much later in 1934 and was developed by Enrico Fermi to describe the decay of the neutron [1]. This theory was developed in close analogy to what had previously been done by Dirac with QED in 1927. Fermi proposed the following Lagrangian for the weak decay of the neutron to a proton electron and anti-neutrino: Lp = - ^ J p l l ^ n y e l ^ v e (1.1) In equation 1.1 ^ p is the proton wave function, \&n is the neutron wave-function, \&e is the electron (fi) wave function, -tyve is the electron neutrino wave-function, -y^ are the Dirac matrices and GF is the weak interaction coupling strength. With the addition of the Hermitian conjugate this Lagrangian (equation 1.1) was extended to accommodate both electron capture and positron decay which were observed shortly thereafter [2],[3]. Fermi's theory of /3-decay successfully explained a large class of observed /^ -decays. However, the Fermi theory predicted all the most probable decays to have no change in nuclear spin, and this was not the case. In order to solve this puzzle Gamow and Teller introduced another term into the Hamiltonian [4] which in modern notation can be written as follows: L/3 = ^ % 7 ^ n % l ^ u e (1-2) All variables in equation 1.2 are the same as equation 1.1. In the 40's and 50's it became apparent that other reactions existed in nature with a coupling strength similar to that of /3-decay. Pion and muon decay were deemed to be caused by the same mechanism, and this led to the idea of a universal weak interaction. It was not until 1956 that parity non-conservation of weak interactions was considered by Lee and Yang [5]. The transformation properties (vector, axial vector, scalar, pseudo-scalar, tensor) of the weak interaction were unknown at the time, so in their ground-breaking paper they considered the most general parity non-conserving Hamiltonian for /3-decay. Hf>= % ( [ * P * n ] [ C S * e ^ + C ^ e 7 5 ^ ] + [* P 75*n] [ C P * e 7 5 * , + Cp*e*v]) (1.3) In equation 1.3 C/'s and Cj's are arbitrary couplings for: vector (V), axial vector (A), 2 Chapter 1. Introduction scalar (S), pseudo-scalar (P) and Tensor (T) weak interaction components. In 1957 parity violation was experimentally confirmed by Wu et al. [6], validating the proposal of Lee and Yang. It remained only to find values for the coupling constants given in the above Hamiltonian to completely describe the weak interaction. Since the nucleus can be described with non-relativistic quantum mechanics, the pseudo-scalar portion must be small or zero, since this disappears when we reduce the Hamiltonian to the non-relativistic limit. A series of experiments on the polarization of the emitted leptons and the /? neutrino correlation in 35Ar allowed the determination that the primary structure was vector and axial vector [7],[8],[9] and showed if there were any tensor or scalar components, they had to be small. Experiments on the decay rates of the 0 + —> 0 + super allowed transitions enabled the determination of the vector coupling constant. These transitions have no Axial vector component, and their matrix elements can be very precisely calculated. The axial vector coupling constant was then derived from the neutron lifetime, which is proportional to Cy + 3C\. Although the theory of weak interactions enjoyed enormous success, it had obvious flaws. The most obvious was that it gave non sensible results for scattering at higher energies. This is caused by the fact that as written, the weak interaction has zero range and is thus a contact interaction. All interactions of this type have similar troubles. If the analogy between QED and weak interactions is pursued further at this point, we can create a spin one massive vector field and include this in the Lagrangian. However, even the introduction of this propagator term in the amplitude doesn't resolve all the non sensible results that arise at higher energies. This problem would go 20 years unresolved. The modern theory of weak interactions dates back to the work of Yang and Mills in 1954 [10]. They developed an SU(2) invariant theory with the hopes of describing proton and neutron interactions. Their theory however, had the problem of requiring three massless spin one bosons which clearly did not exist. In 1964 Higgs showed that by introducing a spontaneously broken symmetry one could give mass to the massless bosons in a gauge theory [11]. The result derived by Higgs was combined with the SU(2) theory of Yang and Mills in 1967 by Steven Weinberg [12]. Weinberg's idea was to group the left handed leptons into isodoublets under SU(2) and the right leptons into isosinglets under SU(2). This was extended to quarks as well using the GIM mechanism in 1970 [13]. This bold theory gave agreement with all current data and predicted weak neutral currents, which were later observed in 1973 [14]. The theory was mathematically validated in 1971 by't Hooft, who proved the theory was renormalizable [15],[16]. 3 Chapter 1. Introduction 1.3 Motivation In 1958 the V - A structure of the weak interaction was not yet firmly established. Using the Hamiltonian in equation 1.3, theorists calculated a variety of experimental observables. The observable of interest in this experiment was calculated in 1958 by Treiman [17]. Treiman calculated the lowest-order angular distribution of recoil-nuclei from spin-polarized /3-decay (equation 1.4). W(9)d6 = d = In equation 1.4, W is the angular distribution, 6 is the angle between the recoils mo-mentum and the polarization axis, J is the nuclear spin of the initial state, Jz is the initial state nuclear spin projection along the axis of polarization, Ap, Bv, c, x\ and xi are defined in Appendix A. In equation 1.5 the term multiplying c to create c' will be referred to as Treiman's alignment. Our current knowledge of the weak interaction dictates that it is predominantly V-A, and any other contributions must be small. In the case of a Gamow-Teller type decay the asymmetry term (Ap + Bv) in equation 1.4 is non-zero only if the weak interaction has a tensor component. Since any tensor component present must be small, this observable offers the great advantage of large signal to background ratio. In the lowest order approximation any asymmetry whatsoever would indicate new physics. In order to measure the spin-polarized recoil-nuclei angular distribution the experimen-talist requires: a source of radioactive isotopes whose recoils can freely escape once they decay, a method of detecting the recoils and a method of inducing polarization in the nu-cleus. The TRINAT apparatus meets all three of these requirements. The initial radioactive isotopes are trapped in a magneto-optic trap (MOT). The M O T is atom (nucleus) selective (selecting only the parent nucleus) thus allowing the daughter recoil to "freely" escape. The recoils are detected using the combination of an electric field and a position sensitive micro-channel plate (MCP). When a /3-decay takes place the daughter atom is frequently left in a charged state. These charged recoils are then accelerated by the electric field onto the MCP. The electric field gives the recoils sufficient energy to efficiently trigger M C P upon impact and increases the solid angle for detecting recoils. The polarization is induced by extinguishing the M O T and optically pumping the 4 {(l + i c ' ) x 2 -J(J + l) <Jz> J 3 < Jl > J(2J - 1) (Ap + Bv)xi cos(0) - c'x2 cos2(0) J dcos(9) (1.4) (1.5) Chapter 1. Introduction Figure 1.1: This figure shows the detection chamber of the TRINAT apparatus. The DI fight shown is the light used for optical pumping, also illustrated are the micro-channel plates and the calcium fluoride /3-detectors. Details about the detectors can be found in Chapter 4. atoms. The induced polarization can be varied by simply changing the polarization of the optical pumping light. 5 Chapter 2 Nuclear Theory This Chapter will outline all the necessary theory to understand /3-decay and the various asymmetries of interest in this experiment. The goal is to show all terms which contribute to the recoil momentum distribution. The Chapter begins with a basic overview of /3-decay, then adds complications to the basic theory introduced by the structure of nucleons. These complications are important because they introduce corrections to the /3-decay rate at the 0.01 level and this is the desired accuracy in measuring the spin-polarized recoil asymmetry. The Chapter then discusses contributions to /3-decay from non-SM sources and finally brings all these things together and shows the complete recoil momentum decay rate. The final section shows plots of the recoil momentum distribution for the specific case of 80Rb. 2.1 /3-decay Overview The goal of this section is to give a basic introduction to the theory of nuclear /3-decay. Nuclear /3-decay is a term that describes three distinct processes: positron emission, electron (/?) emission and electron capture. In positron emission a proton in a nucleus decays to: a neutron, a positron and an electron neutrino. In (3 emission a neutron (either free or bound to a nucleus) decays to: a proton, an electron and an electron anti-neutrino. In electron capture, a nuclear proton and an atomic electron are transformed into a neutron and an electron neutrino. This section will assume protons and neutrons are fundamental particles. Complications introduced by their underlying structure will be addressed in a later section. In the SM, the weak interaction is purely V - A and is mediated by massive ( « 80=^) vector bosons. The energy released in nuclear /?-decay is much smaller ( « lOMeV in the most energetic cases) than the mass of these exchange bosons and so the exact structure of the propagator can be ignored without any significant loss in precision (1 s t order corrections to this approximation are at 10 _ 7% level). With this assumption, and ignoring the structure of protons and neutrons, the nuclear /J-decay Lagrangian is given by equation 2.1. £p-decay = GFVud%Cv ~ ^ 7 5 T ^ n ^ ^ ^ V V V e + H.C. (2.1) 6 Chapter 2. Nuclear Theory Type A I A 7 r (Parity) Fermi 0 no Gamow-Teller 0,1 (0 to 0 forbidden) no Table 2.1: Selection rules for Nuclear matrix elements within the allowed approximation. In equation 2.1, GF is the coupling strength, Vud is a value from the C K M matrix, \J>P is the proton wave-function, \I/n is the neutron wave-function, ipe is the electron (positron) wave-function and ipUe is the (anti-)electron neutrino wave function. Cy and CA are the vector and axial vector coupling coefficients (both 1 for purposes of this section). The electron and neutrino wave-functions are momentum eigenstates (for electron capture, the electron wave-function is the atomic wave function). The operator term i}-^!^) is the gamma matrix representation of V - A (a Vector operator minus and Axial vector operator). The nuclear portion of this Lagrangian (^fp^-^-j^n) is non-relativistic and it is customary to make a non-relativistic reduction of this portion. In the non-relativistic limit (ignoring factors of h) the Lagrangian reduces to equation 2.2. J d 3x{[tf p(x)tf n(x)]M = 0 + [ * ; ( x ) a ^ „ ( x ) ] M = 1 , 2 , 3 } e ( - i ^ - ^ ) I ) u ( p e)^-^7 ^ ( p , e ) (2-2) In equation 2.2 the proton and neutron wave-functions are non-relativistic spinor fields, pe is the electron energy momentum four vector, pVe is the neutrino energy momentum four vector, the u's are Dirac bispinors and the <7M are the Pauli spin matrices. The first term in equation 2.2 comes from the vector portion of the operator and is called the Fermi matrix element (MF)- This term only exists for the /x=0 index. The second term in equation 2.2 comes from the axial vector portion and is called the Gamow-Teller matrix element (MCT)-This term only exists for /x=l, 2, 3. The exponential term e ^ ~ l ^ v ~ P e y x ^ can be Taylor expanded since {p„ - pe} • x « 1 ({pv —pe} « lMeV). Ignoring all higher order terms (taking e^~%^Pv~1Peyx = 1 ) is called the allowed approximation. Within this approximation the decay products have no orbital (as opposed to spin) angular momentum; the selection rules for both Fermi and Gamow-Teller matrix elements are shown in table 2.1. In table 2.1 I is the nuclear spin and ix is parity. In a Fermi decay the (anti-)neutrino and (electron)positron spins are coupled to a singlet state. In a Gamow-Teller decay the spins are coupled in a triplet state. This conserves angular momentum within a given matrix element. Since the selection rules are different for the two decay types it is possible 7 Chapter 2. Nuclear Theory Type AI Air (Parity) Fermi 0,1 (0 to 0 forbidden) yes Gamow-Teller 0,1,2 yes Table 2.2: Selection rules for Nuclear matrix for 1 s t forbidden decays. to have decays that are purely Gamow-Teller, or purely Fermi. These are referred to as Gamow-Teller decays and Fermi decays for obvious reasons. If we look at the next order terms in the Taylor expansion of e^~l^Pv~Peyx^ we can see the matrix element will gain a factor of {p„ — pe} • x. This changes the selection rules because now the leptons (neutrino and electron) have a unit of angular momentum. The selection rules for this "I s* forbidden" decay are shown in table 2.2. Since the nuclear parity must change the "1 s t forbidden" terms make no contribution to 80Rb decay. The selection rules come from all the possible couplings of the mechanical and spin angular momenta. There are other higher order forbidden decays as well but these two examples should give enough information to deduce further selection rules. The nth order forbidden process is suppressed by a factor of {{pu — Pe}-%}n ~ (0.01)n; therefore the lowest n allowed for a given decay completely dominates the rate. 2.2 Parity Violation 2 . 2 . 1 Parity Violation in /3-decay Perhaps the most well-known fact about the weak interaction is that it violates parity. The parity operator (P) takes vectors f and inverts them so they become —r. Violating parity means the physical results are different if the parity operator is applied. This section will outline how this is manifest for nuclear /3-decay (in this section /?-decay refers only to positron and electron emission, not electron capture). In order to study the parity dependence of nuclear /3-decay, we need an expression for the decay rate. The decay rate can be calculated from Fermi's Golden Rule. ^ L ^ W (2.3) f In equation 2.3 T is the total decay rate, and Mif is the transition matrix element between the initial (i) and final (f) state. The summation is over all possible final states. In order to observe parity violation in /3-decay the initial nuclear state must be polarized. This defines an axis along which parity will be violated. The partial decay rate for nuclear 8 Chapter 2. Nuclear Theory Term SM value 2\MF\LCV + 2\MGT\LC\ 2\MF\iCj/-i\MGr\iCiA 0 -{J-^^\{mGT?ciKJIJ} Apt 2{- ± \MGT\'\J,JCa - 26J,JMFMGTCVCA} 2{±\MGT\2\J,JCa - 26j,jMFMGTy/jLICVCA} 0 Table 2.3: Standard model values for /3-decay coefficients. It can be seen that for MF=0 Ap + BV=Q as was previously mentioned. /?-decay with a polarized nucleus was calculated in 1957 by Jackson, Treiman and Wyld (JTW) [18]. The result is given in equation 2.4. The J T W result is a calculation to lowest order. It ignores the energy of the recoil ( IM'J2L—) f ° r purposes of conserving energy and other effects of similar size called recoil order terms. dT GFVud\^\r? ( R \ _ J? \2JT? rin t / i i „Pe-Pi/ ( bm, + (2TT)5 3EeEu \pe\Ee(Q - EeYdEedQ,edClvi{l + a EeEv Ee (2.4) Pe -3PV3 EeEv + <J> J 4 , & + A £ + D * X » EeEv j } In equation 2.4 GF is the coupling strength of the interaction, the subscript e refers to the electron (positron) and the subscript v refers to the (anti-)neutrino, E is the total energy, p is the momentum vector, J is the initial nuclear spin, j is a unit vector along J, and Q is the total energy released in the decay. The values of the /?-decay coefficients: £, a, b, c', Ap, Bv and D within the SM are given in table 2.3. In table 2.3 all ± are + for electrons and - for positrons; CV is the vector coupling strength and CA is the axial vector coupling strength, KJIJ and \J>J are spin dependent quantities (J is initial and J ' is final spin) whose values are given in equations 2.5 and 2.6. 1, J j+i> J -J J+i' J J' = J - 1 J' = J J' = J + 1 (2.5) 9 Chapter 2. Nuclear Theory 1, J J' = J - 1 ~ ( 2 J - l ) ) J-+J> = J •7(2 J - l ) ' r , r / _ T , ! ( 2 - 6 ) (2J+3) (J+1) ' J J — J "I- 1 Since neutrinos are extremely difficult to detect any practical experiment will observe the /?. With this in mind we integrate equation 2.4 over all possible neutrino momenta and are left with equation 2.7. dr = ^^\p-e\Ee(Q - Ee)2dEedQeai + A0^J^ • fr> (2-7) Under the parity operator, momentum vectors change sign, however, spin and all other angular momenta do not. We see then that the -^p- • term changes sign under parity. 2.2.2 Parity Violation in Electron Capture In the case of electron capture, parity violation can be observed in the spin polarized momentum distribution of the recoiling daughter nucleus. This distribution was calculated in 1958 by Treiman for the case of K shell electron capture [17] and is given in equation 2.8. This calculation ignores all recoil order corrections. Wec(Pr)dVtr = {1 - 5+^ 4^  " Pr}dnr (2.8) In equation 2.8 Pr is the unit recoil momentum vector, B+ is defined in equation 2.9. £B+ = -2\MGT\2C2A - SJ.JMFMGHJ^^CVCA (2.9) It is worth noting that in the case of a G T decay the electron capture asymmetry is large (B+ = 1). Electron capture contributes a significant background asymmetry which is excluded using other constraints (see Analysis Chapter for details). 2.3 Proton and Neutron structure effects In the preceding sections we have ignored that fact that the proton and neutron are not elementary particles. The weak interaction is modified by the virtual quarks and gluons that exist due to the strong interaction in the vicinity of the proton and neutron. The modification from these virtual particles would manifest itself by changing the structure of 10 Chapter 2. Nuclear Theory the hadronic portion [iipCv~2A'15Iti^n) of the /?-decay Lagrangian. However these modifi-cations must occur in a way that remains Lorentz invariant. This means the overall current must still have a single index (u) to contract with the lepton current (ipe^-^l^ue)- It is customary ([19]) to replace the operator term ^ - f i t 75 w j th a n effective V - A operator where V - A are the most general possible vector and axial vector operators that can be constructed (equation 2.10, 2.11) from the Dirac matrices (7) and the momentum transfer («)• v- = c ^ - ^ ^ + i ^ - ( 2 1 0 ) = -^pvl\i - lalv\ (2.12) In equations 2.10, 2.11 rrii is the initial state nuclear mass, Cv and CA are the vector and axial vector couplings, and the remaining g^s are arbitrary coupling constants. The calculation of these couplings is in the realm of theoretical QCD. Fortunately there are symmetries that allow us to predict the most important of these terms. The conserved vector current hypothesis (i.e. c^VM=0) predicts that CV=1 and gs=0. The divergence (i.e.c^) of the term multiplying QM is identically zero. This allows C V C to predict a value for gM in terms of the non-Dirac magnetic moment. The fact that electromagnetic charge is conserved leads then to the fact that the weak vector charge must also be conserved, therefore the vector portion of the weak interaction is not affected by the strong interaction's presence. The coupling strength is the same as it would be for a free quark. The CA term is the strength of the axial vector current. This term is influenced by the presence of the strong interaction, but turns out to be partially protected according to the partially conserved axial current theorem (PCAC) [20]. In the limit of zero pion mass the axial vector current would be completely conserved, but because the mass is small compared to the scale of Q C D (lGeV) the value deviates only slightly from unity taking on a value in the range of CA = 1.267—1.276 (for a complete discussion see [21]). There are no symmetries protecting the remaining terms. The calculation of these operators in the case of polarized nuclear /3-decay has been performed by Holstein [22] and the resulting current contracted with the lepton current (I) is given in equations 2.13 and 2.14. 11 Chapter 2. Nuclear Theory r<f\V„\i> = ( a ( q 2 ) ^ + e(q2)(^-)6jj,5MM, (2.13) OA) C GA) € + Z O A 7 UJ'1;J W x vfc + W'2;J 17777 °11;2 'n9n' r<f\A„\i> = C^fj M e «* % A , -J-[c( , 2 ) l i P'> (2.14) + ^ c ^ f c ^ ( f ) ^ 2 " ' ( ^ ) p £ ^ J 2 ( 0 2 ) +cjsr^(f ) * ^ ' M ) ( ^ i s ( ^ + . . . In equations 2.13 and 2.14 the form factors i.e. the lettered functions of q2 (x(q2)) are defined in equations 2.15-2.23 note the g's appearing here are the same as those in equations 2.10, 2.11, P is the sum of the initial and final nuclear 4-momentum, Mave is the average of the initial and final nuclear mass, q is the difference between the initial and final nuclear 4-momentum, J is the initial nuclear spin and J' is the final nuclear spin, M and M ' are the initial and final nuclear spin z-projections, the C - ^ n are Clebsch-Gordan coefficients described by Rose [23] and the Ytk (q) are the vector spherical harmonics described by Rose [23]-o = CV[MF] (2.15) b = A{gMMGT + CvML} (2.16) c = CAMGT (2-17) d = CA{-MGT + AMaL + miMarv\-AguMGT (2.18) e = Cv[MF±Ags] (2.19) / = Cv2miM[rtP] (2.20) G = - C V ^ - M Q (2.21) 12 Chapter 2. Nuclear Theory h = - C A — ^ M l y + gFA2MGT (2.22) IO2 2 M 2 ji = ~CA^f^Miy (2.23) In equations 2.15-2.23 A is the nuclear mass number, the gi are form arbitrary couplings appearing in equations 2.10 and 2.11, rrij is the initial nuclear mass, Mave is the average nuclear mass and all the M's are nuclear matrix elements described in equations 2.24-2.31. MF = < / l l £ r f N > 7" (2.24) M G T J = < f\\Y,Tt<rk\i> j (2.25) MQ = ( f )"<f\\2ZTNY"(fMi> j (2.26) ML = </llE^[^x^]IK> j (2.27) MAL = i < /II E T f °i x fa x Will* > j (2.28) = ^ 7 < /II 2)2 Tf • ri>Pj) + {"J • ft* rJMII* > (2.29) MKY = (^)* < /II E ' f r J C ^ ? x a i B i y (rj)||i > (2.30) M[rtP] = 2m- < "^ E^^ll^' X hnPin' +Pinnn'W > (2.31) In equations 2.24-2.31 j is the summation index over all nucleons (protons and neutrons), / designates the final nuclear state and i designates the initial nuclear state, T± are the isospin raising and lower operators (which change a proton to a neutron or vice versa), a are the Pauli spin matrices Yjq are the vector spherical harmonics, r is the single nucleon position operator and p is the single nucleon momentum operator, and the £ 7 ^ " are Clebsch-Gordan coefficients. In equations 2.30 and 2.31 the spin, position and momentum operators have a second index (n) designating which axis (x, y, z) is to be used in that instance. Note that a and c are the lowest order Fermi and G T terms respectively. The terms which contribute the largest corrections for the case of 80Rb are b, d and / . The total /3-decay rate including leading recoil order corrections within the SM is given by equation 2.32. In equation 2.32: F(Z,Ee) is the Fermi function (see Appendix B) and 13 Chapter 2. Nuclear Theory all other terms have been previously defined. \Mf = 16m f E,£,£„[/ 1 + / 2 § ^ + / 3 [ ( f ^ ) - ^ ] (2.33) , J 'Ve , J • Pe Pe • Pv f J ' Pv f J " Pv Pe • pi Wt \t 'P~e\2 1 , j . ,J-PeJ-Pv Pe • Pv • + [ 1 _ J ( J + l)][/lo[(^T) ~ 3 E 2 ] + / l 2 [ £ e £ „ 3J5eE, ,pe-p1vJ-P^J 'Pv Pe-Pv, , , tfJ-Pvs2 P"v2 + / i 3 l - = r i r - J l— p-p TEr-Erl + / i 4 l ( - p — ) -1/ ' EeEv EeEv 3 e v Ev 3E% . f [Pe -PvyUJ -Pv\2 Pv_n + ^E^^^EV' ~2El]] In equation 2.33 the f s are functions of the lettered functions in equations 2.15-2.23; their exact definition is given in Appendix A. The [1 - j^j+jy] will occur frequently when discussing distributions. It is Holstein's version of the alignment. From this point this is the definition of alignment (T). This decay rate is to be compared with equation 2.4 in which all recoil order corrections are ignored. 2.4 Nuclear /3-decay beyond the Standard Model The preceding sections have all discussed /3-decay within the SM. In order to search for SM extensions we need to develop the theory of /3-decay with non SM interactions. The most general /3-decay Hamiltonian that can be constructed (limited only by Lorentz invariance) is given in equation 2.34. H0= Qj7frL ti*P*n][CS*e*v+ 0^,15**] + [%l^n}[CV%l^v + C'vVe-fl&v] + ^V^n]{CT^e(JpX^v + C^e^TuVv] + [*P75*n] [ C P ¥ e 7 5 # „ + C'pVe*,,]) (2.34) 14 Chapter 2. Nuclear Theory Lagrangian term Non-relativistic matrix element * p * n MF Mp #p7^75*n MGT MGT * P 7 5 * n 0 Table 2.4: This table shows the non-relativistic reduction of each of the fundamental hadronic currents. In equation 2.34 C/'s and C\'s are arbitrary couplings for: vector (V), axial vector (A), scalar (S), pseudo-scalar (P) and Tensor (T) weak interaction components (some texts use Gi in place of C / . In this text the two are used interchangeably.). In the SM Cv=Cv=l and CA=C'A, with the value of CA being in the range 1.267-1.276 [21]. All other coefficients are zero in the SM. In order to calculate the decay rate first the hadronic currents must be reduced to their non-relativistic approximations. The results of this reduction are shown in table 2.4. In the non-relativistic limit the pseudo-scalar term is zero and therefore of no relevance in nuclear /3-decay experiments. The total decay rate for polarized nuclear /3-decay for the most general Hamiltonian is given by equation 2.35. dr = F(Z, Ee)dZPf^fPLJ5\Pe + Pr + PV)\M\* (2.35) xZ/g Jl/j* J-Jy \M\2 = EeErEv£{l + a Pe-Pv EeEv , m, + h t + d Pe -Pu Pe-JPu • J + < J > J A ^ E - e + B ^ + D ^ E : \ 3EeEv } EeEv (2.36) A similar equation was shown previously (equation 2.4) but the parameters in that equation were defined within the confines of the SM. The generalized expressions for the decay parameters are given in Appendix A. 2.5 Recoil Asymmetry If the nuclear recoil is the only observed quantity then equation 2.35 can be integrated over all possible [3 and neutrino momenta. This can be done analytically in the case of a constant 15 Chapter 2. Nuclear Theory Fermi function. The result can be further integrated over all possible recoil momenta to give an angular distribution. This distribution is given in equation 2.37. W{6)de = |(1 + \c')x2 - <Jj> (Ap + cos(0) - c'x2cos2(0)J dcos(0) (2.37) In equation 2.37 8 is the angle between the recoil direction and the polarization axis, c' A@ and Bu are defined in Appendix A (for the most general Hamiltonian, the SM definitions are in table 2.3), the Xi are functions of the energy released in the decay (Q) and appear in Appendix A. In the limit where ^ is much greater than unity (Q is energy release and M is nuclear mass) X\ = | and xi = \. The asymmetry for nuclear recoils is proportional to Ap + Bv which is 0 for a pure Gamow-Teller decay in the SM and will be non-zero if any tensor component is present in weak interactions. 2.6 Sources of Tensor Interactions The goal of this thesis is to set competitive limits on any tensor components present in the weak interaction. This section will first outline how a fundamental tensor interaction at the quark level will manifest itself in /3-decay, then will outline theoretical SM extensions that allow for the existence of tensor interactions. 2.6.1 Tensor Interactions at the quark level The discussion to this point has always written the Lagrangian and Hamiltonian in terms of the proton and neutron fields. If tensor interactions are a fundamental interaction they will occur at the quark level; if instead we write down the fundamental Hamiltonian at the quark level we get equation 2.38 ([24]). H = ^ ^ V u d [alLVL^eLdL^uL + alRVL^eLdR^UR (2.38) (^RLPRIII^BALI^UL + ORppRjueRdR^UR ampLeRdRUr, + asRRpLeRdLuR aSLRvR^LdRUR + afrpReLdRUL ORipLO-^eRdRa^UL + alRVRO-^eLdLO^UR] 16 Chapter 2. Nuclear Theory In equation 2.38 the quark and lepton fields are chiral and the lower index on the field denotes whether it is right (R) or left (L) handed. All the a's are arbitrary coupling constants. The upper index denotes the interaction's type (Tensor, Vector or Scalar), and the two lower indices refer to the chirality of the (3 (e) and the up quark (u). There is a direct relation between the a coupling constants in equation 2.38 and the C coupling constants in equation 2.34. This relation is given in equations 2.39 through 2.46. Cv = 9v(avLL + a l R + a v R L + a v R R ) (2.39) Cy = gv{-aVLL-avLR + avRL + avRR) (2.40) CA = 9 A {aVLL ~ aLR ~ aRL + aRR) (2-41) C'A = 9A{-avLL + avLR-avRL + avRR) (2.42) Cs = 9s(aSLL + asLR + asRL + asRR) (2.43) C's = 9s(asLL + asLR-asRL-asRR) (2.44) CT = 2 g T { a l R + a T R L ) (2.45) C'T = 2gT(aTLR-aTR]-) (2.46) The g's are form factors described by the following equations. < p\u^d\n >= gv(q2) < pYt^n >= gvMF (2-47) < p\wf~Kd\n >= gA(q2) < p\l"l5\n >= gAMGT (2.48) < p\ud\n >= gs(q2) < p\n >= gsMF (2.49) < p\u-^d\n >= gT(q2) < P \ - ^ \ N >= 9TMGT (2.50) 2.6.2 Supersymmetry One of the major problems with the SM comes about when one calculates radiative correc-tions to the Higgs mass [25]. The SM predicts that the Higgs mass gets quadratic correction factors at the cutoff energy scale of the theory. If we take the cutoff scale to be the Planck scale then in order for the Higgs to have a mass in the 100 GeV range we need to have cancellation of factors to 32 decimal places. This is clearly unacceptable and this problem has been called the hierarchy problem. In order to solve this problem another symmetry was introduced at an energy scale of about 1 TeV. The new symmetry called supersym-17 Chapter 2. Nuclear Theory metry (SUSY) creates a new scalar boson partner for every chiral fermion in the SM and also creates a new fermionic partner for every boson in the SM. The theory also requires another Higgs doublet to exist, one for giving up-type particles (up,charm and top quarks) mass and one for giving down type (down quarks, charged leptons) particles mass. If supersymmetry were a good symmetry of nature, all the supersymmetric partners would have the same mass as their SM counterparts. Since this is obviously not true, the symmetry must be broken at some level. The details of how SUSY is broken are addressed in some models (SUGRA, AMSB, GMSB) but this happens at an energy scale much higher than the TeV scale of interest in SUSY, so the details of how the symmetry is broken are not that important. The most general SUSY models have over 100 free parameters. An important consequence of SUSY is the possibility for proton decay on short time scales. There are three terms that would appear in the Lagrangian at leading order which would contribute to proton decay. Cp-decay = XLLEC + \'LQDC + X"DCDCUC (2.51) In the above Lagrangian L contains the lepton and slepton SU2 doublets, Ec contains the lepton and slepton SU2 singlets, Q contains the quark and squark doublets and finally Uc and Dc are the quark squark singlets. The first two terms above are lepton number violating while the third is baryon number violating. If both lepton number violation and baryon number violation are allowed, then the proton would decay very quickly. This is usually avoided by introducing R parity as a new symmetry. R parity is defined as R — (_i)2S+3B+z, Imposing R parity as a good symmetry eliminates all three terms above. However, R symmetry is not required to eliminate proton decay. One can get the same result by simply setting some or all of the coupling constants to zero. In order to generate leptoquarks in this model and still have a stable proton one can set only the coupling X' to be non-zero, then the super partners of the quarks behave as lepto-quarks and give rise to a tensor interaction. There is another way besides tree-level leptoquark exchange to generate tensor interac-tions in a supersymmetric theory. In SUSY each chiral fermion has its own super partner. This means there is a super symmetric left handed electron and a super symmetric right handed electron. Of course these are just labels since the super partners are spinless bosons. In the most general SUSY models the chiral super fields can mix with one another, and this gives rise to box graphs that could contribute to tensor interactions [26] even if R parity is conserved. In the limit of maximal left right mixing between superpartners and a degenerate 18 Chapter 2. Nuclear Theory superpartner mass spectrum (Ms) we get the following tensor coupling constant. T _ aVudM\ a L R ~ 367TMJ ( 2 - 5 2 ) If we take the super partner mass scale to be at the TeV level we get a^L « 1 0 - 7 . The mass limit on the lightest super symmetric particle is 36 GeV [27]. Using this mass value we get a coupling constant of 3 x 1 0 - 5 . 2.6.3 Technicolor Technicolor is another extension to the SM designed to eliminate the hierarchy problem. The idea in technicolor, however, is that there are no fundamental scalar particles. The symmetry breaking mechanism is generated by introducing a new interaction called techni-color and new particles called technifermions. The technicolor force is designed to be very strong at the electroweak energy scale, and so the techifermions only exist in bound states analogous to pions. These bound states can be scalars and thus generate the necessary sym-metry breaking in the SM. The advantage of this theory is that without any fundamental scalars there is no hierarchy problem. The simplest versions of technicolor are unfortunately ruled out by accelerator experi-ments. In these versions of the theory the coupling constant for technicolor runs similarly to that of the color interaction, so calculations can be made by simply scaling observations from color interactions to the appropriate energy scale and cutoff. In extended technicolor theories one can introduce multiple technifermions and this can change the running of the coupling constant. In these theories we don't know how to make accurate calculations of observables, but the theory is then not ruled out by current experiments. In extended technicolor models scalar leptoquarks are expected to exist at the TeV scale [25]. 2.6.4 Current Tensor Limits There exist both experimental and theoretical limits on the value of CT and C'T from previous experiments and calculations. In 1963 a group at Oak Ridge [28] measured the recoil spectrum of He which allowed them to set the following limits: jeja+ic/p < 0.004. This result was revised in 1998 by Gluck who evaluated the recoil order and radiative corrections [29]. The revised limit is el 2 + K / | 2 < ° - 0 0 8 . In 1991 a group at Louvain performed an experiment comparing the positron polarization in pure Fermi ( 1 4 0 ) and pure Gamow Teller ( 1 0 C) decays [30] which allowed them to set the following limits: ( CSQ °S -TC A T ) < 0.003^0.018. There is also a theoretical limit on tensor terms that arises from 19 Chapter 2. Nuclear Theory Limits on Tensor Interactions 0.15 -0.15 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 Figure 2.1: This shows the current limits on tensor interactions. The dashed line at 45 is from the Louvain group's experiment. The solid line at —45 is from Ito and Prezeau. The circle is from the 6 He experiment. the fact that any tensor interaction will induce neutrino mass. Using 0.71eV as the upper limit on the sum of neutrino masses Ito and Prezeau completed an effective field theory estimate [31] based on naturalness arguments which sets the following order of magnitude limits \c2L + c^\< 1 0 • A U t h e s e l i m i t s are combined in figure 2.1. 2.7 Nuclear /3-decay rate including all model extensions In order to search for SM extensions in nuclear /3-decay, the partial decay rate must be calculated including the fully extended Hamiltonian and all recoil order corrections as well as a non trivial Fermi function. There is one other kind of recoil order correction to be included which has not be discussed to this point. This is including the kinetic energy of the recoiling nucleus in the delta function appearing in the total rate. The complete decay rate is given in equations 2.53 and 2.54: dr = F(Z, Ee) 1 d3p"r d3p"e d^pl (2n)464(pr-Pe-Pl/)\M\2 (2.53) 2m* (27r) 32£ r (2n)32Ee (2n)32El/ 20 Chapter 2. Nuclear Theory |M| 2 = + [ 1 - 7 ( T T T y J l s l ( m ^ e 3E% 1 2 EeEv ZEeEv Pe2 i , J -PeJ-Pv fe-fu -4- . Q m — (2.54) + fe -pi J -fj -fv _ pe • fv y P \Y TP TP OTP TP 1 -C/eJ-/f ^e^v o±Je*-Ju + Equations 2.53 and 2.54 are the same as equations 2.32 and 2.33 with the exception that the / functions from 2.33 have been replaced with s functions which now contain non-SM terms as defined in Appendix A. This function cannot be integrated analytically (due to the Fermi function) to get the recoil momentum distribution, but can be integrated numerically. 2.8 Complete Recoil Momentum Distribution Equation 2.54 can be numerically integrated over all possible electron momenta. This integration is final state dependent, since the amount of energy released in the decay changes the final endpoint energy, and also because the final spin appears in some terms. The case of interest in this experiment is 80Rb, so the distribution will be investigated for this particular decay. All the following discussion will be limited to the dominant decay branch in the 80Rb decay. Similar results have been calculated for the other decay branches, the only difference being the endpoint energies and the spin values. In the limit of all recoil order matrix elements and tensor couplings being zero the complete recoil momentum distribution is composed of three distinct terms, much like equation 2.37. Of these three terms one contains no angular dependency (N(#, |P r |) see figure 2.2), one has a cos(0) angular dependence(C(#, \Pr\) see figure 2.3) and the third term has a second order Legendre polynomial dependence on cos(#) (L(#, \Pr\) see figure 2.4). It is worthy of note that there is a finite asymmetry at this junction. This asymmetry comes from including the energy of the nuclear recoil, which destroys the perfect cancellation of asymmetries which was observed in the J T W result. Using these three terms, and given the polarization (P) and the alignment (T) (T = 1—3 j(jli\)the total recoil momentum distribution (in the absence of recoil matrix elements) 21 Chapter 2. Nuclear Theory MSJEJI Figure 2.2: This plot shows the recoil momentum distribution term containing no theta dependence other than the sin(0) term required by dcos(#). 22 Chapter 2. Nuclear Theory Figure 2.3: This plot shows the recoil momentum distribution term containing cos(0) an-gular dependence. It is the cos(f?) terms which are responsible for any asymmetries 23 Chapter 2. Nuclear Theory ygjpji Figure 2.4: This plot shows the recoil momentum distribution term containing second order Legendre polynomial dependence on cos(#). 24 Chapter 2. Nuclear Theory is given by equation 2.55 and shown in figure 2.5. W(6, \Pr\)d9dPr = [N(9, \Pr\) + P C(9, \Pr\)+T L{9, \Pr\)}d9dPr (2.55) Momentum Distribution Figure 2.5: This shows the total recoil momentum distribution (in the absence of recoil order matrix elements) for the decay branch going to the ground state (73%), using P = l and T=0.17. If the condition of al l recoil order matrix elements and tensor couplings being zero is relaxed, the total recoil momentum distribution receives contributions from more terms. Each recoil order matrix element contributes three terms (one appearing with neither the 25 Chapter 2. Nuclear Theory alignment nor polarization, one appearing multiplied by the polarization and one appearing multiplied by the alignment) which vary linearly with the size of the matrix element. To be clear let us define some notation, for a given recoil order matrix element (x) we have three distributions (xa where a is one of N,C or L be analogous to the non recoil matrix element case). The total contribution from a recoil order matrix element to the total momentum decay spectrum is then given in equation 2.56. Xcontribution(0, \Pr\)d6dPr = x{xN(9, \Pr\) + Pxc(0, \PR\) + TxL(6, \Pr\)}d9dR£.56) In addition to the contribution from recoil order matrix elements if the tensor couplings are allowed to be non zero we get two additional terms with cos(#) dependency. Of these two terms is linear in C T ^ T (V* = C T C A T ) ^ n e °ther term is linear in C Q C J T (4> — C c i T )• These distributions which we will denote xpc and <f>c as well as the recoil order matrix element distributions are included in Appendix C. The complete recoil momentum'distribution including all terms is then given by equation 2.57. For the case of 80Rb and the ground state decay only the b and d recoil order terms are important. For the excited state branch b, d and / terms must be included. W(9, \Pr\)d6dPr = {N(6, \Pr\) + P C{6, \Pr\) + T L(d, \Pr\)} (2.57) + YI x{xN{e,\Pr\) + Pxc{e,\Pr\)+TxL{0,\Pr\)} x=b,d(J) +P{ip iPc(0, \PR\) + <$> tc(0, Pr)}d6dPr 26 Chapter 3 Atomic Physics Borrowing methods from atomic physics, one can precisely manipulate the quantum state of an atom. This precise manipulation is a very attractive feature for this experiment. In order to perform asymmetry experiments it is required to have a "cold", localized source as well as a method of polarizing the nucleus. In this experiment the source is achieved by trapping radioactive atoms in a M O T [32], and polarization is achieved by optically pumping the atoms. This Chapter will give a very brief overview of these two important topics. A more complete discussion of these topics can be found in [33]. 3.1 Magneto-Optical Traps (MOTs) A magneto-optical trap (MOT) combines light fields as well as a static magnetic field to form a three dimensional potential well with a dissipative component. A standard M O T comprises of a static quadrupole magnetic field and three retro-reflected circularly polarized beams of light. In order for the M O T to trap atoms the circular polarizations of the reflected beams must be opposite that of the incident beam. 3.1.1 1-D M O T The general proof that a M O T is confining in three dimensions is quite involved and difficult [34], however, a lot of insight can be gained by examining a M O T in one dimension. This section will give an overview of the trapping and dissipation mechanisms of a M O T in one dimension. The one dimensional M O T consists of a magnetic field that is linear in the vicinity of the origin B = BQX and two counter propagating monochromatic beams which are detuned from an atomic resonance by an amount 5 with wavelength A. In order to avoid unnecessary complications we will assume we have only a two state atom. The ground state of the atom will have 0 angular momentum and the excited state will have unit angular momentum and a lifetime r (7 = £) Light incident on the atom will be absorbed, putting the atom in the excited state and giving it a recoil momentum equal to that of the photon. 27 Chapter 3. Atomic Physics Detuned Laser rj beam o beam Ground State Position Figure 3.1: This shows how the energy of the excited state is affected by the magnetic field as a function of position. This also gives a schematic view of the laser detuning. There are then two routes the atom can take back to the ground state spontaneous or stimulated emission. Spontaneous emission is a random process in which the atom de-excites by emitting a photon with isotropic probability. The total process of excitation and de-excitation by spontaneous emission therefore transfers a net momentum equal to that of the ini t ia l photon into the atom. The other process that can take place is stimulated emission, which is caused by the presence of a light field. The light field increases the probability that upon de-excitation the atom wi l l emit a photon wi th the same characteristics (polarization and direction) as the photons making up the light field. The net process of excitation followed by stimulated emission therefore transfers no net momentum to the atom. If we average over many of the above mentioned cycles, we can write the average force felt by the atom as the product of that rate at which it scatters light times the momentum of the photon. Therefore for two counter-propagating beams we get the following equation [33]: In equation 3.1 SQ is the ratio of laser intensity to saturation intensity. The &+/- terms have three components given as follows: 28 Chapter 3. Atomic Physics M O T Restoring Force -0.04 -0.02 0 0.02 0.04 Position [m] Figure 3.2: This is a plot of the M O T force in one dimension. Near the origin the force is nearly linear. The first term is just the detuning of the light beam from the resonance. The second term accounts for the Doppler shift of the light in lowest order where v is the velocity of the atom. The third term is just the Zeeman effect with p, = gMefiB, M B is the Bohr magneton, g is the electron g factor and Me is the magnetic sub-level of the excited state. When both the Doppler and Zeeman shifts are small compared to the detuning of the light beam, we can Taylor expand equation 3.1. This expansion leaves us with a very simple force: F b = —bv — kx _ 8h5so A27 (l + *o+(f): = " I T 6 (3.3) 29 Chapter 3. Atomic Physics Uncoupled Excited State Coupled Ground State Coupled Excited state Uncoupled Ground State Figure 3.3: The blue arrow indicates the transition induced by the resonant laser beam. The red arrows indicate the spontaneous transitions that can take place from the coupled excited state. This is just the force due to a damped harmonic oscillator in one dimension. Intuitively we can see how this could be extended into three dimensions, by adding counter propagating beams along each axis. 3.2 Optical Pumping In order to observe asymmetries in this experiment we require the nucleus to have a finite polarization. In order to achieve this polarization we borrow the atomic technique of optical pumping. Optical pumping is a net transfer of angular momentum to an atom by single photon excitation that results in a net decrease in the atoms entropy. This process is simple to understand wi th the help of an example. For this purpose we wi l l use a simple model atom with an F = \ ground and excited state. We assume the atom begins at thermal equilibrium with equal populations in the two degenerate ground states. The we add a circularly polarized resonant laser field which couples one of the ground states to one of the excited states. The laser wi l l move population from the coupled ground state into the coupled excited state. Once in the excited state there are only two process that can take place: spontaneous emission back to the init ial state or spontaneous emission to the other ground state. Once population is moved into the non coupled ground state there is no mechanism which can cause it to be displaced. We can therefore see how all the population can end up in this state. Optical pumping in a realistic atom has two minor complications, the addition of another ground state manifold (Hyperfine splitting) and the possibility of stimulated emission. The hyperfine splitting is overcome by adding another resonant laser beam with the same circular polarization; however, this beam is tuned to be resonant with the other ground state manifold. Again we have a fully stretched state (Mjr =_ F) which does not couple to the light field. 30 Chapter 3. Atomic Physics When stimulated emission takes place that atoms returns to the same state as it was in before the photon was absorbed. The stimulated emission rate is proportional to the intensity of the laser beam, so by increasing the laser intensity you not only increase the excitation rate but also the stimulated emission rate, which hurts the rate at which polar-ization is achieved. 3 . 3 S h a k e o f f E l e c t r o n s When a /3-decay takes place, there is a sudden change in the nuclear charge and thus the potential binding the electrons. This sudden change can lead to the ejection of some atomic electrons possibly leaving the atom in an ionic state. The electrons ejected by this mechanism are referred to as shakeoff electrons. The mechanism for the electrons ejection can be understood within the sudden approximation. The "sudden approximation" is useful when a time dependent perturbation occurs on time scales which are very small compared to the time dependent changes in the systems wave function. In the case, of moderate energy positron (.EfcjneJjC=lMeV) escaping from a nucleus, the perturbations time scale is about 10 - 1 8 s and the relevant time scale for atomic wave functions 10 _ 1 6 s . This approximation works less well for the inner shell electrons. The sudden approximation is just the statement that the wave functions don't change in the time frame of the perturbation changes, however, since in the final state the potential has changed, the old wave functions are no longer energy eigenstates. The probability the atom is in a particular eigenstate of the final potential is given by projecting the old wave function onto the new basis. The /?~-decay shakeoff probability of various atoms has been calculated by Carlson et al using the sudden approximation [35]. It is also worth noting that the average kinetic energy of a shake electron is approximately 1.8 times its original binding energy [35]. In general the charge-state distribution of the daughter ions observed in the lab will differ from that predicted in the sudden approximation. The reason for this is two fold: first the sudden approximation breaks down for the inner shell electrons and second the ionic daughter atom can be left in an excited state which is unstable to Auger electron de-excitation which leads to further ionization. Overall the relative number of produced charge states is very complicated. 31 Chapter 4 Experimental The goal of this Chapter is to give a detailed explanation of the experiment and to show the data relevant to the final result. The first section will give a broad overview of the preparation of the initial state for this experiment. The second section will describe the isotope used in this experiment. The third section will describe the apparatus in detail. The fourth section will give the timing of the experiment, showing how each piece of the apparatus is used. The fifth section will discuss all important data sets in this experiment. The sixth section will show how these data sets are modified by introducing other constraints (cuts). The seventh section will show auxiliary data sets that were required to make corrections to the final analysis, and the final section will show how this data relates to the variables 9 and Pr described in Chapter two. 4.1 Overview Radioactive isotopes are delivered to the TRINAT apparatus from the ISAC 1 beam facility. A beam of 500 MeV protons from the cyclotron is incident on one of a variety of targets (Tantalum Carbide in this case). There are three types of nuclear reactions caused by the incident proton beam: spallation, fragmentation and fission. In a spallation reaction the incident proton knocks free a number of nucleons from the target nucleus. Fragmentation is somewhat similar to spallation with the exception that some of the nucleons knocked free remain bound to one another thereby making other nuclei. Fission occurs when the incident proton is absorbed into the nucleus, creating a very unstable isotope which spontaneously decays into two smaller fragments and some free nucleons. In this experiment 8 0i?6 was produced predominantly by spallation reactions. The resulting reaction products are surface ionized (the target is in a Tantalum-lined tube) and accelerated from the target area (the target tube is biased by 30 kV with respect to ground). At this point the radioactive beam contains a large number of radioactive isotopes. In order to purify the beam it is passed through a mass separating magnet. After purification the beam is delivered to the TRINAT apparatus (see figure 4.3). The 32 Chapter 4. Experimental beam is incident on a zirconium foil located inside the first M O T chamber. The zirconium is heated resistively using 12A of current. This heat speeds the diffusion of the implanted ions to the surface. Upon reaching the surface if the ion has sufficient kinetic energy it can escape the zirconium and take with it an electron [36]. Zirconium was chosen as a foil due to its low work function. These now neutral atoms form a vapor in the first chamber which is cooled and trapped by the M O T . In principle we could perform experiments in the first chamber using this cold localized source of radio-actives. However, in practice the backgrounds in this chamber are overwhelming, due to the proximity of the incident beam and the vapor which permeates the chamber. The efficiency for capturing atoms in the first M O T from the beam is 0.001. In order to circumvent the background problem the radio-actives are transferred to a second M O T . A laser (push beam) incident perpendicular to the ion beam and tuned to the blue of the D2 transition imparts momentum to the atoms in the first trap. The atoms then coast through two light funnels to the second chamber and are collected in a second M O T . Fortunately the efficiency for transfer between traps is much greater than the first trap capture efficiency (0.75 measured in Potassium) allowing us to capture about a million atoms at any given time in the second trap [37]. 4.2 8 0 Rb The isotope selected for this experiment is 8 0 R b . This choice was made for two reasons: 8 0 R b is a pure G T decay (no Fermi component) and thus to lowest order in the SM the recoil asymmetry will vanish, and it is easily trappable in the TRINAT apparatus because it is an alkali atom. The final state is dominated by two decay branches (see figure 4.1) which can be distin-guished by observing the 616 keV gamma which occurs for all excited state decays. The line width for both the D l and D2 transitions is w 6 MHz. The D2 trapping transition is nearly closed, requiring only a single moderate intensity re-pumping beam. The excited state levels are inverted from the usual order due to the large magnetic moment of the nucleus (see figure 4.2). 4.3 TRINAT Apparatus The TRIumf Neutral Atom Trap (TRINAT) is located in the basement of the ISAC I beam facility. The TRINAT group has performed a series of experiments [37] all with the goal of constraining extensions to the weak sector of the SM. Though some of the detection equipment has been modified from previous experiments, the core of the apparatus remains 33 Chapter 4. Experimental 80 Rb decay (34s) 0+ 1321.1 keV^ 2+ \ / 1256.5 keVJ 2+ 616.8 keV 0+ ^Kr _1+ 34s_ 3?Rb 1.8% P+ 0.068% ec 2.0% p+ 0.071% ec 21.2% P+ 0.44% ec 73% p+ 1.01% ec Q 5708 keV Figure 4.1: The dominant decay branch is to the ground state (73 %), but there is a significant contribution (21.2%) to the first excited state which is followed by a 616 keV gamma ray. 34 Chapter 4. Experimental Rb Ground State and D 2 Energy Levels F=l/2 5S 1/2 232 M H z 5P3/2 F=l/2 I 40 M H z F=5/2 T 1 36 M H z F=3/2 i 780nm F=3/2 Figure 4.2: The atomic levels in 8 0 R b are flipped from the usual configuration due to the large magnetic moment. Chapter 4. Experimental Collection chamber Detection chamber Figure 4.3: Birds eye view of the T R I N A T apparatus. The first trap is located on the left, the second on the right. Once ions are neutralized and trapped they are pushed to the second chamber on the right where the detection equipment is located. intact. This core consists of a large U H V chamber that houses two M O T s as seen in figure 4.3. The detection equipment present for this experiment consists of two opposing micro-channel plates ( M C P s ) (see figure 4.3). Of these M C P s one has position sensitivity and is labeled M C P in figure 4.3, the other has no position resolution and is labeled e~ M C P (electron detector). The chamber is also equipped wi th a Germanium detector and two opposing ^-detectors made of a th in plastic detector and a larger C a F crystal. The chamber has a total of twenty-two view ports which allow laser access. Of those twenty-two view ports, twelve are used by the two M O T s , two are used by a push beam, and eight are used by funneling beams. The push beam is used to transfer atoms from the first trap to the second. The funneling beams in conjunction with magnetic field coils around the view ports are used to control the divergence of the atoms moving between chambers [38]. The M O T beams have a nominal wavelength of 780 nm and are tuned 5 line-widths (w 30MHz) to the red of the D2 transition (see figure 4.2). The M O T beams have a diameter of w 2.5 cm, which causes the M O T to be w 6 cm along the z axis from the position sensitive M C P (so the beams don't clip). 4.3.1 Optical Pumping Details In order to achieve the optical pumping described in sections 3.2, the atoms are immersed in a diode laser field tuned to the D l transition in 80Rb. The beam also has sidebands at 232 M H z added [39] one of which acts as a re-pumper during the optical pumping. The beam is incident along the same axis as the push beam but from the opposing direction. The polarization of this light is varied using a liquid crystal variable retarder. The liquid 36 Chapter 4. Experimental crystal retarder allows one to vary the phase of the light along one particular axis and this can in turn change the lights overall polarization. The degree of retardance is controlled by the amplitude of a 2 kHz square wave fed into the device. The atoms are photoionized using a 355 nm pulsed laser with an average power of 8mW. The pulses are 0.5 ns in width and repeated at 10 kHz. This laser has sufficient energy to singly ionize the excited state with minimum perturbation to the ground-state. This laser gives us a probe of the excited state population without destroying the M O T (ionizes approximately 1 in a million atoms per pulse). The beam is incident along z axis shown in figure 4.3, using the same view port as is used for the M O T beam on that axis. 4.3.2 Electric Field The second trap is immersed in a uniform electric field which accelerates positive charges from the M O T towards the position sensitive M C P and negative charges towards the elec-tron detector. The electric field has two main purposes: it increases the solid angle of accepted decay recoils on the position sensitive M C P and it flattens the M C P sensitivity due to higher impact energy. The field is generated by applying a bias voltage to a series of precisely oriented vitreous carbon hoops (see Appendix E). The hoops are mounted on the same assembly as the position sensitive M C P [see figure 4.4] so that the orientation of the field with respect to the M C P is controlled with great precision. There are a total of six hoops numbered from zero to five sequentially beginning with the hoop nearest the channel plate. Each hoop is electrically isolated from the others, so a unique potential can be applied to each hoop. Each hoop has an outer diameter of 13.5 cm and an inner diameter of 10 cm. The hoop thickness is 1 mm. In addition the zeroth hoop has an inner hoop with an outer diameter of 5.8 cm and inner diameter of 3.6 cm and the fifth hoop has an inner diameter of 2.5 cm and outer diameter of 5.2 cm. The two additional hoops also have a 1 mm thickness. The additional inner hoops help with field uniformity. The field is assumed linear though there is a slight deviation which doesn't affect the experiment (see Appendix E). The hoop bias is applied using high voltage Bertan supplies. The bias on each hoop during the experiment was set to the specifications in table 4.1. These are very similar to the potentials .optimized, in.[37].to.produce a..uniform electric field. .. 37 Chapter 4. Experimental Hoop Number Hoop Bias [keV] 0 inner -3.862 0 outer -3.759 1 -1.990 2 -0.002 3 1.860 4 4.152 5 inner 5.799 5 outer 4.416 Table 4.1: Potentials of Hoops creating the electric field in the T R I N A T apparatus. Figure 4.4: This shows a cross section of the mount assembly holding the field hoops in place. The hoop mount is connected to the M C P mount to ensure precise physical alignment. 38 Chapter 4. Experimental OVERALL TIMING PUSH BEAM OP CYCLE LCVR OP CYCLE 34 jiS 26 tis 30 m I I 3 ^ MOT OP LIGHT 355nm 355nm VETO 37|1S 32 |is | 32 ns Figure 4.5: This shows the t iming for the experiment. The 355 nm veto is generated by an acousto-optic modulator which deflects the beam generating the not signal. 4.4 TRINAT Experiment Timing This section wi l l outline the duty cycle of this experiment. Charged isotopes are delivered to the T R I N A T apparatus at a continuous rate. They effuse from the zirconium foil creating 39 Chapter 4. Experimental a permanent vapor in the first trap. The M O T beams in the first trap are on continuously building up a stable M O T population in the first trap. Both MOTs require a modest magnetic quadrupole field in order to function, which is created by a set of anti Helmholtz coils mounted outside the trap that remain on throughout the experiment. The laser which pushes atoms from the first to second trap cycles on for 30 ms then off for one second. The liquid crystal variable retarder (LCVR) changes its polarization on this same 1 Hz cycle. During the transfer process the second M O T beams are all on, enabling them to trap atoms pushed over. Once in the second trap the atoms need to be polarized. In order to polarize the atoms the M O T is extinguished and the optical pumping diode is turned on. The optical pumping requires a finite time 10 //s)to produce a polarization, but at the same time if the M O T is left off for too long the atoms will drift away. Taking these two effects into consideration the M O T was on for 34 /xs and off for 30 /us. The OP light was on 1 p,s after the M O T light was extinguished. This timing gap was to let the M O T light completely extinguish so the beginning of the optical pumping was clearly defined. In order to determine the polarization the excited state population needs to be known as a function of time. This is achieved using the 355 nm laser. The 355 nm laser ionizes atoms in their excited state which are then accelerated and collected on the MCP. Although pulsed at 10 kHz the 355 nm beam was gated with an A O M (acousto-optic modulator) to only reach the M O T during the OP cycle. The A O M deflects the beam so it doesn't enter the M O T . Without this gate, a population of ions can implant in the M C P and subsequently decay creating a background signal. 4.5 Detectors 4.5.1 M C P A micro-channel plate (MCP) is an array of miniature electron multipliers (channels) ori-ented parallel to one another; typical channel diameters are in the range 10-100 /xm and have length to diameter ratios between 40 and 100 [40]. MCPs are usually fabricated from lead glass and treated so as to render them semiconducting. A metal coating is added to the front and back on the channel plate which allows one to apply a bias. With a bias in place, each channel acts like an independent continuous dynode, and one can typically get electron multiplication factors between 104 — 107. In order to increase the amplification generated by a channel plate, they can be stacked on top of one another. The two most common configurations are called a Z stack and a 40 Chapter 4. Experimental lllllll/ll'lllll/lllllll \\\\\\\\\\\\\\\\\\\\\\\ /////////////////////// Figure 4.6: This is a schematic view of three M C P s in a Z stack configuration. The channels on different plates are angled so as to maximize collisions of electrons with the channel walls, thereby maximizing amplification. A chevron is similar to a Z stack, the difference being there is no third M C P . chevron. A chevron is a stack of two channel plates on top of one another in such a way as to maximize electron transfer. A Z stack is a similar structure but made of three channel plates (see figure 4.6). There are two micro-channel plates used in this experiment distinguished by their po-sition and output signals. The first M C P is a Z stack positioned at a potential lower than the trap position, which allows positive ions to be collected on this channel plate. The output signal consists of a fast component for t iming signals and a slower output from a resistive anode giving position information. The resistive anode output consists of four signals coming from the four corners of that anode. The strength of each signal depends on the proximity of the electron cloud strike. Each of the four raw signals is passed through a preamp before reaching the N I M modules. These outputs can be combined to obtain the coordinates where the ion hit wi th a resolution of 0.25mm [37]. The t iming signal is read directly off the channel plate, and is passed through a delay before reaching the NIMs . The second M C P is a chevron at a higher potential than the trap with the intention of collecting atomic electrons shaken off in the decay process. It is important to note the efficiency of detection for electrons varies with energy (see figure 4.7 from [41]). A metallic mesh is mounted in front of the channel plate and held at a fixed voltage to help with the electric field uniformity in the trap. The mesh voltage can also be kept constant to keep the electric field the same while the entire M C P is floated to different potentials. For example the entire M C P was floated to a potential lower than the trap so no atomic electrons could reach the M C P , and we therefore know all signals are from betas. This M C P only provides a t iming signal which is decoupled from the applied high voltage via a bias tee. 41 Chapter 4. Experimental Channel Plate Efficiency 1 0.95 0.9 & 0.85 c g> o W 0.8 <t> .2 © or. 0.75 0.7 0.65 0.6 0 500 1000 1500 2000 Electron Energy (keV] Figure 4.7: Theoretical efficiency curve for electron detection in an MCP. 4.5.2 Bias Tee In order to uncouple the timing signal from the M C P a bias tee was constructed. A bias tee is an R F component that isolates A C and DC signals from one another. In this case we wish to bias the anode of the M C P (DC) but still need a way to send the timing signal (AC) to the NIM modules. The bias tee isolates the power supply behind an inductive load. Since the impedance of an inductor increases with frequency, this prevents the timing signal from propagating down the inductive part of the bias tee. The A C portion of the bias tee is isolated with a 6kV rated capacitor, which resembles an open circuit to a DC signal but short circuit to a high frequencies. Reasonably careful impedance matching must be done to suppress reflections from the fast signal. The raw timing pulse is shown in figure 4.8. 4.5.3 /3-detectors If the beta asymmetry is known we can look at the asymmetry of beta events in the two detectors and use this to extract a value for the polarization [42] (see figure 1.1). The /?-detector is a thin plastic scintillator positioned in front of a larger calcium fluoride crystal. The plastic scintillator provides a fast signal for timing purposes, and the calcium fluoride crystal gives a much slower signal but allows for energy resolution. The /3-detectors are oriented at an angle of 30 degrees with respect to the optical pumping axis. Calcium fluoride was chosen as an economical detector with high enough Z to stop 10 MeV positrons [43]. 42 Chapter 4. Experimental Raw Electron MCP Signal 0 1 2 3 4 5 6 7 8 9 10 time [ns] Figure 4.8: Timing signal from the electron MCP. The events for this signal were created with a gamma ray source. The ringing is due to multiple reflections in the bias tee. These detectors allow us to distinguish beta events from gamma events. When a (3 enters the detector it will deposit energy in both the plastic and calcium fluoride crystal. Gamma rays in the relevant energy range preferentially Compton scatter are will thus deposit energy only in a single detector then scatter away. This difference allows us to eliminate most of the gammas from the beta asymmetry. 4.6 Main Data The data for this experiment were collected over two separate one week periods in December 2005. Only the second week's data are included in the analysis due to the fact that the first week's data had large systematics; in particular there was significant cloud motion between polarizations. The cloud motion is a complicated function of the local magnetic field and the light frequency and polarization. It was found in the second data set that the motion could be nearly eliminated by adjusting the light frequency for the two opposing polarizations. This resulted in a total of 38 runs of viable data and 3 runs of background data. In the background runs the bias on the electron detector was modified so that no shakeoff electrons could penetrate the detector. With this modification in place only /3's or gammas can trigger the electron detector. An important concept for this and future sections is the definition of an event. Through-43 Chapter 4. Experimental Figure 4.9: Distr ibution of all events striking the position sensitive M C P . out the experiment each detector is coUecting information, however this information is only read-out when a specific set of conditions are met. This set of read-out conditions is what defines an event. In this experiment there are two kinds of events. The first is any signal being registered in the position sensitive M C P and the second is any coincidence between a /3-detector and the electron M C P . 4.6.1 MCP distribution The position sensitive M C P is at a lower potential than the M O T and therefore al l positive charged particles are collected on this detector. The total distribution is the sum of al l photoion events, all charged recoiling nuclei from beta decay and any /?'s which happen to go towards this detector. The total distribution is shown in figure 4.9, the projection of this distribution along the polarization axis is shown in figure 4.10. 44 Chapter 4. Experimental MCP SINGLES T—•—i—1—i—1—i—•—i—1—i—1—r X POSITION [0.1mm] Figure 4.10: Distribution of al l events striking the position sensitive MCP projected onto the axis of polarization. The spike is caused by an overactive (hotspot) on the MCP. Chapter 4. Experimental Time of Flight Spectrum Time of Flight Spectrum 300032003400360038004000420044004600 3000 3200 3400 3600 3800 4000 4200 Channels Channels Figure 4.11: This time of flight spectrum shows clear peaks at each charge state. Also visible is the prompt peak. The physical flight time of an ion is given by tprompt — t. Each channel corresponds to 1 ns. The right hand figure is shown on a linear scale 4.6.2 Time of flight [TOFe] (e~detector stop) The spectrum is generated by taking the time difference between an event trigger and the signal from the electron detector using a T D C . In real time the electrons strike the electron detector before ions hit the position sensitive MCP. However, there is a long delay on the electron detector so the signal actually arrives later than the ion signal. This creates a reversed time of flight spectrum with faster ions showing at greater times see figure 4.11. Different charge states arrive at different times. There are two kinds of events that can contribute to the prompt peak; the first is from a beta striking the ion channel plate and the corresponding shakeoff electron being detected in the electron detector, the second is an electron is released from the ion detector on impact, which can only contribute if the initial shakeoff electron goes undetected. The physical flight time of an ion can be calculated from tprornpt — t. The electron takes just a few ns to reach the electron M C P so this signal determines the ion T O F with good accuracy. This spectrum was also obtained from the background data sets and is shown in figure 4.12. This is a few hours worth of data. The normalization determination will be considered in the following Chapter. In the background data set there is an anomalous peak appearing at 3200ns. This peak is believed to be an apparatus timing artifact. It has also been observed in the absence of radio-actives using only a gamma ray source placed outside the 46 Chapter 4. Experimental TOF Background 40 35 30 25 10 h 15 5 1 o u-i 3000 3200 3400 4200 time [ns] Figure 4.12: Time of flight spectrum with electron detector biased so no shakeoff electrons are accepted. detector. It appears as though the number of counts in the peak scales linearly with overall count rate. 4.6.3 Photoion Time of Flight (photodiode stop) The photoion time of flight spectrum is generated by taking the time difference between an event trigger and the signal from the 355 nm laser's photodiode. The photodiode has a long delay, making this a reversed time of flight spectrum as well. There are two peaks of interest in the spectrum. The first occurs at a time of « 2300 ns (see figure 4.13). This is a prompt event peak, which determines the effective zero time. The peak is caused by some of the 355 nm light being scattered on to the ion channel plate and causing it to trigger. The second peak at « 1150 ns is due to the photoionized excited state atoms hitting the ion channel plate. The time is determined by the trap position and the strength of the electric field. The width of the peak contains information on both the field strength and the width of the M O T distribution along the time of flight axis. The number of events in the photoion peak is proportional to the number of atoms in the trap and the total time data was taken. 4.6.4 Optical Pumping Excited state population [OPex] During the optical pumping cycle a 10 MHz pulse generator sends pulses into a scaler. The scaler is then reset a fixed time after the optical pumping light is extinguished. The scaler 47 Chapter 4. Experimental 4000 3500 3000 500 Photodiode TOF — i — • — i — • — i — • — i — 1 — i • No Cuts i • i — • I—• i > i ' I—• I — 500 750 1000 1250 1500 1750 2000 2250 2500 Time [ns] Figure 4.13: The two peaks of interest are caused by photoions (wl 150ns) and laser triggers («2300ns) both in the ion channel plate. 48 Chapter 4. Experimental OP Photoions 3000 2500 2000 w c Q 1500 1000 500 0 100 200 300 400 Time [100ns] Figure 4.14: This is a timing spectrum of events that occur during times when the atoms are polarized. is read out whenever an event occurs. This gives a timing spectrum of events which occur during the times when the atoms are being polarized (see figure 4.14). The majority of these events are caused by the 355 nm laser ionizing atoms in the excited state, which are then collected on the position sensitive MCP. Since the laser can only ionize excited state atoms, this signal is proportional to the number of atoms in the excited state as a function of time. With the help of a rate equation model, this signal can be fit to extract a value for the nuclear polarization [42]. The signal from « 100 - « 350 is the times during which the M O T is off and the OP light is illuminating the atoms. The large spike that appears is caused by the M O T light coming back on. n • I 1 r -• Photoions during O P 49 Chapter 4. Experimental M C P S I N G L E S Conditions: GoodMCP X POSITION [0.1mm] Figure 4.15: This is the distribution of events on the M C P with al l hotspots removed. 4.7 Data Cuts 4.7.1 M C P distribution (with cuts) Unfortunately the detection efficiency of the position sensitive M C P is not entirely uniform. There are a few locations that show a disproportionately large number of events which are unrelated to the /3-decay. In order to correct for these "hot spots" a condition was defined call " G o o d M C P " which does not include any events from these locations. Another event type that was vetoed in the " G o o d M C P " condition was events in which one of the preamp outputs had saturated. The distribution of events with the " G o o d M C P " cut is shown in figure 4.15. The distribution shown in figure 4.15 contains events that occur when there is no po-larization. In order to examine events during times when the atoms are polarized another cut was introduced called "cgplus". This cut requires the event to occur within times of 125-300 on the OPex spectrum. The cut also requires the L C V R to be set for positive polarization and includes the G O O D M C P cut. There is a similar cut called "cgminus". 50 Chapter 4. Experimental Figure 4.16: This is the distribution of events on the M C P during the optical pumping cycle for positively polarized light. A t iming cut is introduced so only singly charged ions are present. The difference between the two conditions is the polarization of the light as determined by the l iquid crystal variable retarder. The height of the square wave fed into the L C V R is recorded on each event and this allows the ability to distinguish the polarization. The distribution in figure 4.15 also contains events from all possible charge states and from electron capture. Since each charge-state must be analyzed individually their distribu-tions must be separated. The electron capture t iming is determined from a Levenberg Mar-quardt fit [44] described in the analysis section. In order to define individual charge-states a t iming cut is made on the T O F e spectrum (see table 4.2). The cuts are called "krinec" where i is the charge state number for that particular cut. A n example of the distribution on the M C P with both the "cgplus" and the "krinec" in addition to the " G o o d M C P " cut is shown in figure 4.16. The projections of all the 2D M C P distributions along the polarization axis is shown in figure 4.17 Although a M O T does provide a well localized source of nuclei, it is sti l l of finite ex-tent. The relevant M O T size can be determined by looking at the position distribution of 51 Chapter 4. Experimental MCP SINGLESfwith cuts) 1000000 100000 o o -KRInec -GoodMCP • RAWMCP •200 -150 -100 -50 0 50 100 150 200 X POSITION [0.1mm] Figure 4.17: This shows all distributions on channel plate events as projected along the axis of polarization. 52 Chapter 4. Experimental X distributions Y distributions 2 c Position [mm] 0 5 Position [mm] Figure 4.18: Trap distributions during optical pumping. The backgrounds are from betas hitting the channel plate. photoions on the ion M C P during the optical pumping portion of the duty cycle. To get the photoion events, a timing cut was made to the photoion T O F isolating only the peak at « 1150ns. It is also important to know the exact location of the M O T cloud because incorrectly identifying the position of the cloud can generate a false asymmetry. This result is included in the final analysis. Background events in this data are mostly betas hitting the plate during the optical pumping cycle. 4.7.2 Asymmetries An asymmetry can be constructed for each produced charge state by comparing the M C P position distributions along the axis of polarization of recoiling ions for two opposite po-larizations of light. The asymmetry is defined as the difference between the positive polar-ization and negative polarization distribution divided by their sum. We can extract these spectra by requiring the following: a time of flight cut in the TOFe spectrum so we know the charge state of the ions ("krinec" see table 4.2), a timing cut in the duty cycle so we know the atoms are polarized ("cgplus"or "cgminus"), and a timing cut in the photoion T O F so we aren't observing photoion events. In order to minimize systematics from different numbers of events in each of the two polarization states each the position distributions are first divided by the total number of counts in the distribution. 53 Chapter 4. Experimental Condition Low cut High cut krlnec 3159 3559 kr2nec 3599 3749 kr3nec 3814 3869 Table 4.2: Timing cuts in TOFe spectrum which define each charge state. Kr" Asymmetry EC removed -(00 -so 40 MCP position (0.1mm) Figure 4.19: Asymmetry for charge state one. Figure 4.20: Asymmetry for charge state two. 54 Chapter 4. Experimental Kr Asymmetry EC removed -too -eo -eo -40 -20 0 20 40 60 00 100 MCP position [0.1mm] Figure 4.21: Asymmetry for charge state three. 4.7.3 TOFe with cuts The "GoodMCP" cut also has an effect on the TOFe spectrum which can be seen in figure 4.22. However, the desired final dataset however needs to include only events which are polarized. It also needs to exclude events where photoionization is caused by the 355 nm laser, since these are not /3-decay events. With this purpose in mind another cut called "Polarized" was created. This cut is just the sum of the "cgplus" and "cgminus" cuts with an additional constraint that the time in the photoion T O F not fall within its peak (see figure 4.22). The bump appearing at 3250 in figure 4.22 is caused by 355 nm laser photoion events and disappears once the cut is put in place (see figure 4.22). Note that the ion events which have zero velocity appear at this time. 4.7.4 Photodiode TOF with cuts The photodiode T O F spectrum has a significant background signal caused by the large number of unwanted events. The photoionizing light is only present for a small portion of the total experiment timing. In order to increase the signal to noise a timing cut can be made on the OPex spectrum so only the events during the MOT-on time are included. This spectrum will then be dominated by atoms which were photoionized from the M O T (see figure 4.23). 55 Chapter 4. Experimental eMCP TOF Spectrum 40000 35000 30000 25000 c 20000 O o 15000 10000 5000 0 3000 3250 3500 3750 4000 Time [ns] Figure 4.22: This plot shows how the T O F e spectrum changes as the " G o o d M C P " and "Polarized" cuts are introduced. 4 . 8 Auxiliary Data Sets 4.8.1 CaF T O F (/3-detector stops) The calcium fluoride time of flight spectrum is generated by taking the time difference between an event trigger and the calcium fluoride signal. The calcium fluoride is delayed longer than the event trigger so we again observe a reversed time of flight spectrum (see figure 4.24). The prompt peak appearing at « 1100ns is created by calcium fluoride electron M C P coincidences. These create an event and subsequently a stop signal for this spectrum. The majority of the interesting events are generated by a 0 firing one of the calcium fluorides and an ion subsequently hit t ing the ion M C P . The charge-state peaks have a different shape than observed in the T O F e spectrum due to the kinematic constraint of requiring a beta in one of the calcium fluoride. There is also a background observed in this spectrum. Since not all decays produce a charged ion, there wi l l be some betas in the calcium fluoride that do not coincide with an 56 Chapter 4. Experimental Photodiode TOF 4000 3500 2500 c 2000 o O 1500 1000 • No Cuts - MOT cycle I' ' I ''I '"I i I' ' I ' "I1 1 500 750 1000 1250 1500 1750 2000 2250 2500 Time [ns] Figure 4.23: This plot shows how the photodiode T O F signal to noise can be increased by requiring the M O T light to be present. ion. We can estimate the size of this background wi th a simple argument. We begin by deriving an expression for the event rate expected in the charge state region of figure 4.24. If we define the following variables: Pp is the probability of a beta triggering one of the calcium fluorides, Pim is the detection probability of an ion, r is the decay lifetime, N is the number of atoms and f is the fraction of decays that produce a charge-state. Then the decay rate is given by equation 4.1. - (4.1) r The background rate (equation 4.2) wi l l be the rate of betas triggering the calcium fluoride multiplied by the average ion time of flight ( T O F ) and the rate of ions with no beta in calcium fluoride (equation 4.1 wi th Pp replaced by l-Pp)-R b k g = W ~ PfiPicn) NPbetgTof ( 4 2 ) 57 Chapter 4. Experimental CaF Time of Flight Spectrum Channels Figure 4.24: The charge-state peaks have a different shape than observed in the TOFe spectrum due to the kinematic constraint of requiring a beta in the calcium fluoride. If we divide these we get the fraction of events expected to be background. NTof BKG = (4.3) This amounts to a 1% flat background. This effect is too small to account for all of the background observed in this spectrum. This is a next to leading order correction to the electron M C P efficiency as a function of charge-state; as such, the uncertainty in the background is a negligible error in the final result. 4.8.2 M C P T O F (ion M C P stops) The ion M C P time of flight is generated by taking the time difference between an event trigger and the ion M C P timing signal. This is the only time of flight spectrum that natu-rally appears in real time order (faster ions at lower times). The prompt events at «300ns are M C P event and then a self stop. The events of interest come from an electron MCP, calcium fluoride coincidence and are then stopped by the ion M C P (see figure 4.25). Again the kinematic constraint changes the charge-state peaks as compared to TOFe; however, the shape is the same as that given in the CaF T O F spectrum. 58 Chapter 4. Experimental 350 MCP Time of Flight Spectrum (ion MCP stops) 800 1000 Channels Figure 4.25: The events of interest come from an electron MCP, calcium fluoride coincidence and are then stopped by the ion MCP. 4 . 9 C o n n e c t i o n b e t w e e n o b s e r v a b l e s a n d 6, Pr In Chapter 2 the distribution of recoil nuclei as a function of both momentum magnitude (Pr) and angle with respect to the polarization axis (9) was shown. The observables in this experiment do not allow us to directly measure 9 and Pr however, we do have sufficient information to determine their value. This Chapter will outline the connection between the available observables and 9,Pr. When a decay takes place the daughter nucleus is left with a recoil momentum Pr. Since the recoil momentum is small ( « 5MeV) compared to the rest mass of the nucleus ( « 80GeV) we can use classical physics to relate the velocity and momentum. We define the x axis to be the axis of polarization and z to be going from the position sensitive M C P to the M O T cloud (see figure 4.26). Then the three components of the recoils velocity can be defined as in equation 4.4. Vx — ^Nucleus \Pr\ ^Nucleus cos(6>) sin(f?) cos(</>) sin(#) sin(0) (4.4) 59 Chapter 4. Experimental Electric | Field OP LIGHT Figure 4.26: This figure shows how the recoil angle is defined wi th respect to the electric field and optical pumping (OP) light. In equation 4.4 Msucieus is the mass of the daughter nucleus, 6 is the angle between PT and the x axis and <p is the angle from the PT projection in the yz plane and the y axis. If the daughter nucleus is neutral it w i l l freely propagate within the chamber; however, i f it was left in a charged state it w i l l be accelerated by the electric field in the negative z direction. The magnitude of the acceleration is easily derived from classical physics (see equation 4.5) MNucleus In equation 4.5 a is the acceleration, q is the net charge, E is the electric field and ^Nucleus is the nuclear mass. The time (t) for the ion to strike the position sensitive M C P if the M O T is a distance d away is given by equation 4.6. t = -vz + yjv\ + 2ad (4.6) In this experiment the t iming is dominated by the acceleration but the t iming width contains information about the ini t ial velocity. This can be seen by Taylor expanding equation 4.6 for small vz. t (4.7) 60 Chapter 4. Experimental The position on the channel plate is also required to gain full information about the recoils initial momentum. These are related by equation 4.8 x = vxt (4.8) V = vyt Armed with the knowledge of how the experiment funcitons, we are prepared to describe how all the data was analyzed to extract meaningful values. 61 Chapter 5 Analysis The goal of this Chapter is to explain how the data was analyzed. The experiment was modeled with a Monte Carlo simulation written in F O R T R A N . In order to accurately describe the dynamics the simulation requires input to describe the M O T and the electric field. This initial input data consists of nine input parameters: the M O T centroid and width in 3 dimensions; the electric field strength (assumed constant along the z axis therefore requiring only a single parameter); the polarization and the alignment. 5 .1 Fitting Photoion TOF In order to model the kinematics of ions in the TRINAT apparatus, it is important to know the distribution of atoms before decay. This is equivalent to how the M O T is distributed in space. Both its location and extent have relevant implications on the final asymmetry. The dimensions parallel to the ion M C P face (x,y) can be analyzed by examining the photoion distribution on the MCP. These atoms are cold (lmK) and therefore have a negligible initial velocity. Any photoionized atoms will be accelerated in the electric field and hit the plate at the same x,y position as they had in the M O T . The observed distribution on the M C P is then a convolution of three distributions: 1) the profile of laser light exciting the atoms (the 355 nm can only photoionize atoms in the P state), 2) the profile of the 355 nm laser and 3) the M O T cloud itself. In this analysis both 1) and 2) are assumed uniform. This is justified since both these beams have a 1" diameter. Assuming Gaussian beam profiles, this results in a 2% variation in laser intensity over 98% of the M O T cloud. The dimension perpendicular to the plate (z) is more complex. The information about the trap in this dimension in contained in the photoion time of flight spectrum. This spectrum not only contains information about the trap width but also the electric field strength and the trap distance from the ion MCP. In order to extract this information a Levenberg Marquardt fit was performed with three fit parameters: the electric field (EF), the trap distance (ztrap) and the trap width (az) assuming a Gaussian trap. The algorithm generates initial start positions from a Gaussian distribution (initial ztrap and az must be given) with zero initial velocity. The algorithm assumes a constant electric field and 62 Chapter 5. Analysis Parameter Value Uncertainty E F 81369 & 144 £ ztrap 6.4536cm 0.0115cm oz 0.952mm 0.072mm Reduced x'2 13.16 Table 5.1: Fit parameters for Photoion T O F analysis. Photoion Fit -T 1 1 1 -if X I /* \ - & i t \ Jf 5 -111 i T Tn =fT* / \ / \ 1100 1120 1140 1160 1180 1200 time [ns] Figure 5.1: This shows the fit to the photoion time of flight spectrum. The points with error-bars are data and the dashed line shows the fit. trivially calculates the T O F for the event. This is repeated for five million events. The derivatives (required for a Levenberg Marquardt algorithm) are numerically generated by incrementing the appropriate parameter by 5% of its initial value and generating another T O F distribution. Once all the derivatives are generated the parameters are incremented according to the Levenberg Marquardt algorithm. This is repeated until the variation in X2 between successive steps is less than 1%. The final parameters and x 2 are in table 5.1. The time axis was converted to absolute time by inverting the spectrum and setting the peak of the prompt peak as the origin. A background subtraction as also performed by averaging over 800 time bins (channel 200-1000 in figure 4.13). The x2 for this fit is very poor. This is caused by the irregular shape of the trap which deviates from the simple Gaussian assumed in the model. The uncertainties quoted have been inflated to account for the poor fit. The discrepancy between the real trap shape and 63 Chapter 5. Analysis the Gaussian assumed in the fit code will lead to a small systematic uncertainty in the final result. This trap width is used as an input parameter for future fits. The field and distance will be checked against other results for consistency. 5.2 Polarization In order to analyze the asymmetry data a value of the polarization is required. A detailed analysis of the polarization in this experiment is given in [42] and will be discussed only briefly in this section. A similar analysis was also performed in [43]. During the optical pumping cycle the 355 nm laser photoionizes a small fraction of atoms in the excited state. These photoions are then collected on the M C P giving an event trigger. The number of events as a function of time is therefore proportional to the number of atoms in the excited state, which changes as the atoms are optically pumped. Using a rate equation model this signal was fit to extract a polarization value. The value of polarization obtained using this method is P=0.88. This method of analysis is sensitive to inhomogeneities in the magnetic field. The local magnetic field determines the Zeeman shifts and a finite trap samples multiple values of the field. This method is also sensitive to the light profile of both the optical pumping beam and the photoionizing beam. This method neglects the re-absorption of light as well. Given a high trap density it is possible for light emitted by one atom to be reabsorbed by another, which is not accounted for in the rate equation model. The polarization can also be obtained by examining the /? asymmetry. Examining the number of counts in the two /3-detector detectors as a function of energy allows one to track how the asymmetry varies with beta energy. The most high energy betas should be the least sensitive to scattering systematics. Using the upper energy betas which were coincident with and signal in the electron detector, the asymmetry was found to be P=0.545 ± 0.031. The polarization was also examined by looking at electron capture events of high charge state. These events have a large asymmetry which can be calculated in the SM. Modelling the asymmetry from these events was done by John Behr [45] and the resulting asymmetry was found to be 0.57 ± 0.04. These two later methods of analysis are clearly in grave disagreement with the first. Noting the potential systematics from the first method a value of P=0.55 ± 0.05 was accepted. The polarization value is also constrained by the alignment term. The alignment (T) is related to the expectation value of squared z angular momentum (< m2 >) by equation 5.1. 64 Chapter 5. Analysis 3<m2j> T ' 1 ~ J(J + 1) ( 5 - 1 } In a J= l system rrij sets an upper limit on the value of the polarization. The values of the alignment from the time of flight fit therefore restrict the possible values of the polarization. The systematic effect of the polarization on the asymmetry fit is therefore correlated with the alignments uncertainty. The details of how this systematic effect will be handled are in section 5.3.4. 5.3 TOFe fit In order to fit the TOFe spectrum, another Monte Carlo-Levenberg Marquardt algorithm was employed. A detailed discussion of the Levenberg Marquardt algorithm can be found in [44]. This spectrum contains information about the electric field (EF), the trap to M C P distance (ztrap), the nuclear alignment (T) and trap width (crz). The fit code fits for all these values except cr2, which is taken from the photoion T O F fit. The TOFe spectrum is very complex, because each charge state creates a different TOFe component. In addition to all the /3-decay contributions, it was found that electron capture also makes a significant contribution in this spectrum. Although electron capture occurs in only s=s 1% of decays, it is much more likely to leave the daughter atom in a charged state than a regular /3-decay. The TRINAT apparatus therefore has increased sensitivity to electron capture events. The amount of each /3-decay charge state and electron capture charge state observed is a product of the number of each charge state produced and the overall apparatus efficiency. Since the charge state production numbers are not known a priori the relative contribution of each (3 — decay and electron capture TOFe are left as fit parameters. In addition to /3-decay and electron capture events there were also background events which required a correction to the data. This correction is described in the following section. 5.3.1 T O F e background subtraction Since the timing of TOFe is controlled by the electron MCP, any non-shakeoff electron signals in the M C P will cause a background. As mentioned previously /3s can trigger the electron channel plate giving events with a timing distribution observed in figure 4.12. The fact that a ft travelling towards the electron M C P can create a signal, even if the shakeoff electrons don't trigger the MCP, means there is increased sensitivity for these kinds of events. The model used to fit the time of flight assumes uniform sensitivity to events 65 Chapter 5. Analysis irrespective of the beta direction. The goal then is to subtract off the excess events created by the f3 firing the electron M C P when the shakeoff electrons did not. This can be achieved by subtracting an appropriately scaled background distribution (figure 4.12) from the TOFe data. The subtraction is complicated by the fact that the probability to trigger the electron M C P varies with the charge state produced (number of shakeoffs). In order to make the subtraction two numbers are required: 1) the overall scaling of the background, since the background data was taken for a much shorter period of time than the asymmetry data, and 2) a scale factor for each charge state describing the probability that the electron detector was not triggered by a shakeoff electron. The overall scaling of the background can be obtained by comparing the number of events in the photoion T O F photoion peak with the M O T beams illuminating the trap. The number of photoions produced is given in equation 5.2. Nphotoion ~ &ionizefexl355nmNa,tomsTrun (5-2) In equation 5.2: cxit^e is the ionization cross section, fex is the fraction of atoms in the excited state, / 3 5 5 n m is the 355 nm laser intensity, N a t ( m i s is the number of atoms in the trap and Trun is the acquisition time. After subtracting the backgrounds in both photoion T O F spectra, we take the ratio of Nphotoian (sum of all events in photoion peak) for the real data and the background data, we are left the desired scale factor (S). S = 17.12 ±0 .38 (5.3) Calculating the scale factor this way assumes both the excited state distribution and the 355 nm laser profile remained the same during the background runs and the regular data runs. In order to deduce the probability of detecting electrons for a given charge state, both the CaF T O F and M C P T O F spectra were required. The spectra were divided into separate charge state regions according to table 5.2, and all events in each division were summed. In order for a charge state event to appear in the M C P T O F spectrum, we must have a trigger in the electron detector, the ion M C P and the calcium fluoride. The most likely events of this type are a [3 triggering the calcium fluoride and both the electron and ion being collected. If we let Pe- be the probability of triggering the electron detector for a given charge state, Pion be the probability of triggering the ion M C P for a given charge state, Pp be the probability to trigger the calcium fluoride, N a t o m s be the number of atoms in the trap and Trun be the length of data collection, then the number of events in a particular charge state in CaF T O F is given by equation 5.4. 66 Chapter 5. Analysis Charge-state Low cut High cut 1+ 3 371 2+ 450 608 3+ 611 728 4+ 728 806 5+ 806 855 6+ 855 870 Table 5.2: Division of charge-states in CaF T O F spectrum in relative time units. Charge-state Low cut High cut 1+ 1025 1408 2+ 808 981 3+ 697 746 4+ 618 696 5+ 570 618 6+ 530 569 Table 5.3: Division of charge-states in M C P T O F spectrum in relative time units. Nmh5 — NatomsTrunPo-PionPp (5-4) Similarly the number of a events in a particular charge state in CaF T O F is given by equation 5.5. Nmh3 = N a t o m s T r u n ( l — Pe-)P%(mPj3 (5-5) The CaF T O F spectrum has no events with the electron triggering the electron detector. These events would be prompt in this spectrum since it is stopped by the calcium fluoride, and a coincidence of calcium fluoride and electron detector is required to register as an event. Therefore the quotient of M C P T O F and the sum of both spectra over a given charge state will give the probability of the electron detector being triggered when said charge state is produced (see figure 5.2). In order to use these numbers the background spectrum itself had to be divided into charge-state components. Fortunately the uncertainty in the background is dominantly statistical, and so the subtraction is very insensitive to how the background is divided. The divisions used are given in table 5.4 (they are given in real time). 67 Chapter 5. Analysis Shake-off Detection Efficiency —i 1 1 1 r I I i I 5 " , j I I I I I " " 0 1 2 3 4 5 6 7 Charge state j Figure 5.2: Electron detection efficiency for Krypton charge-states 1 through 6. Charge-state Low cut High cut 1+ 950 1200 2+ 725 950 3+ 610 725 4+ 530 610 5+ 475 530 6+ 475 400 Table 5.4: Division of charge-states for background spectrum to be subtracted from TOFe. 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 n oc 68 Chapter 5. Analysis X distributions Y distributions c s Position [mm] -5 0 5 Position [mm] Figure 5.3: This shows the fits (dashed) to the photoion distribution data (cross) in x and y for both polarizations. The anomalous peak in the spectrum was observed offline to scale with count rate and so this portion of the spectrum was only multiplied by the scale factor. Each charge state division was then weighted by the appropriate 1 — Pshake {Pshake being the values in figure 5.2). This corrected spectrum was then subtracted from the raw TOFe data before the fit was performed. 5.3.2 Position Distribution fit In order to correctly model the TOFe spectrum the x and y distributions are required. In order to fit these distributions a Levenberg-Marquardt algorithm was used. The photoion distributions were only fit over a limited range to eliminate contamination from background events. The x distributions were fit from -20mm to 20mm and the y distributions were fit from -20mm to 10mm. 5.3.3 TOFe fit algorithm In order to begin, the algorithm requires an initial electric field (EF), trap position (ztrap), alignment (T) and polarization (P) as well as x,y and z trap distributions. The algorithm also requires the /3-decay momentum distribution. The decay momentum distribution is a function of the magnitude and angle with respect to the polarization axis of 69 Chapter 5. Analysis the recoil momentum and cannot be expressed analytically due to the complexity involved in integrating the Fermi function over electron energy. In order to overcome this hurdle, a discrete distribution was generated by discretizing the magnitude of the momentum and possible angles to 100 points each. The complete recoil momentum distribution was further divided into three components as discussed in Chapter 2 section 8. These discrete distributions are generated by another code and are input parameters for the fit code. The algorithm begins by constructing the total recoil distribution with positive and negative polarization for both the ground and excited final state. This is done by summing the three distributions as was shown in equation 2.55. The recoil order matrix elements are ignored for the purpose of this analysis. This approximation is justified because recoil order terms have very little effect on the T O F . Once the recoil distributions are constructed the simulation begins. For each of the charge-states one through nine one million events are generated with both positive and negative polarization using both the ground and excited state momentum distribution. The excited state contribution is scaled to account for the smaller branching ratio. This gives a total of four simulated data sets. The two ground state simulation sets are summed and the two excited state simulation sets are summed as well, this reduces the total to two sets. These two sets are summed by first scaling the excited state contribution to correct for the smaller branching fraction. The events are generated by first generating x, y and z starting positions and then a recoil momentum value. The latter are generated by a simple rejection algorithm. Three random number are used: one in the range of zero to the maximum momentum magnitude; one in the range of zero to 7r; and the final between zero and the maximum value of the momentum distribution. If the last value is larger than the momentum distribution value at the point described by the other two random numbers (as determined by a linear inter-polation between grid values), the value is rejected and the process repeated. If the value is smaller, then the momentum angle and magnitude are accepted. Since the momentum distribution is insensitive to the other angle needed to fully describe the momentum direc-tion, this value is generated from a flat distribution once a magnitude and angle have been accepted. When generating events which decay to the excited state the gamma ray must also be included. The gamma ray has a fixed energy (616keV) and its direction is generated using equation 5.6 and another rejection algorithm. W(0) = M 2 ( l - 3cos2(0)) + 2(1 + cos2(0)) (5.6) 70 Chapter 5. Analysis The initial state now has a fully defined position and velocity, and the kinematics can be worked out to find its final position and T O F . The T O F simulation is binned in one nanosecond bins. Once completed, this portion of the simulation has created nine T O F functions. In order to simulation electron capture x, y and z values are again generated. The electron capture events are simulated much more quickly because the final state in electron capture has only a single momentum defined by the kinematics of a two body decay. The angular distribution is defined by equation 2.8, and the angle of the recoil momentum was generated using another rejection algorithm, although this time there is an analytic function in the place of a discrete grid. Events were generated again for both polarization states and to both branches (ground and excited). These four distributions were summed in a similar way. All the different polarizations were again added directly, but the excited state was appropriately scaled before being added. Electron capture events were simulated up to charge state eight giving a total of 17 spectra to fit the TOFe. A linear fit is then performed to find the coefficients for each spectrum, and chi square is evaluated. The Levenberg-Marquardt algorithm works within a parameter subspace, only affecting E F , ztrap and T. The numeric derivatives are generated by a finite difference method for each of these three fit parameters. This requires the simulation of all charge state events and electron capture events to take place once for each of the three fit parameters. The derivatives are used to calculate new tentative values for these three fit parameters [44]. The simulation is then run again at these updated values of the three parameters. At this point another linear fit is performed giving tentative linear fit parameters as well, then the chi squared is evaluated. If the chi squared is better than the currently accepted chi squared value, then all the tentative fit parameters are accepted as the new best values and the chi squared value becomes the new currently accepted chi squared value. If the chi squared is worse than the currently accepted value then all the tentative values are rejected and new tentative values must be calculated from the same derivatives (process described in detail in [44]). This process is repeated until the newly accepted chi squared value is within 1% of the previously accepted chi square. The resulting parameters are displayed in table 5.5. The data and fit are shown in figure 5.4. 71 Chapter 5. Analysis Parameter Value Uncertainty (Statistical) E F 81354 £ 59 ztrap 6.5117cm 0.0047cm Alignment 0.13 0.17 Reduced x2 6.546 Table 5.5: Time of flight fit parameters and statistical uncertainties. Fit to Time of Flight Spectrum (stmhl) 400 500 600 700 800 900 1000 1100 1200 1300 1400 Time [ns] Figure 5.4: This is the simulation fit to the TOFe spectrum. The total fit is obscured by all the data points however, the electron capture and ion contributions are very clear. 72 Chapter 5. Analysis Parameter Value Uncertainty (Statistical) E F 81369 v- 60 £ ztrap 6.5144cm 0.0048cm Alignment 0.548 0.181 Reduced %2 6.553 Table 5.6: Time of flight fit parameters and statistical uncertainties for fit with trap width minus one sigma. Parameter Value Uncertainty (Statistical) E F 81367 £ 6 0 ^ ztrap 6.514cm 0.0048cm Alignment 0.095 0.17 Reduced x2 6.553 Table 5.7: Time of flight fit parameters and statistical uncertainties for fit with trap width plus one sigma. 5.3.4 TOFe systematics The most obvious systematic effect on the fit is from the width of the trap, which was determined in the photoion T O F fit. In order to account for this effect, the time of flight fit was performed at the 68% confidence extremes. The parameters for these fits are displayed in tables 5.6 and 5.7 The values in tables 5.7 and 5.6 suggest the only systematic effect is in the alignment term. This analysis suggests we add an asymmetric systematic error bar to the alignment of +0.41 which is correlated with the smaller trap width. Upon inspection of the fit parameters for the photodiode T O F fit (table 5.1) and the fit parameters for the TOFe fit (table 5.5) we see excellent agreement with the electric fields but there appears to be a 6cr disagreement between the trap position values. A large portion of this systematic is due to the irregular shape of the trap along the z axis. If we look at the fit to photodiode T O F (figure 5.1), the peaks are misaligned by 3ns, which translates into 0.3mm at the given electric field. This would push the photodiode T O F fit value to 6.48cm, giving only a 3a deviation. In order to account for this discrepancy in trap position, the average value of the two positions will be taken (6.484cm) and the extreme values will be taken as the systematic uncertainty range. This will be seen below to have little effect on the final asymmetry. The electric field could also be changed to accommodate for this disagreement, but 73 Chapter 5. Analysis Data Set H Value photodiode T O F 1260831.164 X TOFe 1249351.168 X, Table 5.8: Hvalues for two runs Parameter Value with Systematic Uncertainty E F ztrap 6 . 4 8 4 ± ^ c m Alignment 0 .13^ 4 i Table 5.9: Final values and systematics. there is a strong correlation between these two variables and they should not be varied independently. It is important we also find the corresponding systematic uncertainty in the electric field. In the limit of no initial velocity and trap width, the photodiode T O F spectrum becomes a delta function at time tnv given by equation 5.7. Guided by this relation we define a new parameter H to be the ratio of electric field and trap distance (see table 5.8). The systematic extremes of the electric field are determined by using the two H values and the systematic extremes of the trap position. This results in the following values to be used for the one a confidence interval in the final asymmetry analysis code. The electric field and trap distance are correlated in such a way that the largest electric fields will be used with the smallest trap distance and vice versa. 5.4 Asymmetry analysis This section will overview how the asymmetry data was analyzed. The background from electron capture proved to be much larger than originally an-ticipated, and since the asymmetry for electron capture is large (+0.52 averaged over all branches) this could lead to a very large correction in the analysis. Since the electron capture cannot be analyzed independently to check the accuracy of the analysis model, it was excluded from the analysis by making a timing cut in the data. Unfortunately for 74 Chapter 5. Analysis Charge-state Low cut High cut 1+ 950 1350 2+ 760 910 3+ 640 700 Table 5.10: Timing [real time] cuts used to define each charge state and remove electron capture events. all charge-states greater than three the electron capture portion is too large to exclude by making a timing cut and so this data was excluded from the asymmetry analysis. In each charge state a time of flight (TOFe) cut was made on the data to exclude electron capture. The cut removed any time bins for which electron capture events accounted for more than 1% of the total. Since a charge state is defined over multiple time bins these cuts reduced the overall electron capture contribution to be less than 0.1%. This cut was made by examining the TOFe fit (figure 5.4), in any given charge state the relevant electron capture appears at the extremes of time of flight. The electron capture recoils are monoenergetic with a total energy of one electron mass greater than the highest energy available in the /3-decay events, the detectors acceptance for these high energy events is limited since the velocity is sufficient for the recoils to escape the M C P if they have a sufficiently large non axial (axial being along the electric field) component. Since the apparatus will collect events travelling along the electric field axis these will come as either fast events (those travelling towards the detector) or slow events (those travelling away from the detector) and this creates two T O F peaks for E C events. As the charge state increases so does the solid angle of detection and the gap between the two peaks begins to fill. Therefore by eliminating the fastest and slowest ions in a particular charge state we can effectively reject electron capture events. The cuts made in time of flight (real time) are given in table 5.10. The data used for a given charge state is between the low and high cut. This data is also affected by the background that occurred in the TOFe time of flight spectrum. In order to correct for the increased sensitivity to events where the f3 travels towards the electron channel plate, the scaled position distribution of these events was subtracted from each position spectrum before the asymmetry was generated. The scaling factor for each charge state is the same as was used in the time of flight fit. 5.4.1 Asymmetry Fitting Algorithm The algorithm to fit the asymmetry data is very similar to the one used to fit the TOFe spectrum. 75 Chapter 5. Analysis a" Polarizations 10000 r 1000 r 100 r 10 r 1 -200-150-100 -50 0 50 100 150 200 Position [mm] -200-150-100 -50 0 50 100 150 200 Position [mm] -200-150-100 -50 0 50 100 150 200 Position [mm] a + Polarizations 10000 F 1000 100 10 1 -200-150-100 -50 0 50 100 150 200 Position [mm] 10000 1000 100 10 1 —1 1 1 1 1 1 r-_ J I I L . -200-150-100 -50 0 50 100 150 200 Position [mm] 10000 1000 100 f-10 1 —1 1 1 1 r _I i i i_ •200-150-100 -50 0 50 100 150 200 Position [mm] Figure 5.5: Position spectra for recoiling nuclei of different charge states, both total events and the background from /?'s with a normalization correction in place. The left and right columns have opposite polarizations. The first row is charge state one, the second is charge state two and the third is charge state three. The fitting program requires the following initial parameters: the electric field, the alignment (T), the polarization (P) as well as x,y and z trap distributions. The fit program can be used to extract values of the recoil order matrix elements or if the recoil order matrix elements are known the fit code can be used to extract values for tp — C t c ¥ ^ t and <j> = C ^ T . If the program is being used to fit for values of the recoil matrix elements (RMEs) then initial guess values must be input as initial parameters. In fitting the RMEs the tensor coefficients are assumed to be zero. The fit code begins by creating the initial recoil momentum distribution for both po-larization states and for both the ground and excited state. This is done using the initial guess values of the RMEs and summing the distributions according to equation 2.57. Events are generated using the method described in the TOFe fitting algorithm for both the ground and final state. Once the T O F is calculated the final positions on the M C P trivially follow from equations 4.8. In this case fifty million distinct events are generated for each charge state at each polarization. Events are rejected if they fall on a portion of the M C P excluded by the GoodMCP cut. An asymmetry is generated by first binning the simulations M C P position data along the polarization axis into 32 bins (like the data). Once the data is binned the asymmetry is created by taking the difference between the number of events at a given bin with positive polarization and negative polarization and 76 Chapter 5. Analysis Parameter Value with Statistical uncertainty hgs CQS -1140 ± 597 hex Cp.T 2836 ± 926 dex Cf"r -245 ± 1065 Jex Cprr. 513 ± 63 Table 5.11: Final values and uncertainties. dividing by the sum number of events. An accepted chi squared value is calculated by comparing the simulated asymmetry to the asymmetry from the data. Sample asymmetries from simulation are included in Appendix D. Once again a Levenberg-Marquardt algorithm is used to update the parameters. The numeric derivatives in the asymmetry are created by recreating the momentum distribu-tion with each R M E incremented from its current value by a small portion epsilon and then running the simulation again recreating an asymmetry. The difference between this asymmetry and the initial guess is the derivative. These derivatives are used to construct a matrix and the parameters are appropriately incremented [44]. The simulation is again run with the newly incremented parameters and a chi squared is evaluated. If the chi square is better than the current accepted value the new value becomes the accepted value and the updated parameters are accepted. If the new chi squared is worse than the accepted value, the matrix used to increment the parameter is modified ([44]) and the parameters are again incremented until a better chi squared is achieved. The fit code continues to improve the chi squared until successive improvements are less than 1% of the accepted chi squared value. Using this fit method the following results were achieved. The dgs R M E was excluded from the fit because it has very little sensitivity to any asymmetry as can be seen by examining the sample asymmetries shown in Appendix D. The most naive "theoretical" approximation would be to set \ = 4.7A(376). This particular fit does not agree with the naive assumption. The fit parameters were very highly correlated in this fit and a single point in parameter space with error bars is not really representative of the fit senstivity. The acceptable range of parameters is better described by a complicated hyper surface in parameter space. . 77 Chapter 5. Analysis 0.05 0.04 0.03 0.02 ? 0.01 y o -0.01 -0.02 -0.03 -0.04 Charge state 1 MCP Position [0.1mm) Figure 5.6: This shows the charge-state 1 fit (dashed line) and data Charge state 2 -50 o MCP Position [0.1mm] Figure 5.7: This shows the charge-state 2 fit (dashed line) and data 78 Chapter 5. Analysis Charge state 3 0.08 | 1 1 1 1 1 0.06 - T -0.04 -? 1 | I • T £ 0.02 -'? T T 1 ts. T T 1 -0.02 - | i 1 1 .0.04 I 1 ' 1 ' ' -100 -50 0 50 100 MCP Position [0.1mm] Figure 5.8: This shows the charge-state 3 fit (dashed line) and data. 5 . 5 A s y m m e t r y s y s t e m a t i c s False asymmetries can easily be created by variations in trap position between polarization states. Fortunately the trap position is known to 0.1 mm and any systematics introduced into the asymmetry from this are on the order of 0.00004. The statistical uncertainties were very large in performing this analysis. This is not sur-prising considering the fact that the excited state and ground state terms generate opposing asymmetries for a given recoil order coefficient. This creates a large parameter space that can generate a given asymmetry. The statistical uncertainties were found to be much larger than the systematics for this analysis. It is hoped in the future that the recoil order coefficients can be calculated theoreti-cally using a shell model code. With this hope in mind I will outline how the systematic uncertainties should be assessed. The systematics from the trap position and electric field are easily found by running the simulation with the largest acceptable electric field and smallest trap distance and vice versa (5.9). Examining the range of variation of the fit parameters under these circumstances can be used as the systematic uncertainty. The systematic from the polarization and alignment term is more complex. In order to handle this systematic the alignment value needs to be varied to its confidence extremes. The value of polarization used with a given alignment must be physically consistent. In 79 Chapter 5. Analysis Parameter Value with Statistical and Systematic uncertainty 2.1 x IO- 1 ± 2.2 x 1CT1 + 2 x l 0 ~ ' 2.2 x IO" 2 ± 4.1 x 1CT2 + 2 ^°I^ Table 5.12: Potential statistical limits on tensor coefficients. order to meet this physical requirement the polarization is then varied between 0.5 and 0.6. If however, the alignment restricts part of this range as unphysical, then the polarization is only varied over the extremes physically allowed. If the alignment restricts the polarization to a value outside the allowed range, then this value is used in place. Examining the range of variation in the fit parameters from these four fits should give an estimate of the systematic uncertainty due to the polarization and alignment. Since the polarization value appears linearly with the asymmetry, it is likely to be the dominant uncertainty when setting limits on tensor coefficients. 5 . 6 P o t e n t i a l S e n s i t i v i t y Although a theoretical calculation of the required matrix elements is still in the works, it is still important to get an idea of what kind of limits can be set with this data set. In order to achieve this goal an analysis was performed using | = 4.7A for both the ground and excited state and setting all other coefficients to zero. This is the contribution to the b coefficient from the anomalous nucleon magnetic moment. The two variables fit were labeled ip and <p and they are defined in equations 5.8 and 5.9. 0=^F (5-9) The results of the fit are shown in table 5.12. The statistical limit of this analysis is on the same order of magnitude as similar experimental results 5.9. The systematic uncertainties in this experiment are dominated by the alignment term. Using the procedure outlined previously to quantify this systematic we arrive at the uneven systematic error bars shown in table 5.12. The largest systematic uncertainty is four times smaller than the current statistical error bars, so the experiment can still be improved by simply increasing the run time. With the help of a theoretical calculation this experiment has potential to be directly 80 Chapter 5. Analysis Limits on Tensor Interactions 1 j 1 X 1 1 1 r~7" 1 1 -I I _ __i i i -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 G y G A Figure 5.9: This light lines show the current limits on tensor interactions from theory and experiment. The heavy lines are the statistical limits from this experiment. The allowed region is bounded by the opposing hyperbola and the parallel straight lines, everything else is excluded. competitive with the current world limits. 81 Chapter 6 Improvements This Chapter will highlight the methods that can be used in the future to improve the results of this experiment. The first goal is to get a theoretical calculation of the recoil order coefficients completed. This will then allow the analysis to be completed which will set limits on tensor couplings. 6.1 Polarization The polarization and alignment uncertainties are the dominant systematic effects. In order to improve the experiment a polarization of much better than 0.68 must be achieved and with a much better uncertainty. In order to achieve better polarization and alignment the magnetic field from the M O T needs to be extinguished during the optical pumping cycle. The downside of this increased polarization is the possibility that the atom cloud moves during the optical pumping. In order to eliminate this movement the TRINAT apparatus needs only minor modification. Currently there are two opposing view ports at 150 and -30 degrees with respect to the horizontal, which are used by cameras viewing the trap position. If the cameras were removed two counter propagating optical pumping beams could be incident through these view-ports. The power in the two beams could be varied independently and this could be tuned in such a way as to eliminate any trap motion. Previous results of optical pumping without a magnetic field have resulted in polarization in the high 90s. 6.2 Gamma detection Although 80Rb provides a clean G T decay it does unfortunately have two dominant decay branches. Ideally we would like an isotope with a single decay branch. The next best thing however, would be to examine the secondary branch and make the required correction in the data. Requiring a coincidence in a gamma ray detector for a 616 keV gamma would isolate the excited state events in question. The germanium detector was used with this goal in mind but unfortunately the detector didn't have a high enough count rate to achieve 82 Chapter 6. Improvements the desired statistics. This deficiency can be overcome by using a LaBr^Ce) crystal. This crystal has a higher count rate and still sufficient energy resolution to resolve both the 511 keV and 616 keV peaks. An analysis similar to the one completed in this thesis could be performed on the gamma ray coincidence recoil events. This would allow an experiment determination of the excited state asymmetry coefficients (b,d,f). TRINAT is currently investigating the possibility of acquiring a LaBr^Ce) crystal for another experiment. 6.3 Charge State Separation In this experiment there is significant overlap of the individual charge states. The overlap can be reduced or even eliminated by increasing the distance of the trap from the M C P (as can be seen by inspecting equation 4.7). TRINAT is currently upgrading the main apparatus with one goal being to increase the trap distance from the M C P to produce better separation of charge states. 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J T W correlation coefficients £ = | M F | 2 ( | C 5 | 2 + | C V | 2 + \C'S\2 + \C'V\2) + | M G T | 2 ( | C T | 2 + \CA\2 + \CT\2 + \C'A\2) (A.l) & = 2Re[\MGT\2{CTC*A + C'TC'X) + \MF\2(CSCV + C'SC$)} (A.2) ^ = \MF\2(\CV\2 + \C'V\2 - \CS\2 - \C'S\2) + ^ f ^ ( | C T | 2 + \CT\2 - \CA\2 - \C'A\2) (A.3) dB± = 2Re[\M2GT\\jjl{±){CTC^ + CAC'X)(±)(CTC'X + C'TCA) - (A.4) SjjMpMGTij^-r)1* (CsC'X + C'SC*A + CvC'f + C'VC$) + (CyC'X + CVCA + CSC'T + C'sCT)\ c> - c i ( J + !) - 3 < M j > (A.5) J(2J - 1) £c = \MGT\2AJ,J(\CT\2 + \C'T\2 - \CA\ ~ \C'A\2) (A.6) ZA = 2Rel±\MGT\2\j'J(CTCTf-CACX)+6jj,MFMGT(j^)H (A.7) 87 Appendix A. Decay correlation functions 2Re[\MGT\2\j,J{r^)(CTC'Z + C'TC%) ± {CTC'{ + CAC'X) 8jj-MFMGr{j^i)kCsC'f + C'SCT - CVC'X - C'VC\) ±—^-(CsC'A* + C'SC*A + CyC'f + C'VCT)] (A.8) 1, 1 •H-l' -J 1, J+l ' •7(2.7-1) (2J+3 ) (J+1 ) ' J —y J' = J — 1 J->J' = J J J ' = J + 1 J J ' = J - 1 J J' = J J ^J> = j + \ (A.9) (A.10) The Q dependent functions describing the recoil asymmetry are included here: i5mS , „ ^Q+v / Q r " xi(Q) x2{Q) = j 2(4Q4 - 28m 2 Q 2 - 81m V Q 2 - m 2 + (90m4Q + ^ M ^ I ^ I ™ ) ^ 8(Q 4 - 4m 4)VQ 2-m 2 + 3 0 m 4 Q / n ( Q + v ^ ^ ! ) - 36m2QVQ2 - m2 Q-VQ* 5(Q 5 + 3Qm 4 + ^ + 1 2 m 4 Q Z n ( £ ) ) - 6 Q 3 m 2 8(Q 4 - 4m4)VQ2-m^ + 30m 4 QZn (^2p|) - 36m 2 Q 2 VQ2 - m2 Q—y/Q*—m2 (A.11) (A.12) The S i functions from Holstein are explicitly written here including contributions from the J T W (non-SM) Hamiltonian. This is combined in a hybrid notation. The fi functions are similarly defined with the additional constraint CT = C'T = 0 2M2GTC\ 2 E ° |H (lOC2AMGT - 2V2MGTCAb) [W2CAMGT (y2CAMGT + b-d))+ (A. 13) me ZMEe (4MGTC% + V2CAMGT (d - 2b)) 88 Appendix A. Decay correlation functions ^ = -2MlTC\ + | | ^ m 1 t C a + ^2M C T ^(d - 6)) - (A.14) 4Be 3M ( 6 M G T C i - \f2MGTCAb) s4 fi3 = 2-WMGTC\ (A.15) = 7 J J ' f ( V ^ M O T C ^ - ^(V2MGTCA + d-b))+ (A.16) AJJ, -V2MGTCAfE0 _ IMITCTC'^JJ, v n u ' T i l j + r 2M ' J + I M (j^~MGTCA(5V2MGTCA -d-Zb)- j^LV2MGTCA^fj S5 S6 = z g . {^^f^^uGTcA -d-b)- j^^GTcAfyl7) = j^V2MGTCA(V2MGTCA-^(y/2MGTCA + d-b))- (A.18) j^LV2MGTCA^ - j^2MGTCTC'T + ^Cj\\^MlTC\2V2MGTCA + d-3b) + j^yfiMercJl) + S7 = y ^ ( > / 2 M O T C A ( | ^ _ | ( ^ ( ^ £ A { 1 J - 2 M g t C a + d - b)) - jf-i{V2MGTCA)3-l) 89 Appendix A. Decay correlation functions 810 = ^(^KjJ'MGrCASf + ^VlMGTCAiVlMGTCA+d + b)) (A.20) S l 2 -eJrV2MGTCA(V2MGTCA - ^(V2MGTCA + d-b)) KJJ,V2MGTCA^^ + (A.21) ^(Kjj>V2MGTCASf - ejj,V2MGTCA(3V2MGTCA b)) Sl3 M djj,mGTc2A (A.22) s i 4 = Kjj<V2MGTCA^f-ejj,^V2MGTCA(V2MGTCA + d-b)+ (A.23) ^{9JJ,V2MGTCA(V2MGTCA + d-b)- KJJ^MGTCAY) U'j J + 1, J -> J = J' + 1 1, J —> J = J' -J, J -+ J = J' - 1 V3(J 2 - 1), J —> J = J ' + 1 V ( 2 J - l ) ( 2 J + 3), J ^ J = J ' V3J(J + 2), J - f J = J ' - 1 1, J^J = J' -J j _> / = 7' _ i 2 J + 3 ' J —> J — J i (A.24) (A.25) (A.26) 90 Appendix A. Decay correlation functions 2J-1 ' 3 ^ / ( 2 J - l ) ( 2 J + 3 ) ' > /J (J+2) j 2J+3 J = J' + l J = J' J = J' - 1 (A.27) 91 Appendix B Fermi function coefficients B.0.1 Fermi function We have to this point derived all the corrections to the weak interaction matrix element that arise due to the strong interaction. There is, however another important component to include when one is calculating a decay spectrum, which is called the Fermi function. The Fermi function is included to account for a variety of effects including: the Coulomb field of the nucleus on the positron, the finite size of the nucleus and the screening of nuclear charge due to atomic electrons. The Fermi function used for this analysis was derived by D.H. Wilkinson and a detailed discussion can be found in his papers [46],[47],[48]. The result of the papers is a product of Taylor expanded functions along with their accuracy for each order. In order to be consistent with the precision of the other theoretical considerations an accuracy of 1 part in 104 was used. Rather than repeat that discussion I will simply quote the Fermi function used and briefly describe what each portion attempts to describe. In what follows there are a variety of algebraic expressions which will be defined here for clarity: E is the electron kinetic energy in MeV, me is the electron mass in MeV,W = 1 + m\,p= VW2 — 1, Eo is the max electron kinetic energy, WQ = l + ^ P o = \JWQ - l ,a is the fine structure constant, 7 = A / 1 — (aZ)2, R is the nuclear charge radius in fmx2.5896xl0~3 and Z is the charge of the daughter nucleus (negative for positron decay). The first portion of the Fermi function describes the effects of the nuclear charge on the final state positron. We will label this portion C. C = £ ZaKbLc(aZ)a(W/P)b(ln(p))c (B.l) a,b,c The ZiKjLk are numeric coefficients which are tabulated in Appendix B. The next portion of the Fermi function accounts for the finite size of the nucleus and the effect this has on the final state positron wave function. The nucleus is approximated as a uniformly charged sphere. Then the Dirac equation can be solved to give the value of 92 Appendix B. Fermi function coefficients the wave function. This correction is labeled Lo. 0 . 4 1 ( 7 ? - 0 . 0 1 6 4 ) ( a Z ) 4 5 (B.2) 6 x = l The bnx terms are numeric coefficients that are tabulated in Appendix B. Finally we must also consider the convolution of the positron wave function and the wave function of the nucleus. This gives us the following term. AC(Z,W) = 1+A Co+A CiW+A c2w2 A _ - 2 3 3 ( a Z ) 2 (WpR)2 2W0RaZ 0 630 5 + 35 A -21RaZ 4W0R2 AC, = (B.3) The total Fermi function is just the product of all the terms that have been derived in this section. The net effect of the Fermi function for positrons is to shift the positron spectrum to higher energies and to reduce the overall rate of the decay. This experiment is not directly sensitive to the overall rate, so the only portion of interest is how the (3 energy distribution is shifted. The shift in energy is illustrated in'figure B . l . This Appendix contains all the relevant numeric coefficients for calculating the Fermi function derived by D.H. Wilkinson [46],[47],[48]. 93 Appendix B. Fermi function coefficients Figure B . l : This plot shows the (3+ energy spectrum with (green) and without (red)the inclusion of a Fermi function. The endpoint energy is the same as in this experiment (5.209 M e V ) . The two spectra have been normalized to the same value. Coefficient Value -0.0701 &-12 -2.572 &-13 -27.5971 6_i4 -128.658 &-15 -272.264 &-16 -214925 Table B . l : Coefficients making a _ i term. Coefficient Value 0.002308 0.066483 &03 0.6407 &04 2.83606 &05 5.6317 &06 4.0011 Table B.2: Coefficients making ao term. 94 Appendix B. Fermi function coefficients Coefficient Value 6 1 1 -0.07936 bu -2.09284 bn -18.45462 bu -80.9375 bis -160.8384 bm -124.8927 i.3: Coefficients making a Coefficient Value 6 2 1 0.93832 6 2 2 22.02513 6 2 3 197.00221 ' 6 2 4 807.1878 6 2 5 1566.6077 6 2 6 1156.3287 Table B.4: Coefficients making 0 2 term. Coefficient Value &31 -4.276181 6 3 2 -96.82411 & 3 3 -835.26505 & 3 4 -3355.8441 & 3 5 -6411.3255 &36 -4681.573 Table B.5: Coefficients making 0 3 term. Coefficient Value 6 4 1 8.2135 6 4 2 179.0862 &43 1492.1295 6 4 4 5872.5362 6 4 5 11038.7299 &46 7963.4701 Table B.6: Coefficients making 0,4 term. 95 Appendix B. Fermi function coefficients Coefficient Value 651 -5.4583 &52 -115.8922 &53 -940.8305 654 -3633.9181 655 -6727.6296 ^56 -4795.0481 Table B.7: Coefficients making 05 term. Coefficient Value ZQKQLQ 1 3.141593 0.577216 Z^KQLI -1 Z2K2L0 3.289868 Z3K1L0 1.813376 ZzKxLx -3.141593 Z^KQLQ 0.722126 Z4K0LI -0.827216 Z4K0L2 0.5 Z4K2L0 0.696907 Z4K3LX -3.289268 Z4K4L0 -2.164646 Z5K1L0 2.268627 ZsKiLi -2.598775 ZZKXL2 1.570796 Z5K3L0 -3.776373 Z^KQLQ 0.730658 ZQKQLI -0.991430 ZQKOL2 0.538608 ZQKQLZ -0.166667 ZQK2LQ 0.569598 Z§K2L\ -1.519374 ZQK2L2 1.644934 ZQK^LQ -4.167149 ZQK^LI 2.164646 ZQKQLQ 2.034686 Table B.8: Coefficients appearing in Taylor expansion of Fermi function. 96 Appendix C Recoil Order Corrections to the Momentum Distribution This Appendix shows all the recoil order matrix element recoil momentum distributions as well as the <j> and ip distributions which are relevant for the decay branch to the ground state of 80Kr. All z axes are in relative number, which is the same scale used in figure 2.5 97 Appendix C. Recoil Order Corrections to the Momentum Distribution bgsN(e,|Pr|) Figure C . l : This figure shows the momentum and angular dependance of the term for the groundstate decay branch. 98 Appendix C. Recoil Order Corrections to the Momentum Distribution bgsc(e,|Pr|) Figure C .2: This figure shows the momentum and angular dependance of the be term for the groundstate decay branch. 99 Appendix C. Recoil Order Corrections to the Momentum Distribution bgsL(e,|P|) Figure C.3: This figure shows the momentum and angular dependance of the bi term for the groundstate decay branch. 100 Appendix C. Recoil Order Corrections to the Momentum Distribution dgsN(e,|P|) Figure C.4: This figure shows the momentum and angular dependance of the term for the groundstate decay dranch. 101 Appendix C. Recoil Order Corrections to the Momentum Distribution Figure C.5: This figure shows the momentum and angular dependance of the dc term for the groundstate decay dranch. 102 Appendix C. Recoil Order Corrections to the Momentum Distribution dgsL(e,|Prl) Figure C.6: This figure shows the momentum and angular dependance of the di term for the groundstate decay dranch. 103 Appendix C. Recoil Order Corrections to the Momentum Distribution vgsc(e,|P|) Figure C.7: This figure shows the momentum and angular dependance of the ipc term for the groundstate decay dranch. 104 Appendix C. Recoil Order Corrections to the Momentum Distribution <i>gsc(e,|P|) Figure C.8: This figure shows the momentum and angular dependance of the <j>c term for the groundstate decay dranch. 105 Appendix D Simulated Asymmetry Distributions The following figures show simulated asymmetry distributions for various values of both recoil order matrix elements and tensor coefficients. A l l of the simulations used charge-state 3. This gives a sense of the size of these terms required to generate the asymmetries observed in the data. b Asymmetry -100 -50 0 50 100 X Position [0.1mm] F i gure D . l : Asymmetry for three different values of bgs. 106 Appendix D. Simulated Asymmetry Distributions Figure D.2: Asymmetry for three different values of b, 107 Appendix D. Simulated Asymmetry Distributions dgs Asymmetry 5 0.002 * c 0.000 -0.002 -0.004 -0.006 -100 -50 50 100 X Position [0.1mm] Figure D.3: Asymmetry for three different values of dg3. 108 Appendix D. Simulated Asymmetry Distributions d Asymmetry —ex 0.008 \-0.004 C 5 0.002 c r o -0.002 -0.006 d /c=-600 d jc=0 d /c=600 -100 -50 50 100 X Position [0.1mm] Figure D.4: Asymmetry for three different values of d€ 109 Appendix D. Simulated Asymmetry Distributions f Asymmetry -0.10 I 1 1 1 1 1 1 1 1 1—I •100 -50 0 50 100 X Position [0.1mm] Figure D.5: Asymmetry for three different values of / , 110 Appendix D. Simulated Asymmetry Distributions X Position [0.1mm] Figure D.6: Asymmetry for three different values of tp. Appendix D. Simulated Asymmetry Distributions cb Asymmetry 0.016 -100 -50 0 50 X Position [0.1mm] Figure D.7: Asymmetry for three different values of <f>. 112 Appendix E Electric field Generation and Systematics E . l Electric Field The purpose of the electric field in the second M O T is threefold: to collect all photoions on an MCP, to collect ionized decay products on the M C P and to collect shakeoff electrons on an opposing MCP. Ideally the electric field would be uniform over the entire trap. This becomes extremely useful for simulation purposes. It allows an analytic expression for the time of flight. Unfortunately the electric field is not constant in this experiment. The goal of this section is to show the size of the systematic error made by assuming the electric field is uniform. E . l . l Field Source The field is generated by applying a bias voltage to a series of precisely oriented vitreous carbon hoops. The hoops are mounted on the same assembly as the position sensitive M C P [see figure E.l] so that the orientation of the field with respect to the M C P is controlled with great precision. There are a total of six hoops numbered from zero to five sequentially beginning with the hoop nearest the channel plate. Each hoop is electrically isolated from the others, so a unique potential can be applied to each hoop. Each hoop has an outer diameter of 13.5 cm and an inner diameter of 10 cm. The hoop thickness is 1 mm. In addition the zeroth hoop has an inner hoop with an outer diameter of 5.8 cm and inner diameter of 3.6 cm and the fifth hoop has an inner diameter of 2.5cm and outer diameter of 5.2 cm. The two additional hoops also have a 1 mm thickness. The additional inner hoops help with field uniformity. The hoop bias is applied using high voltage Bertan supplies. The bias on each hoop during the experiment was set to the specifications in table E . l . 113 Appendix E. Electric field Generation and Systematics Hoop Number Hoop Bias [keV] 0 inner -3.862 0 outer -3.759 1 -1.990 2 -0.002 3 1.860 4 4.152 5 inner 5.799 5 outer 4.416 Table E . l : Potentials of Hoops creating the electric field in the T R I N A T apparatus. Figure E . l : This shows a cross section of the mount assembly holding the field hoops in place. The hoop mount is connected to the M C P mount to ensure precise alignment. 114 Appendix E. Electric field Generation and Systematics E0 -81991.63 £ Ei -70340 Xr E2 - 1 . 4 5 x l 0 6 - £ £ 3 1 -7xl0 8 & Table E.2: Coefficients of the polynomial used for modeling the electric field. Electric Field -0.02 0 Distance [m] Figure E.2: Electric field from Simlon calculation. Ions are collected on a channel plate at -6cm. The trap is located at 0. E.1.2 Field corrections Given the potentials and the materials making up the chamber the electric field is uniquely determined and can be solved numerically. In order to estimate the systematics of a non uniform field, the field was simulated using Simlon. The resulting field is shown in figure E.2 and is accurately described by a third order polynomial whose coefficients are given in table E.2. E{z) =E0 + Exz + E2z2 + Ezzz (E.l) The dynamics of ions in our electric field are governed by equation E.2. d2z m-^ = q(E0 + Exz + E2z2 + E3z3) (E.2) This equation can be solved by a power series method for cases of interest in this experiment. 115 Appendix E. Electric field Generation and Systematics ' si' -• \ 1 1 1 1 1 1 1 1 1 T O F 1 ' ! " • •20 -10 0 10 20 velocity 11C1 m/s) Figure E.3: The lower plot shows the time of flight in ps for various init ial velocities. The upper plot is the absolute difference from the 20th order expansion and the second order approximation used time is in ns. The values of the field constants are taken from the Simlon simulation in order to evaluate the size of the induced systematic errors. The solution used in the simulation corresponds to truncating the power series at second order terms. In order to compare with the real solution the series was taken to 20th order and time of flights were compared in figure E.3 . It is worth noting that if E\ = E2 = E3 = 0 then the solution is exact at second order. The field given in E.2 doesn't satisfy Gauss' law, however, introducing the necessary radial terms to satisfy Gauss' law turns out to be a negligible contribution. This shows agreement in time of nights at the part per thousand level, which wi l l be sufficient for determining the electric field and trap position. In order to evaluate how this systematic effect wi l l affect the final asymmetry we need to first examine how the time of flight systematic error affects the final position on the M C P . This is displayed in figure E.4. The goal is not to calculate the exact systematic error but only to give an upper bound on its value. W i t h this in mind we see from figure E.4 that it is the ions that decay with the largest total velocities that give the largest systematic errors. The determination of the position is made for both the 20th order approximation and the second order approximation and the deduced asymmetries are compared. This was repeated for multiple velocities in order to verify the assumption that the maximum velocity wi l l have the largest systematic error in the asymmetry. The assumption that the largest total velocities would have the largest systematic was verified and the systematic effect was found to be 0.6% of the value of the asymmetry. This is a negligible systematic. We can also examine the contribution of neglected radial electric field terms. In order 116 Appendix E. Electric field Generation and Systematics Systematic Position Errors 0.00014 I 1 1 1 1 1 1 1 1 r -25 -20 -15 -10 -5 0 5 10 15 20 25 Z velocity [1000 m/s] Figure E.4: Systematic differences in time of flight translate into systematics in position on the MCP. This shows the absolute difference between the 20th order expansion and the second order approximation for three different total velocities with various projections along the z axis. to satisfy Gauss's law there must be a radial gradient term of the same order as E\. Using the longest time of nights and,the charge to mass ratio of Rubidium we can estimate that the maximum positional systematic is 0.24mm for ions hitting the outer edge of the channel plate. The resolution of the channel plate is limited to 0.25mm [37]. Therefore in the worst case scenario the systematic effect is still smaller than the instruments inherent resolution and can be safely neglected. 117 Appendix F Germanium F.0.3 Germanium The germanium detector used in this experiment is an Ortec pop-top model GEM-80215-P-ST. The detector itself is made of a high purity germanium crystal (less than 10 1 0 impurity atoms per cubic centimeter) in a coaxial geometry. The detector is fabricated from a NP junction. Electrons from the n side and holes from the p side can migrate across the junction setting up a region that is neutral called a depletion region. The range of this migration is enhanced by applying a bias field. Ionizing radiation passing through the depletion region of the crystal creates particle hole pairs, the number of which is proportional to the energy deposited in the crystal. The number of pairs created follows a modified poissonian distribution the modification is called the Fano factor. The Fano factor is just a multiplicative factor that appears in the width of the distribution they are unique to each detector material, for germanium the Fano factor is 0.129 [49]. On average it takes about 3 eV [50] to produce a pair so large numbers are produced. It is these large numbers of events that give germanium its excellent energy resolution. In a sodium iodide crystal fewer signal events are generated due to higher energy required to excite the crystal and so statistical fluctuations on these numbers are more significant, leading to a poorer resolution. Germanium has a small band-gap (0.7eV) and so electrons can be thermally excited to the conduction band at room temperature which leads to excessive signal noise. This is overcome by cooling the crystal to liquid nitrogen temperatures. The crystal is housed in a vacuum sealed cryostat to minimize its thermal contact with surroundings. The cryostat is connected to a large dewar that is filled with liquid nitrogen during operation. In this experiment the Germanium was used to distinguish decays that went to an excited state of Krypton from those that decayed to the ground state. In order for this to be effective we need to be able to correctly identify the peaks in the Germanium spectrum (see figure F.l) . The spectrum is affected at high count rates by pile, up and this was corrected by increasing the shielding surrounding the detector. 118 Appendix F. Germanium 10° 80, Kr Gamma Spectrum 511 keV annihilation peak 616.7 keV peak Gamma Spectrum • sum peak 1257.1 keV 1320.5 keV 1000 1500 2000 Channels Figure F . l : This is the gamma ray spectrum from 8 0Rb decay. The decay proceeds to the ground state or to one of three excited states which then decay via E2 or M l transitions. The strong 511 keV peak is due to the positrons from the /3-decay annihilating. There is also a sum peak from the 511s and the 616s. 119 Appendix G Glossary of common abbreviations AOM-acousto optic modulator CaF-calcium fluoride CVC-conserved vector current G T -Gamow-Teller JTW-Jackson, Treiman and Wyld LCVR-liquid crystal variable retarder MCP-microchannel plate MOT-magneto-optical trap OP-optical pumping P-polarization or Parity PCAC-partially conserved axial current SM-standard model T-alignment most commonly Holsteins version (1 — or tensor TOF-time of flight 120 


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