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Reconstructing ν̅ₑ energy spectrum and ground state branching fraction of laser trapped ⁹²Rb McNeil, James Cameron 2019

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Reconstructing ν¯e Energy Spectrum and Ground State BranchingFraction of Laser Trapped 92RbbyJames Cameron McNeilB.Sc. Simon Fraser University, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2019c©James Cameron McNeil, 2019iiThe following individuals certify that they have read, and recommend to theFaculty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Reconstructing ν¯e Energy Spectrum and Ground State Branching Fraction of LaserTrapped 92Rbsubmitted by James Cameron McNeil in partial fulfillment of the requirementsfor the degree of MASTER OF SCIENCE in Physics.Examining Committee:Dr. John BehrSupervisorDr. HasinoffAdditional ExamineriiiAbstractThe anti-electron neutrino ν¯e energy spectra from the fission product 92Rb isof interest for reactor neutrino physics and short baseline neutrino oscillationexperiments. A study of the beta-neutrino correlation parameter a1 was per-formed in laser trapped neutral 92Rb, constraining the strong ground-state toground-state (GS) first-forbidden 0− → 0+ rank-0 nuclear matrix element ratioξo/ω. The first-forbidden rank-0 correlation parameter depends non-linearlyon beta energy W with a1(ξo,ω,W) for a decay end-point of Wo. Two distinctanalysis were performed via: 1) recoils coincident with atomic shake-off elec-trons (SOE); and 2) recoils coincident with the beta and atomic SOE. In the for-mer case-1, no distinction can be made between transitions to the GS or excitedstates and we find the two possible solutions (ξo/ω)RSOE+RSOE− =+0.437−0.512 ±(0.0050.008)stat±(0.010.01)sys with a1(W → Wo) →0.300.36, respectively. In the later case-2, however, GSevents are isolated through kinematic constraints on the reconstructed transi-tion Qexp-value, with two possible solutions being (ξo/ω)BRSOE+BRSOE− =+0.541−0.739 ±(0.0080.018)stat ± (0.010.01)sys with a1(W → Wo) →0.500.58, respectively. These are all signifi-cantly different from the naively expected |ξo/ω|  1, and a1(W → Wo) = 1.From the beta-recoil 3-momenta, the ν¯e energy spectra was also reconstructedfor decays through the GS, and eventually compared with theory. We alsopresent our future proposal to measure the GS branching ratio from the recon-structed Qexp-value distribution.ivLay SummaryEvent excesses observed in neutrino energy spectra in the 5-7 MeV range fromnuclear fission reactors may suggest there is an incomplete understanding ofthe main β-decaying progeny in this energy range, the dominant of which be-ing 92Rb. In this experiment we directly reconstruct Eν of 92Rb using energyand momentum conservation from measured momenta of the beta and recoil-ing daughter. β-decays to the daughter’s strong ground state transition arekinematically isolated in total transition energy Q-value to separate β-decaysto highly excited states. This ground state isolation enables us to both mea-sure the affiliated Eν spectra, and perform angular correlation measurementsbetween the beta and neutrino momenta to constrain parameters driving theground state transition. We also propose measuring the ground state transitionbranching fraction from the reconstructed experimental Q-value, which are rel-evant for constraining theoretical predictions in reactor neutrino experiments.vPrefaceI, James Cameron McNeil, declare that this thesis titled, “Reconstructing ν¯e En-ergy Spectrum and Ground State Branching Fraction of Laser Trapped 92Rb”and the work presented in it are my own, with the contributions of others de-tailed in the acknowledgments section. My contribution included developingthe analysis code specific to this experiment, development/characterization ofthe in-vacuum mirrors and position sensitive electron detector, all detector cal-ibrations, conditioning our electrostatic hoops to operate at high-voltage, andled the experiment prior to and during data acquisition.viContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Nuclear Beta decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 92Rb Level Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Recoil Kinematics & Angular Correlation aβν . . . . . . . . . . . . 72.4 Forbidden Transitions . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.1 Correlation Parameter . . . . . . . . . . . . . . . . . . . . . 92.4.2 More information about 0− to 0+ transitions, particularly92Rb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Spectrum shape . . . . . . . . . . . . . . . . . . . . . . . . . 12The “ξ approximation” doesn’t work . . . . . . . . . . . . 123 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1 TRIUMF-ISAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 TRINAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Recoil Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.1 Recoil Event Selection . . . . . . . . . . . . . . . . . . . . . 163.3.2 Absolute Event Timing . . . . . . . . . . . . . . . . . . . . 183.3.3 rMCP Spatial Calibration . . . . . . . . . . . . . . . . . . . 203.4 Non-Uniform ~E(~r) Field . . . . . . . . . . . . . . . . . . . . . . . 223.5 Recoil Kinetic Energy and Momenta . . . . . . . . . . . . . . . . . 253.6 Scintillator Energy Calibration . . . . . . . . . . . . . . . . . . . . 263.7 ν¯e Energy Spectrum and Qexp-value Reconstruction . . . . . . . . 293.8 Photo-Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.9 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 35vii4.1 Recoil-SOE Coincidence . . . . . . . . . . . . . . . . . . . . . . . . 354.1.1 TOF Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1.2 ~E-Field Corrections . . . . . . . . . . . . . . . . . . . . . . . 374.1.3 rMCP Pulse-Height & Detector Efficiency . . . . . . . . . . 374.1.4 rMCP Detector Response Function . . . . . . . . . . . . . . 434.1.5 Background Estimation . . . . . . . . . . . . . . . . . . . . 454.1.6 Recoil-SOE Kinematic Spectra and Averaged ξo/ω . . . . 484.2 Recoil-SOE-SCINT Coincidence . . . . . . . . . . . . . . . . . . . . 514.2.1 Kinematic Observables and Constraints . . . . . . . . . . . 514.2.2 Reconstructed Eν and Qexp-value Distributions . . . . . . . 544.2.3 GS Transition ξo/ω . . . . . . . . . . . . . . . . . . . . . . . 584.3 Recoil-SOE-SCINT Coincidence GS Transition ξo/ω . . . . . . . . 624.4 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . 655.1 Recoil-SOE Coincidence . . . . . . . . . . . . . . . . . . . . . . . . 655.1.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.1.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66A Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73viiiList of Tables3.1 Timing Offsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Scintillator calibration parameters from fits to calibrants in Fig-ure(3.10) using the saturation equation(3.8). . . . . . . . . . . . . . 274.1 Nuclear Matrix Element ratio ξo/ω at 90% C.L. from fits of therecoil Kr spectrum of the specified charge state (CS) using equa-tion(4.5) assuming the dominant transition is the First-forbiddenGS branch in the Recoil-SOE coincidence. See Appendix A for fitresults from ξo/ω > 0. ( ∗Non-linear ~E-field corrected recoil hitcoordinate.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Nuclear Matrix Element ratio ξo/ω at 90% C.L. from fits of the re-coil Kr spectrum of the specified charge state using equation(4.5)from decays through the First-forbidden GS branch in the Recoil-SOE-SCINT coincidence. See Appendix A for fit results fromξo/ω > 0. ( ∗Non-linear ~E-field corrected recoil hit coordinate.) . 62ixList of Figures2.1 92Rb level scheme (Compiled 2012 Nuclear Data Sheet) . . . . . . 52.2 Toy model of a Giant-Resonance where a continuum of highly ex-cited and fragmented states become accessible via β-decay withbeta a) transition strength function Sβ(Ex), b) phase-space func-tion f (Q− Ex, Zr), and c) transition intensity Iβ(E f ). . . . . . . . . 72.3 (left) 92Rb allowed β spectrum with fractional corrections (right)to the Fermi-function from higher order corrections Ro,Lo,CA. . . 93.1 Isotope Separator and Accelerator facility as of 2003 at TRIUMF(see ISAC). The main cyclotron is off to the page towards bottom-right of the image. The TRINAT experiment is correctly indicatedabove the ISAC pre-separator room (LEBT). . . . . . . . . . . . . . 133.2 TRIUMF Neutral Atom Trap (TRINAT) with electrostatic hoopsfor recoil charge state separation with opposing recoil-ion (rMCP)and shake-off electron (SOE) detectors, and orthogonally orien-tated, symmetric E− ∆E β-telescopes along the vertical ±z-axis.The vertical trapping beam path is indicated in red reflected from70 nm thin gold mirrors backed by 4 µm Kapton. (Horizontaltrapping beams are not shown). 92Rb is trapped in a Magneto-Optic-Trap (MOT) at the center of the apparatus where it β-decayswith outgoing tracks of the β− (blue), recoiling 92Sr daughter(grey), and atomic SOE (red) shown. . . . . . . . . . . . . . . . . . 143.3 a) rMCP-DLA timing sum gate for recoil-event selection (rMCP)utilizing the x (red), and z (blue) DLA detector planes with Lead-ing Edge (LE) constant-fraction trigger. b) SOE-Scintillator eMCP-PMT TOF spectrum of the Recoil-SOE-SCINT coincident eventsusing the upper (red), and lower (blue) scintillators with indi-cated timing bounds chosen to suppress false coincidence events. 163.4 Recoil-SOE coincident TOF spectrum. . . . . . . . . . . . . . . . . 19x3.5 Recoil-SOE-SCINT coincident TOF spectrum with the a,c) upper,and b,d) lower scintillators, respectively. The expanded region ofa,b) in c,d) show the centering in TOF about (eMCP - PMT, rMCP- PMT) = (0,0) [ns]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6 a) Rough recoil hit position calibration of the rMCP from delay-line-anode (DLA) timing signals. b) Hit position following corre-lated XZ-mapping via Fourpt mapping algorithm. Projections ofb) are shown in c) x, and d) z with overlaid fits to local minimato determine piecewise-linear spatial scaling in x/z required forfinal calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.7 Corrected recoil hit position after uncorrelated piecewise-linearscaling in x, and z from fits performed in Figure(3.6c,d) respectively. 223.8 a) COMSOL simulated ~E-field component values as a functionof displacement from the trap center in the y-axis (rMCP at y =+97 mm) for given translational offsets in x, and z.〈Ey〉is thesimulated mean field strength of 998.5 V/cm along the y-axis. b)Expanded view of minimally deviating field component profiles. 243.9 Mapping recoil hit coordinates in non-uniform ~E-field X′ to coor-dinates X in uniform field via COMSOL-based mapping of sim-ulated recoil events. a) Simulated recoil hit coordinate space unitcell in the non-uniform field (S′), and uniform field (S). b) Illus-trated Fourpt stretching transformation mapping recoil hit coor-dinate (x′, z′)→ (x, z). . . . . . . . . . . . . . . . . . . . . . . . . . 253.10 Top and bottom scintillator calibration using a combination ofgamma-ray Compton edges and β endpoints. A significant non-linear response in the lower scintillator can be traced to a defec-tive PMT base. Additionally, the 3.5 cm thick scintillators arethin enough that a small fraction of the most energetic beta’s canleave the volume before depositing their full energy, or generatebremsstrahlung photons which escape detection. The result isless charge deposition at the largest beta energies and this maybe another source of non-linearity in the detector response. . . . . 263.11 a) UPMT-LPMT coincident events in the absence of recoil’s andSOE’s with cosmic muon in the dashed region having correspond-ing energy spectrum projections in b) and relative timing differ-ence in c), respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 28xi3.12 Photo-Ion event distribution fit with superposition of three Gaus-sian ellipsoids per 5 min of acquisition to check for small timedependent trap drift in the xz-plane. . . . . . . . . . . . . . . . . . 303.13 Photo-Ion distribution a),b),c) amplitude, d),e),f) x-centroid andwidth (coloured bounds), and g),h),i) z-centroid and width (colouredbounds) of inner I(red), and outer O1 (blue), O2 (green) elliptical-Gaussian fit parameters, respectively, vs. cumulative run time ofthe 1 kV/cm dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . 313.14 Photo-Ion a) SOE-PD TOF, b) Recoil-SOE TOF, c) y-displacementfrom trap centroid examples per 5 min of acquisition with over-laid Gaussian fits. Cumulative run time dependent Gaussian cen-troid (black-markers) and width (red-bounds) per 5 min of acqui-sition are displayed in d), e), and f), for the respective coincidence. 324.1 Recoil-SOE coincident a) drift corrected ion impact radius vs.TOF, and b) uncorrected recoil hit xz-position for +2(red), 3(blue),and 4(green) charge state events in respectively coloured TOF-bounds in (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Recoil-SOE coincident random background levels vs. TOF inte-grated over the bounding (teal) lines in Figure(4.1). Fits withinthe charge state bounds +2(red), 3(blue), 4(green) provide a ran-dom background rate normalization relative to signal free region(grey) from which time random backgrounds are estimated. Thelocation of the TOF peaks are understood to be real backgroundsand discussed later. . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 Drift corrected recoil xz-coordinate displacement distribution (X)following mapping above in Figure(3.9) X′ → X (mean field of〈Ey〉= 998.5 V/cm) of a) +2(red), b) +3(blue), and c) +4(green)charge states within TOF bounds in Figure(4.1a). The correspond-ing un-corrected recoil event distribution (X′) are shown as blackpoints in the background for each of the respective charge states.The corresponding distribution of recoil hit xz-displacement cor-rections in mapping from X′ → X are shown in d),e),f), respec-tively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38xii4.4 Charge state +2, 3, 4 (left-right) a), b), c) mean rMCP pulse heightdependence on transverse recoil momentum with, d), e), f) pulseheight vs. prx, and g), h), i) pulse height vs. prz distributions,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.5 rMCP pulse height distribution vs. recoil kinetic energy Kr ofcharge state +2, 3, 4 (left-right) a), b), c) charge states with over-laid mean pulse height. Normalized mean pulse height to thelinear fit of that of the +4 charge state vs Kr are are shown in d),e), f) for the respective charge state. . . . . . . . . . . . . . . . . . . 404.6 rMCP pulse height dependencies for charge states +2, 3, 4 (left-right) on a),b),c) angle to rMCP channel axis, d),e),f) azimuthalangle in the plane of the rMCP, g),h),i) maximum rMCP channelpenetration depth for (prx < 0 black-line), and (prx > 0 red-line). 424.7 Recoil kinetic energy Kr spectrum integrated over indicated de-tector hemisphere for charge states +2, 3, 4 in a),b),c), respec-tively, with (teal) corresponding the left/right and (magenta) up/downhemispheres. The corresponding Left/Right (LR) and Up/Down(UD) counting rate asymmetries parameterize the relative rMCPdetection efficiency vs. recoil Kr dependence is shown in d),e),f)with overlaid fit functions parameterizing the relative efficiencyeLR(Kr), eUD(Kr). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.8 Simple MC of recoil ion in recoil-SOE coincidence assuming re-coils with uniform randomly distributed momenta which tra-verse a uniform ~E-field of 998.5yˆ V/cm within the given chargestate TOF bounds in Figure(4.1a). The recoil +2 charge state pass-ing these criteria and within the active area of the rMCP havetransverse momentum distribution shown in a), and impact hitposition distribution shown in b). Azimuthally integrated radialmask transmission efficiency (markers) and finite-density rela-tive event rate effects (lines) of the +2(red) charge state are shownin c). Integrated mask transmission efficiency em(Kr) (markers)and finite-density event rate efficiency e f d(Kr) (lines) vs. recoilKr of the +2(red), 3(blue), 4(green) charge states, respectively areshown in d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.9 Recoil ion azimuthal angular distribution of +2(red), 3(blue), 4(green)charge states. Each charge state was fit with the superpositionA+ B cos φr + C cos2 φr. . . . . . . . . . . . . . . . . . . . . . . . . 46xiii4.10 a),b),c) Transverse recoil momentum distribution of +2, 3, 4 chargestates (left-right), respectively. d),e),f) Recoil kinetic energy Krspectra for respective charge states extracted from ±x quadrants(teal), ±z quadrants (magenta) with difference (black) an esti-mate of background levels dominantly from decays from hoopsurfaces in the ±z orientation relative to the ±x orientation. Theoverlaid black diagonal lines are the chosen boundaries for suchquadrants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.11 Recoil kinetic energy Kr spectrum for +2(red), 3(blue), 4(green)charge states a) before non-uniform ~E-field ion impact coordinateX′ correction (assuming uniform field of Ey = 998.5 V/cm), andb) following correction in mapping X′ → X where recoils wouldtraverse such a uniform ~E-field. Included are expected randombackgrounds and projected hoop backgrounds necessary for ξo/ω <0 model fits. (See Appendix A for ξo/ω > 0 model fits.) . . . . . . 494.12 χ2/do f distribution of equation(4.5) fit to recoil Kr spectra in Fig-ure(4.11) after floating the nuclear matrix element ratio ξo/ω < 0and NORM parameters for +2, 3, 4 charge states a),b),c) withoutnon-uniform ~E-field corrections, and d),e),f) with non-uniform~E-field corrections in mapping X′ → X. Boundaries of these dis-tributions represent 90 % statistical C.L. for the respective chargestate. From the large first-forbidden GS branching we assumethe beta-energy dependence in the correlation a1 parameter ofequation(2.22). The range in ξo/ω was chosen for comparisonof the Recoil-SOE, and the GS isolated Recoil-SOE-SCINT coinci-dent data to see the large change in ξo/ω between these channels. 504.13 Recoil-SOE-SCINT triple coincident event a),c) drift corrected im-pact radius vs. TOF (Recoil-PMT) spectrum of +2, 3, 4, and highercharge states for coincidence with upper, and lower SCINT re-spectively. Overlaid are the respective TOF bounds for the +2(red),3(blue), 4(green) events. (+2) Recoil momentum vs. scintillatorenergy are shown in b),e) with overlaid kinematic boundariesfor Q = 8.1 MeV. Scintillator energy spectra are shown in c),f)with (dashed-line) and without (line) kinematic boundaries ap-plied for indicated charge states. . . . . . . . . . . . . . . . . . . . 52xiv4.14 Recoil-SOE-SCINT coincident Qexp-value vs. reconstructed anti-electron neutrino energy Eν spectrum utilizing the a),b),c) Upperand d),e),f) Lower beta scintillator events within the respective+2, 3, 4 recoil charge state TOF bounds shown in Figure(4.13a,d). 544.15 Recoil-SOE-SCINT triple coincident events with transition Qexp-value distributions of the +2(red), 3(blue), 4(green) charge stateswith beta’s coincident in the a) Upper, and c) Lower scintillators,respectively (See Appendix A for linear scaled plots). Overlaidare the time random coincident backgrounds (dashed-lines), andkinematically forbidden events assuming Q = 8.1 MeV (solid-lines) for the respective charge states. Corresponding anti-electronneutrino energy spectra Eν are shown in b), and d) both withkinematically constrained domain and events within 1.5 MeV ofQ = 8.1 MeV GS transition (bold-markers), and without suchbounds (markers) are shown for each charge state. Expected timerandom coincident background event distribution without theabove bounds are overlaid for each charge state (dashed-lines). . 554.16 a),d) Recoil-SOE-SCINT coincident transverse recoil momentumdistribution of +2 charge state within TOF bounds shown in Fig-ure(4.13a,d) for event subset within ±1.5 MeV of the GS transi-tion Qexp-value centered at Q = 8.1 MeV, and within the definedkinematic boundaries. The transverse momentum distribution ofevents outside the kinematic boundaries are shown in b),d) forcoincident events with the Upper, and Lower scintillator, respec-tively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.17 Simulated transverse recoil momentum distribution from a point-like source with encoded trap drift, within the TOF bounds, andwith conical restrictions placed on the β-solid angle (22◦) definedby our collimator geometry for coincidence with the a) upper,and c) lower scintillators, respectively. Corresponding mask trans-mission functions em(Kr) (markers) and finite event density e f d(Kr)are shown for +2(red), 3(blue), 4(green) charge states for coinci-dence with the b) upper, d) lower scintillator, respectively. . . . . 58xv4.18 Recoil Kr spectra for GS Recoil-SOE-SCINT coincident events incharge state +2(red), 3(blue), 4(green) with beta’s coincident withthe a) upper, and b) lower scintillator, respectively. GS eventswere isolated after applying kinematic bounds, TOF bounds, andgating in Qexp-value on the events within 1.5 MeV of the 8.1 MeVgrounds state transition. Overlaid are the time random coinci-dent background’s (dashed-lines) for the respective charge statesalong with model fits taking ξo/ω < 0. . . . . . . . . . . . . . . . . 594.19 χ2/do f distribution of equation(4.5) fit to recoil GS Kr spectra inFigure(4.18) after floating the nuclear matrix element ratio ξo/ω <0 and NORM parameters for +2, 3, 4 charge states with beta coin-cident with the a),b),c) Upper, and d),e),f) Lower, scintillators, re-spectively. Boundaries of these distributions represent 90 % sta-tistical C.L. for the respective charge state. The range in ξo/ωwas chosen for comparison of the Recoil-SOE, and the GS iso-lated Recoil-SOE-SCINT coincident data to see the large changein ξo/ω between these channels. . . . . . . . . . . . . . . . . . . . 614.20 First-forbidden (rank-0) beta-neutrino correlation function prod-uct a1(Eβ) · (v/c) in equation(2.22) vs. Eβ from (90 % C.L.) boundsset in Table(4.1) for Recoil-SOE events in a,c) and GS isolatedRecoil-SOE-SCINT coincident events from limits set in Table(4.2)in b,d) for ξo/ω > 0, and ξo/ω < 0, respectively. In (b,d) thesolid and dashed line bounds are from event streams coincidentwith the Upper and Lower scintillators. . . . . . . . . . . . . . . . 64A.1 TRINAT DAQ timing setup with indicated delays into the ac-quisition. Strip detectors are not shown nor are the QDC data-streams from the rMCP, Wedge and Strip Anode (WSA) backingthe eMCP, or scintillators. Only relative timing differences areneeded for xz-recoil hit position reconstruction of the delay-line-anode (DLA) so delay-line lengths are not important. The 355 nmUV laser incident on the trap was split off to a photo-diode (PD)to trigger our acquisition. . . . . . . . . . . . . . . . . . . . . . . . 67A.2 Drift corrected photo-ion distribution in xz-plane over entire datasetfit with three independent elliptical-Gaussian’s parameterizingpopulations of strongly bound atoms and those in the process ofbeing collected in the trap along the beam-line axis. . . . . . . . . 68xviA.3 Recoil Kr spectra for +2(red), 3(blue), 4(green) charge states a) be-fore correction from non-uniform ~E-field, and b) following ionimpact coordinate correction in a uniform ~E-field of 998 V/cm.Included are expected random backgrounds and projected hoopbackgrounds necessary for ξo/ω > 0 model comparison. . . . . . 68A.4 χ2/do f distribution of equation(4.5) fit to recoil Kr spectra in Fig-ure(4.11) after floating the nuclear matrix element ratio ξo/ω > 0and NORM parameters for +2, 3, 4 charge states a),b),c) withoutnon-uniform ~E-field corrections, and d),e),f) with non-uniform~E-field corrections in mapping X′ → X. Boundaries of these dis-tributions represent 90 % statistical C.L. for the respective chargestate. From the large first-forbidden GS branching we assume thebeta-energy dependence in the correlation a1 parameter of equa-tion(2.22). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69A.5 Recoil Kr spectra for GS Recoil-SOE-SCINT coincident events incharge state +2(red), 3(blue), 4(green) with beta’s coincident withthe a) upper, and b) lower scintillator. GS events were isolatedafter applying kinematic bounds, TOF bounds, and gating in Q-value on the events within 1.5 MeV of the 8.1 MeV grounds statetransition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70A.6 χ2/do f distribution of equation(4.5) fit to recoil GS Kr spectra inFigure(4.18) after floating the nuclear matrix element ratio ξo/ω >0 and NORM parameters for +2, 3, 4 charge states with beta coin-cident with the a),b),c) Upper, and d),e),f) Lower, scintillators, re-spectively. Boundaries of these distributions represent 90 % sta-tistical C.L. for the respective charge state. The range in ξo/ωwas chosen for comparison of results between Recoil-SOE, andRecoil-SOE-SCINT coincidence. The range in ξo/ω was chosenfor comparison of the Recoil-SOE, and the GS isolated Recoil-SOE-SCINT coincident data to see the large change in ξo/ω be-tween these channels. . . . . . . . . . . . . . . . . . . . . . . . . . . 71xviiA.7 Recoil-SOE-SCINT triple coincident events with transition Q-valuedistributions of the +2(red), 3(blue), 4(green) charge states withbeta’s coincident in the a) Upper, and c) Lower scintillators, re-spectively. Overlaid are the time random coincident background’s(dashed-lines), and kinematically forbidden events assuming Q =8.1 MeV (solid-lines) for the respective charge states. Correspond-ing anti-electron neutrino energy spectra Eν are shown in b), andd) both with kinematically constrained domain and events within1.5 MeV of Q = 8.1 MeV GS transition (bold-markers), and with-out such bounds (markers) are shown for each charge state. Ex-pected time random coincident background event distributionwithout the above bounds are overlaid for each charge state (dashed-lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72xviiiList of Abbreviationsp+ Protonn0 Neutronβ± Beta electron/positronν¯e, νe Anti-electron, electron neutrinoDAQ Data Acquisition SystemTOF Time-Of-FlightSOE Shake-Off Electron (atomic)MCP Micro-Channel PlateeMCP Electron MCPrMCP Recoil MCPrMCP rMCP hit satisfying DLA timing sum in x and zDLA Delay-Line AnodeWSA Wedge-Strip-AnodePMT Photo-Multiplier TubeUPMT Upper-PMT(Scintillator)LPMT Lower-PMT(Scintillator)DSSD Double-Sided Silicon Strip DetectorE− ∆E Scintillator-DSSD beta telescope detectorGS Ground State to Ground State TransitionHV High VoltagexixPhysical ConstantsTDC Dispersion δtTDC = 0.097 656 25 ns/chSpeed of Light c = 299.792 458 mm/nsh¯c h¯c = 197.326 963 1(49)MeVfmFine Structure Constant α = 1/137.035 999 150(33)Electron Rest Mass m = 0.510 998 910(13)MeVUnit Nucleon Rest Mass u = 931.494 095 4 MeV/nucleon92Rb Nucleon number A = 9292Rb Parent atomic Number Z = 3792Rb Parent mass Difference ∆M = −74.772 MeV92Rb Parent mass M = ∆M+ uA+mZ = 85.642 MeV92Sr Daughter atomic number Zr = Z+ 1 = 3892Sr Daughter mass difference ∆Mr = −82.867 MeV92Sr Daughter mass (neutral) Mr = ∆Mr + uA+mZr = 85.634 MeV92Sr Daughter RMS-radius RRMSr = 4.2949 fm92Sr Daughter radius Rr =√5/3RRMS = 5.5447 fmQ-value Q = ∆M− ∆Mr = 8.095 MeVBeta End-point Eo = Q+m = 8.606 MeVxxList of Symbolsf Lepton phase space integral a.u.f t Phase-space corrected lifetime sgV Vector coupling constant a.u.gA Axial-vector coupling constant a.u.BF Fermi strength function a.u.BGT Gamow-Teller strength function a.u.I Nuclear spin h¯ωIi/ f Initial/final nuclear spin h¯ωτ Iso-spin operator a.u.Ti/ f Initial/final iso-spin a.u.pi Parity: spatial transformation operation ~x → −~x a.u.Iβ Beta transition intensity a.u.Sβ Beta transition strength a.u.E f Daughter excited state energy MeVEβ Beta total energy (W = Eβ/m, Wo = Eo/m) MeVEν Neutrino energy MeVpβ Beta momentum MeV/cpν Neutrino momentum MeV/cpr Recoil momentum MeV/cKr Recoil kinetic energy eVφr Recoil azimuthal angle (w.r.t +x-axis in xz-plane) rads.θr Recoil angle (w.r.t +y-axis) rads.aβν β− ν correlation parameter a.u.θ β− ν momenta angle rads.Ωθ β− ν solid-angle a.u.F∗, F Uncorrected,corrected Fermi Function a.u.R Finite recoil mass correction a.u.Lo Finite recoil volume correction a.u.C Beta-recoil phase-space convolution correction a.u.xxiM Degree of transition forbiddenness a.u.R First-forbidden transition rank a.u.ξo Time-like rank-0 first-forbidden nuclear matrix element a.u.ω Space-like rank-0 first-forbidden nuclear matrix element a.u.NORM Arbitrary normalization of theory to data a.u.em Simulated mask transmission efficiency function a.u.e f d Simulated finite recoil event density function a.u.eLR Left/Right rMCP efficiency function a.u.eUD Up/Down rMCP efficiency function a.u.IBckRnd Time-random(TOF) background distribution countsIBckHoop HV Hoop background distribution counts+2, 3, 4 All respective recoil charge states a.u.+2, 3, 4 Respective TOF-bounded recoil charge states a.u.xxiiAcknowledgementsI would like to thank Alexandre Gorelov for all his help with TRINAT’s hard-ware development/upgrades including the in-vacuum pellicle mirrors, upgradesto the recoil TOF-spectrometer, and ion detector developments; all of whichwere critical for this experiment. I would also like to thank Alexandre for hisinsight into the data analysis and building the recoil hit position transforma-tion arrays needed to map between the real non-uniform ~E field and uniformfield. This was again critical for the semi-analytic analysis of the data presentedin this thesis. I would like to thank Melissa Anholm for all her help with trac-ing down light leaks in our scintillators, helping me get started with buildingmy analysis code, and helping me generally with ROOT & GEANT4. I want tothank John Behr for making the proposal’s for this experiment and defendingthem for beam-time at the EEC, trapping the 92Rb at TRINAT, review of my the-sis, along with the funding he has provided me through my MSc. I also want tothank each of the above collaborators for their contributions in running this ex-periment, and sharing beam-time shifts. I also want to thank Dan Melconian forhis analysis recommendations and comments on my work during group meet-ings. Danny Ashery was the initial proponent for reconstructing the neutrinospectra and Q-value distributions in 38mK, who has unfortunately past awaybefore completion of this work, but has been with us in spirit.1Chapter 1IntroductionIn reactor neutrino osculation experiments two anomalies are observed: 1) a6% deficit in anti-electron neutrino ν¯e flux, and 2) a bump-like excess in 5-7MeV ν¯e-energy range of nearly 10% [1] at multiple short-baseline experimentsincluding Daya Bay[2], Reno [3], and Double Chooz [4]. The model’s fit wereβ−-conversion Huber model[5], but this excess was also observed in the MuellerModel[6]. These β−-conversion models are known to have some shortcomingssuch as the forbidden shape corrections often not incorporated despite the sig-nificant fraction of neutrinos in the 5-7 MeV energy range being from severalfirst-forbidden non-unique decays [1]. The shape corrections are largely un-constrained as many of the nuclear matrix elements governing the forbiddentransitions are experimentally unconstrained. The role of forbidden transitionswas revisited by Hayen who attempted to parametrize them with shell-modelcalculations from the dominant contributors, and this has reduced the signifi-cance of the bump feature in the ν¯e spectra[7]. The flux deficit may suggest aneV scale sterile neutrino that the known standard model ν¯e mix with, while thebump like feature may result from an incorrect/incomplete model of the reac-tor fuel/cycle. Unconstrained systematics from missing data, biased branchingfractions (Pandemonium Effect), or forbidden transition beta shape correctionsmay each contribute to this model disagreement[1].A limited set of first-forbidden ground-state to ground-state (GS) transitionswith large Q-values have dominant contributions to the ν¯e spectral shape inthe 5-7 MeV range (96Y,92Rb, 142Cs, 97Y, 93Rb, 100Nb, 140Cs, 95Sr) collectivelyaccounting for 48% of the total ν¯e flux in this energy range [1]. Estimates of92Rb first-forbidden 0− → 0+ GS transition alone contributes conservativelyup to 16% of the total flux in this energy range, driven by comparatively smallChapter 1. Introduction 2log( f t) = 5.75 assuming a GS branching of 95% (2012 Nuclear Data Sheet). To-tal absorption spectrometer experiments performed with 92Rb have consistentlyyielded smaller GS branching fractions of 91(3)%[8],[9], which would slightlyreduce its contribution to the ν¯e excess on the order of a few percent[1] withinthis energy range. Accurate and independent determination of the GS branch-ing fraction is thus important to correctly constrain the expected population ofhigh energy ν¯e neutrinos, particularly in the case of 92Rb. On the order of 1100other beta decays contribute at the sub-percent levels (< 2%) and do not signif-icantly impact the spectrum shape in the 5− 7 MeV region.In the case of 92Rb a continuum of highly excited states is believed to be fed onthe tail of a Giant resonance with very low transition intensities making directGS branching ratio extremely challenging to measure. Total absorption spec-trometers use high-Z scintillators, unlike conventional Ge detectors, and thusnearly completely absorb the gamma-ray energy within the detector volume,but they suffer from known systematic uncertainties. Amongst the systematicsare: 1) "inner" bremsstrahlung, in which photons are radiated during the decayof the nuclei; and 2) "outer" bremsstrahlung, in which the photons are radiatedby the beta interacting with detector material. In our experiment described be-low the gamma-ray photons from excited states and bremsstrahlung are notdetected, but they are inferred from missing transition energy with respect tothe strong GS beta transition energy (Q-value). We will rely on simulations toconstrain the missing energy distribution from bremsstrahlung photons, in or-der to separate the population resulting from transitions to excited states in thedata, which will be necessary for GS branching measurement in the final analy-sis.The reality of short-baseline neutrino oscillation experiments coupled with powerreactors is that they often utilize a mix of fuel 235U, 238U, 239Pu, and 241Pu, com-plicating the analysis of the fission progeny beta decay chains contributing tothe ν¯e flux. A more significant problem for the power reactor experiments is thatthe relative proportion of these fuels is generally assumed to be constant overthe fuel cycle, which need not be true in general, and would be extremely diffi-cult to measure directly. A great deal of recent experimental progress has beenmade in understanding the time dependent antineutrino flux from the domi-nant 235U, and 239Pu fuels by the RENO collaboration [10]. The above modelChapter 1. Introduction 3deviations and experimental challenges have motivated the PROSPECT reac-tor ν¯e experiment. PROSPECT is designed to both search for extremely shortbaseline (∼ 7.9 m) neutrino oscillations [11] expected assuming eV scale ster-ile neutrino, as well as utilize the highly enriched 235U research reactor coreat Oakridge National Lab for precision studies of ν¯e spectra [12]. With limitedstatistics PROSPECTs ν¯e spectra are in good agreement with both the Hubermodel and Daya-Bay excess observed in the 5-7 MeV range [12].Given the importance of constraining 92Rb GS branching, we developed a com-plementary method utilizing direct Q-value measurements through momen-tum resolved coincident beta-recoil kinematics. The TRINAT neutral atom trapat TRIUMF is employed to cool and confine 92Rb using a Magneto-Optic-Trap(MOT), complete with recoil ion Time-Of-Flight (TOF) spectrometer, and op-posing E − ∆E beta detectors. About 100k 92Rb atoms are nominally trappedat mK temperatures within 1mm3 to provide a backing free, zero momentuminitial state. We will show below that the strong first-forbidden 0− → 0+ GSbranching yields a large Q-value resonance at 8.1 MeV, with a low energy tailresulting from transitions to excited states. In the weak decay of a 0 → 0 tran-sition, the outgoing beta and neutrino momenta (angularly displaced by θ) areangularly constrained by a cos θ dependence, weighted by an angular corre-lation parameter aβν, and can be extracted from the recoil kinetic energy (Kr)spectrum. In the first forbidden rank-0 GS transition, aβν acquires a beta en-ergy dependence, and a dependence on the two nuclear matrix elements ξo,ωmediating the decay. With the GS events kinematically isolated in transition Q-value, direct measurement of the ratio ξo/ω is performed via aβν where we find|ξo/ω| < 1, and aβν 6= 1. The 3-body decay kinematics and constrained beta-recoil momenta also enable direct reconstruction of the GS ν¯e energy spectra foreventual comparison with theoretical predictions.The thesis below is organized sequentially beginning with relevant theory, fol-lowed by experimental sections detailing the setup, and detector calibrations.The experimental results section details our analysis of the aβν parameter inrecoil atomic shake-off electron coincidence (Recoil-SOE), and triple coincidentanalysis (Recoil-SOE-Scintillator) for multiple recoil charge states. We detail ourevaluation of the Q-value, GS rank-0 nuclear matrix element ratio ξo/ω, and re-constructed ν¯e spectra. We conclude with a comparison between the aβν with,and without GS isolation.4Chapter 2Theory2.1 Nuclear Beta decayIn nuclear β±-decay, the weak force mediates the semi-leptonic decay of valencequarks (u, d) contained in protons (uud) and neutrons (udd) within atomic nucleithrough the virtual W± vector-boson, where u, d→ d, u, respectively such thatp+ → n0 + β+ + νe,n0 → p+ + β− + ν¯e.(2.1)The charged weak interaction currents mediating the decay have been experi-mentally determined to be Vector and Axial-Vector (V-A) [13]. If |ψi〉 and∣∣ψ f 〉are the initial and final nuclear wave-functions then the vector component of thedecay is parameterized by the Fermi strength function BF = |〈ψ f |τ|ψi〉 |2 whereτ is the isospin transformation matrix (p ↔ n), while the axial-vector currentis encoded in the Gamow-Teller (GT) strength function BGT = |〈ψ f |στ|ψi〉 |2where σ is the Pauli spin matrix. In a pure Fermi transitions, the change innuclear spin(I)/parity(pi) satisfy (∆I = 0/∆pi = 0 with Ii = I f = 0), whilethe pure GT transition encodes the spin/parity dependent transitions (∆I =0,±1/∆pi = 0,+1 no 0 → 0). The lepton phase-space ( f ) rapidly increaseswith available energy in the decay, while the transition lifetime (t) decreases,making the phase space corrected lifetime f t-value a value of comparison be-tween differing nuclear transitions. The beta phase-space is also perturbed bythe coulomb interaction with the nuclear system impacting transition lifetimesand will be discussed further below. If M f i =〈ψ f |M|ψi〉is the general transi-tion matrix element, then the transition strength |M f i|2 ∝ g2VBF + g2ABGT andwill scale inversely with the transition f t-value with a larger strength corre-sponding to smaller f t-value, and vice-versa such thatf t =2 ln 2pi3h¯7m5c4· 1g2VBF + g2ABGT. (2.2)Chapter 2. Theory 5The above holds for allowed transitions where there is no change in the nuclearparity in the initial and final states. In forbidden transitions, however, nuclearparity transformation is permitted where the Fermi and GT operators are scaledby operators which transform under parity, and this will be discussed below.2.2 92Rb Level SchemeThe tabulated level scheme of 92Rb from 2012 Nuclear Data Sheet is illustratedin Figure(2.1), demonstrating strong first-forbidden ground state 0− → 0+ feed-ing, along with several weak branchings to excited state transitions. The excitedstate Gamow-Teller (GT) transitions (0− → 1−) along with additional excitedstates not listed aid in diluting GS decay strength.Figure 2.1: 92Rb level scheme (Compiled 2012 Nuclear Data Sheet)In the 92Rb beta-decay a continuum of excited states are believed to be fed inthe 92Sr daughter of energy E f above the GS on the tail of a so-called giant res-onance. Giant multi-pole resonances result from collective coherent-motion ofnucleons often within excited, or deformed nuclei. One example is the giantdipole resonance resulting from collective uni-axial oscillations of protons withrespect to the neutron population. A toy model of a beta-transition strengthSβ(Eβ) function is shown in Figure(2.2a), with the giant resonance largely abovethe Q-value. The low energy tail of the resonance, however, has states whichleak into the energetically allowed region (E f < Q) and are fed through beta-decay. If the level density of the daughter’s excited states is sufficiently lowChapter 2. Theory 6between E f and E f + ∆E, then for the given transition f t-value the beta transi-tion intensity Iβ(E f ) satisfyf (Q− E f , Zr) · T1/2/ f t = Iβ(E f ) (2.3)where T1/2 is the parent lifetime, and f (Qβ − E f , Zr) the beta phase-space inte-gral [14]. The beta phase-space integral f with transition end-point Wo = Q/mf (Q− E f , Zr) =∫ (Q−E f )/m1F(W, Zr)W√W2 − 1(Wo −W)2dW (2.4)encodes the phase-space accessible to the beta of energy W = Eβ/m along withthe modification to the beta wave-functions through the Coulomb interactionwith the daughter’s (Zr) charged nuclear wave-functions via the Fermi functionF(W, Zr) (detailed below). In 92Rb, the total absorption experiments mentionedabove report a smaller GS branching of 91 % than that reported in the tabulatedlevels (95 %), reducing the GS beta transition intensity Iβ(E f = 0). Since theGS f t-value scales inversely with Iβ(E f ), the log f t should correspondingly in-crease by log(0.95/0.91) = 0.04 to that reported in in Figure(2.1).However, when the level density of the daughter’s excited states is large thenan average beta transition intensity is sampled between E f and E f + ∆E f withthe average beta transition strength Sβ(Ex) for Ex on this interval satisfyingSβ(Ex) f (Q− Ex, Zr) · T1/2 = ∑E f∈∆EIβ(E f )/∆E. (2.5)As shown in Figure(2.2c) the phase-space integral can either enhance, or sup-press excited state transition intensities relative to the low-lying states by equa-tion(2.5) in β±-decay, respectively. In the case of β−-decay, even with the sup-pression of excited state transition intensity, there can still be considerable decaystrength to such states. The result is a dense continuum of highly excited stateswith very low transition intensity which often have complicated γ-decay feed-ing to lower-lying states. Since these initial excited states are weakly populatedand feed multiple low lying states, it becomes increasingly difficult to resolvesuch transitions directly, making absolute branching fractions difficult to con-strain. Total γ-absorption experiments sacrifice γ-energy resolution from largedetector volumes with the goal of collecting all the photon energy from whichbranching fractions can be extracted.Chapter 2. Theory 7a) b) c)Figure 2.2: Toy model of a Giant-Resonance where a continuum of highly excited andfragmented states become accessible via β-decay with beta a) transition strength functionSβ(Ex), b) phase-space function f (Q− Ex, Zr), and c) transition intensity Iβ(E f ).In 92Rb one consequence of both the large beta-decay Q = 8.1 MeV and, likely,the daughter’s dense excited state level-set above the neutron separation energyof Sn = 7.287 MeV, is the opening up of prompt beta-delayed neutron emissionchannels. In principle such events could distort the low energy recoil kineticenergy Kr spectrum by driving events to larger Kr, although that requires scin-tillator thresholds to be made sufficiently small to accept the correspondinglylow energy beta’s from the 92Rb parent decay. Significant efforts were under-taken to lower our beta detection thresholds to increase our sensitivity to thehighest energy neutrinos by reducing the thickness and Z of our in-vacuummirrors which the beta’s must penetrate prior to detection; these effects will bedetailed below.2.3 Recoil Kinematics & Angular Correlation aβνIn the beta decay of nuclei of mass M, the 3-body final state yields energeticbeta (mass m), ν-neutrino, and low energy recoiling daughter (mass Mr). The4-momentum vectors of the β, ν, and recoil are defined by (Eβ,~pβ), (Eν,~pν), and(K + Mr,~pr) for the recoil of kinetic energy Kr, respectively. In the weak de-cay parity violation is maximal at the lepton vertex ensuring only chiral-lefthanded neutrinos, or chiral-right handed anti-electron neutrinos are producedin β±-decay, respectively. The consequence of such helicity constraints on theoutgoing neutrino produce the angular correlation (angle θ) between the betaand the neutrino 3-momenta. In the nuclear spin-0 system (I f = Ii = 0), theChapter 2. Theory 8probability of a beta with energy between [Eβ, Eβ + δEβ] and within [θ, θ + δθ]is dN = P(Eβ, θ)dEβdΩθ such thatdN = F(Eβ/m, Zr, Rr)pβEβp2ν[1+ aβν(pβEβ)cos θ]dEβdΩθ (2.6)with F(Eβ/m, Zr, Rr) a modifying Fermi function (detailed below), Ωθ the unitsolid angle of the beta-neutrino angular phase-space, and aβν the beta-neutrinocorrelation parameter. From momentum conservation p2r = (~pβ + ~pν)2[p2r − (p2β + p2ν)]/2 = pβpν cos θ (2.7)where after differentiating both sides it follows that pβpνdΩθ = prdpr. If Eois the beta end-point energy and Q the transition kinetic energy available tothe beta then Eo = Q + m = Eβ + Eν + Kr. Since the recoil Kr  Eβ, Eν thenEo − Eβ ∼= Eν ∼= pν where the last equality assumes negligible neutrino mass.Given the recoil is non-relativistic with the recoil Kr = p2r/2Mr, it follows thatMrdKr = prdpr. After substitution of the above into equation(2.6) we find therecoil K distribution[15]dNdKr=∫ EomedEβMr2F(Zr, Eβ/m, Rr)··[Eβ(Eo − Eβ) +aβν2(2Mr · Kr +m2 − E2β − (Eo − Eβ)2)].(2.8)The recoil K spectrum has a linear dependence on Kr, up to a modifying Eβdependent Fermi function F(Eβ/m, Zr, Rr) from the beta wave function inter-acting with the nuclear environment. One of the most basic Fermi functionsF∗(W, Zr, Rr) for β−-decay is given byF∗(W, Zr, Rr) = 2(γ+ 1)(2pβRr)2(γ−1)epiαZrWpβ|Γ(γ+ iαZrWpβ)|2|Γ(1+ 2γ)|2 (2.9)with γ =√1− (αZr)2. The Fermi-function F∗ parameterizes solutions to theDirac equation for the outgoing beta wave-function at finite recoil radius Rr,where solutions are non-divergent. Corrections to the outgoing beta wave-function must be applied as it interacts with the nuclear wave-functions re-quiring recoil mass correction R(W, Zr, Mr), finite nuclear volume correctionLo(W, Zr), and phase-space convolution correction C(W, Zr)with modified Fermifunction given byF(W, Zr, Rr) = F∗(W, Zr, Rr) · R(W, Zr, Mr) · Lo(W, Zr) · C(W, Zr). (2.10)Chapter 2. Theory 90 2 4 6 8E  (MeV)00.511.522.5Normalized Prob.10-4F*(W,Zr,Rr)F*(W,Zr,Rr)*RF*(W,Zr,Rr)*R*LoF*(W,Zr,Rr)*R*Lo*C0 2 4 6 8E  (MeV)-0.08-0.06-0.04-0.0200.020.040.06F( W,Zr,Rr )/F*( W,Zr,Rr ) - 1F*(W,Zr,Rr)*RF*(W,Zr,Rr)*R*LoF*(W,Zr,Rr)*R*Lo*Ca) b)Figure 2.3: (left) 92Rb allowed β spectrum with fractional corrections (right) to the Fermi-function from higher order corrections Ro,Lo,CA.The parameters R, Lo,C have unique dependencies depending on whether thevector or axial-current mediate the nuclear transition, as outlined by Wilkinson[16]. Warburton [17] pointed out that in a 0− → 0+ decay the two operators,γ5 (time-like), and σ · r (space-like) drive the axial-current of the decay, therebyrequiring the axial-current parameterization of R, Lo,C in equation(2.10) to beapplied in this analysis. The corresponding beta energy spectrum parameter-ized by the phase space differential d f/dW above in equation(2.4) for the 92RbGS decay is shown in Figure(2.3a), with the sequentially implemented correc-tions to the basic Fermi function F∗ above overlaid. The fractional correctionof each successive component modifying the Fermi function in equation(2.10)is shown in Figure(2.3b), demonstrating energy dependent corrections whichare maximally of order 5%. The modifications to F∗ in applying these correc-tions drive beta decay intensity to lower Eβ as the outgoing beta wave-functionexchanges momenta with the nuclear volume.2.4 Forbidden Transitions2.4.1 Correlation ParameterIt was shown above that the matrix element mediating the allowed axial-currentin the weak decay was στ. In first-forbidden decays, the hadronic and leptoniccurrents are further modified from the allowed theory by additional transitionoperators includingChapter 2. Theory 10r, [r, σ]R, R = 0, 1, 2 (2.11)from the leptonic weak currents acquiring spatial dependencies and,γ5, α (2.12)from the hadronic weak currents[18]. Computing the matrix elements in for-bidden transitions is much more difficult compared to the allowed transitions,which only depend on spin/isospin, as they now acquire a dependence on theradial wave-functions of the nuclear system; particularly for systems with largeradii and Z. We will closely follow the notation of Warburton[17][18]. In thisformulation nuclear spin I and isospin T are considered to be good quantumnumbers with initial and final nuclear wave-functions |Ii, Ti〉 and∣∣I f , Tf 〉. Thetensor rankR accessible to the forbidden transitions are integer values between|I f − Ii| ≤ R ≤ |I f + Ii|. (2.13)In introducing the above transition operators, the phase space accessible to theoutgoing leptons in the allowed decay of equation(2.4) is modified by pow-ers of (p/W)M whereM is the degree of forbiddenness of the transition. Re-expressing the allowed form of dN in equation(2.6) where if d f is the leptonphase-space unit elementd f = F(W, Zr, Rr)pβW(Wo −W)2dW (2.14)and Co(W, θ) the lepton phase-space convolution functionCo(W, θ) = 1+ aβν(pβW)cos θ, then (2.15)dN = d f · Co(W, θ)dΩθ (2.16)Warburton[17] proposed to expand the lepton phase-space convolution func-tion Co(W, θ) → C′o(W, θ) in powers of (p/W)N for N = 0, 1, 2 with C′o(W, θ)satisfying[∑M,N ,RK(MNR)WM(pβW)NPN (θ)]/[∑M,RK(M0R)WM]. (2.17)Here PN (θ) is the N th order Legendre polynomial, and K(MNR) indepen-dent of W with a more general form in [17] (A ∼ 40) following Behrens &Chapter 2. Theory 11Buhring formalism[19]. The formalism for small Z is presented in [18] specifi-cally for the first-forbidden beta-neutrino correlations in 11Be. Important in thisfirst-forbidden formalism is the linearity in rank-R of the transition with nu-clear matrix elements only interfering within their rank, and not outside theirrank. We restrict discussion to N = 0, 1 contributing to the leading order termsin equation(2.17)∑M,R K(M0R)WM +∑M,R K(M1R)WM(pβW)cos(θ)∑M,R K(M0R)WMwhich reduces to the familiar convolution expressionC′o(W, θ) = 1+ a1(W)(pβW)cos(θ). (2.18)The distinction from aβν in the allowed decay is the forbidden correlation pa-rameter a1(W) acquires an energy dependence where [17][18]a1(W) =∑M,R K(M1R)WM∑M,R K(M0R)WM. (2.19)Further restricting our discussion to the 92Rb first-forbidden 0− → 0+ rankR = 0 transition the requisite coefficients K(MN 0) are(N = 0) : K(000) = ξ2o +19ω2, K(−100) = −23µ1γ1 · ξoω (2.20)(N = 1) : K(010) = ξ2o −19ω2 (2.21)where γ1 =√1− (αZr)2 with α the fine structure constant and µ1 = λ1 = 1[19]. Expanding equation(2.19) and normalizing with respect to ωa1(W, ξo,ω) =((ξoω)2 − 19)((ξoω)2+ 19)− 23(ξoω)· γ1W. (2.22)Naively if either of the matrix elements are small then a1 = 1, which cannotbe assumed in 92Rb. Inserting equation(2.22) into the recoil Kr spectrum inequation(2.8) one can perform a two parameter fit to the experimental spec-trum floating the nuclear matrix element ratio ξo/ω and arbitrary normaliza-tion NORM. The nuclear matrix elements of interest theoretically encoding therank R = 0 axial-current can be broken into the time-like MTo , and space-likeMSo components where their relation to ξo, and ω satisfyChapter 2. Theory 12ξo = (MTo + ξMS′o ) +13MSoWo, MTo ∝〈I f , Tf |γ5τ|Ii, Ti〉, (2.23)ω = MSo ∝〈I f , Tf |σ · rτ|Ii, Ti〉. (2.24)Here ξ = αZ/2Rr and MS′o = eS′o MSo where eS′o∼= 0.7 but can be determinedfrom integrating radial wave-functions weighted by (2/3)I(1, 1, 1, 1, r) as per[19]. It follows that the time-like to space-like matrix element ration is linear inξo/ω withMToMSo=ξoω−(ξeS′o +13Wo). (2.25)2.4.2 More information about 0− to 0+ transitions, particularly 92RbSpectrum shapeIn the parameterization used by Ref.[20], the two main operators contributing to0− → 0+ transitions have different β energy spectra, hence different ν spectra.One operator has an allowed spectrum, while the other is multiplied by theshape factorC(Eβ) = p2β + E2ν + 2(pβ/Eβ)2EνEβ, (2.26)which would make fewer high-energy neutrinos by about 5% if it dominatedthe transition. We are finding it very hard to make the correspondence betweenthe notations to let us contrain this particular linear combination of operatorsfrom the β− ν correlation, which would be simpler than trying to fit the Eβ spec-trum. Nevertheless, this has been a strong motivation for us to independentlymeasure the ν spectrum.The “ξ approximation” doesn’t workA review by Hayes and Vogel [21] has much useful information about forbiddenbeta decay and reactor neutrinos. For the high β Q-values of interest, a usefulapproximation does not hold. The “ξ approximation” asserts that the β energyspectra are close to allowed if the Coulomb energy of the emitted β is muchlarger than its total energy at the nuclear radius, αZ/R E0/m. In 92Rb decayαZ/R = 19.2 while E0 = 16.8mβ, so this inequality does not hold at all. It’sstill notable that the measured β energy spectrum for 92Rb decay has close to anallowed shape [22] within the accuracy of the measurement.13Chapter 3Experiment3.1 TRIUMF-ISACFigure 3.1: Isotope Separator and Accelerator facility as of 2003 at TRIUMF (see ISAC).The main cyclotron is off to the page towards bottom-right of the image. The TRINATexperiment is correctly indicated above the ISAC pre-separator room (LEBT).The main cyclotron at TRIUMF supplies a 500 MeV proton beam with a 9.8µA beam current incident on a uranium carbide (UCx) target within the ISACtarget modules [23]. With the proton beam on target, the uranium undergoesnuclear spallation producing many secondary daughter species, most of whichare unstable, decaying through either α, β,γ-decay, proton/neutron emission,Chapter 3. Experiment 14or electron capture processes. Progeny diffuse from the hot bulk target materialand effuse through a hot cylindrical "ionizer" lined with rhenium, a metal witha high work function. Alkali progeny in particular leave their valence electronat the ionizer metal surface, so are "surface ionized" and then extracted with anapplied electric potential. The extracted beam is then purified with a magneticspectrometer separating species in A/q to select for 92Rb prior to delivery to theTRINAT experiment.It should be noted during our original 2017 test run the magnetic spectrom-eter was run with slits-in for improved beam purity, but this led to unstablebeam due to electrostatic charging from the high particle flux of 108/sec pro-duced from the ISAC targets. The TRINAT experiment selectively traps only92Rb and so isotopic purity is not a significant constraint. Consequently, theseslits were retracted in the subsequent 2018 run yielding a largely stable supplyof radioactive beam for the data presented in this thesis.3.2 TRINATzyxFigure 3.2: TRIUMF Neutral Atom Trap (TRINAT) with electrostatic hoops for recoil chargestate separation with opposing recoil-ion (rMCP) and shake-off electron (SOE) detectors,and orthogonally orientated, symmetric E − ∆E β-telescopes along the vertical ±z-axis.The vertical trapping beam path is indicated in red reflected from 70 nm thin gold mirrorsbacked by 4 µm Kapton. (Horizontal trapping beams are not shown). 92Rb is trapped in aMagneto-Optic-Trap (MOT) at the center of the apparatus where it β-decays with outgoingtracks of the β− (blue), recoiling 92Sr daughter (grey), and atomic SOE (red) shown.Chapter 3. Experiment 15The TRINAT experiment consists of two adjacent Magneto-Optic-Traps (MOT),the first within the collection chamber, and the second within the experimen-tal chamber. Ion beams from the ISAC targets typically are delivered at 15-20keV incident on an Zirconium neutralizing foil. The foil is heated to just abovea phase transition in the lattice structure to optimize release from the neutral-izer [24]. The neutral 92Rb then diffuses out into the vacuum and thermalizeon a glass cell with special polymer coating to aid in cooling the atoms beforetrapping. The thermalized neutral 92Rb atoms are collected in the collectionMOT prior to transfer to the experimental chamber. A resonantly detuned pushbeam is then used to transport the atoms from the collection trap to the mea-surement trap at the center of the experimental chamber as shown in Figure(3.2)where they subsequently decay [25]. The TRINAT experimental chamber con-sists of rectangular electrostatic hoops for recoil ion charge-state separation bytime-of-flight(TOF) along the +y-axis, with opposing recoil ion detector (rMCP)and shake-off-electron (SOE-eMCP) detectors. These detectors are microchan-nel plate (MCP) based analog amplifiers for single ion/electron event trigger-ing with ∼ns timing resolution. Position sensitive delay-line-anode (DLA), andwedge-and-strip detectors (WSA) back the rMCP, and eMCP, respectively.Two opposing symmetric E− ∆E β-telescopes along the ±z-axis face the trap.Each β-telescope consists of a trap-facing (∆E) Double-sided Silicon Strip Detec-tor (DSSD) backed by a plastic (E) scintillator. The β-telescopes are separatedfrom the vacuum by a thin 229 um Be foil. The vertical trapping beams of theMOT are necessarily brought in at 19◦ from the ±z-axis and reflected along the±z-axis via high-reflectivity (93 %) pellicle mirrors mounted at 9.5◦ to normalincidence on the trap facing side of the Be foils within the vacuum. The pelli-cles consist of 70 nm thin Au film deposited on 4 µm thick Kapton film, whichwas stretched and epoxied (vacuum compatible Masterbond EP30-2 epoxy) toa beveled stainless-steel ring (National Photo Color Inc.), and mounted withinour β-collimator assembly. In the previous 37K experiment the in-vacuum mir-rors consisted of a thin high reflector ( TiOx + SiOx ) dielectric stack depositedon 273 µm of SiC, which led to non-negligible β-scattering and energy depo-sition, particularly for low energy β’s in the mirrors prior to detection in theE− ∆E detectors, necessitating 400 keV scintillator threshold and 40 keV DSSDthresholds. The goal of the pellicle mirrors in this experiment is to drive thescintillator thresholds down to our DSSD thresholds to ensure the majority ofthe highest energy ν¯e events are accepted by the DAQ system.Chapter 3. Experiment 163.3 Recoil EventsFull details of the rMCP spatial and temporal calibration, and the recoil eventselection are detailed in the subsections below.3.3.1 Recoil Event Selection(DLA_I[0] + DLA_O[0])/2 - rMCP[0] [ns]0 2 4 6 8 10 12 14 16 18Counts/0.2 ns05000100001500020000250003000035000xDLA(LE)zDLA(LE)(eMCP[0] - PMT[0]) TOF [ns]190− 180− 170− 160− 150− 140−Counts/0.5 ns0200400600800100012001400160018002000UPMT(LE)LPMT(LE)a) b)Figure 3.3: a) rMCP-DLA timing sum gate for recoil-event selection (rMCP) utilizing the x(red), and z (blue) DLA detector planes with Leading Edge (LE) constant-fraction trigger.b) SOE-Scintillator eMCP-PMT TOF spectrum of the Recoil-SOE-SCINT coincident eventsusing the upper (red), and lower (blue) scintillators with indicated timing bounds chosento suppress false coincidence events.The recoil detector consists of a Z-stack (3-plate) microchannel plate (MCP)backed by a xz-position sensitive delay line anode (DLA). The rMCP hit tim-ing signals are picked off the front-MCP electrode high-voltage feedthrough.The MCP is an analog amplifier consisting of lead-glass with micro-channelsin a hexagonal unit cell with parallel channels having an axis oriented at 20◦to the plates’ normal. Incident charged particles on the MCP channels excitesurface secondary-electrons within the channel. With a large bias of 1 kV/mmacross the MCP plate, incident charge particles scatter off the channel surfacesuccessively producing secondary electrons which themselves produce subse-quent secondary electrons, producing electron multiplication resulting in anelectron shower out the rear of the MCP channel. A drift space is placed be-tween the plates to allow for space charge repulsion and spreading of the elec-tron shower over several channels of the following MCP. The rMCP consisted ofan impedance matched Z-stack over three plates (diam. 86.7 mm, active diam.77 mm, width 1 mm, chnl. spacing 32 µm, chnl. angle θCH = 20◦, chnl. diam.DCH = 25µm). An amplification of 1011 is nominally achieved, distributed overthe Z-stack MCP. The first MCP plate in the Z-stack facing the trap is mountedChapter 3. Experiment 17such that the micro-channel is θCH = 20◦ to the normal incidence in the xy-plane. This aids in reducing pulse-height dependent efficiency effects along thez-axis, which will be important in future polarized 37K experiments requiringa quantization axis along this axis. The subsequent two plates in the Z-stackare incrementally oriented at 60◦ and 120◦, to average out potential orientationdependent microchannel non-linearities of individual plates over the detectorsurface.The final electron shower out of the rear of the 3rd MCP plate then expandsas a cone prior to collection on the DLA. The DLA consists of two orthogonalplanes (xz) of counter-wound pairs of copper wires. The counter-wound wirepairs have opposing polarity such that induced potentials from the depositedcharge propagate in opposing directions required for absolute 1-dimensionalposition reconstruction. Timing signals are extracted from the x1, x2, and z1, z2DLA anodes, with their difference x1 − x2, and z1 − z2 having a direct propor-tion to the x, and z-position hit coordinates, respectively. A 2D metal grid maskin-front of the MCP permanently blocks some of the MCP channels, providingan in-situ spatial calibration of the timing differences with hit coordinates in thex and z-planes, and this will be discussed below.Due to the low signal thresholds placed on our DLA (4 timing channels) andrMCP (1 timing channel) constant fraction discriminators, random electronicnoise and discharges can fire each channel independently. In the case of thecounter-wound copper wires of the DLA, the net transit time is independentof impact position in the x and z-dimension independently with respect to thetiming signal of the rMCP ion hit. This is a consequence of the counter-woundwires having comparable length and the charge deposited over a localized num-ber of turns on the DLA. Consequently, candidate recoil events recorded as thefirst event entering acquisition in the DLA (x1, x2, z1, z2[0]) and rMCP[0] mustsatisfy the timing sum criteria where (x1[0] + x2[0])/2− rMCP[0], and (z1[0] +z2[0])/2 − rMCP[0] with timing bounds (5.5,12.0) ns (red), and (4.0,11.5) ns(blue) as shown in Figure(3.3a), respectively. Events with timing sums outsideof these windows are considered time random false triggers. The TDC linearchannel dispersion of δtTDC from our DAQ was assumed in this analysis. ThisDLA-rMCP summing criterion on rMCP events significantly aided in suppress-ing false triggers present in the event stream of the recoil detector. The FWHMChapter 3. Experiment 18of the DLA-rMCP summing criterion is ∼ 2ns in x and z-dimensions and rep-resents their convolved nominal timing resolution.In the Recoil-SOE-SCINT event stream the timing is obtained from the rMCP[0]and the PMT[0] timing differences, as the restricted solid angle of the scintil-lator aides in suppressing false coincidences from decays originating from theelectrostatic hoops present in the Recoil-SOE coincident data stream; this will bediscussed later. One property of this triple coincident data-stream is the require-ment of one or more atomic SOE. The largely mono-energetic SOE’s (eV scale)are accelerated by the ~E-field and impinge on the eMCP, yielding a clean SCINT-SOE TOF resonance as shown in Figure(3.3b) providing an additional constrainton true recoil events. SCINT-SOE TOF gates using the UPMT (red) with TOFwithin (-175,-160) ns, and LPMT (blue) within (-180,-165) ns are shown in Fig-ure(3.3b) and applied to our Recoil-SOE-SCINT coincident data. Similarly, acorrelated event timing resolution of ∼ 2ns FWHM in the eMCP-PMT coinci-dence is achieved with UPMT and LPMT detectors.3.3.2 Absolute Event TimingIn the case of the Recoil-SOE coincident events, the TOF is determined from thetiming difference between the rMCP[0] and the eMCP[0], which neglects theSOE TOF through the vacuum, along with electronic delays of the respectivesignals into acquisition. Efforts were made to trace delays from detectors intoacquisition as drawn schematically in Figure(A.1); however, in the end thereremained some unknown delays. Fortunately, a subset of Recoil-SOE events re-sult from the accumulation of neutral 92Rb, along with its progeny, on the sur-face of the rMCP. These subsequently beta decay where the beta fires the rMCPand the atomic SOE and/or secondary electrons drift through the ~E-field andtrigger the eMCP. The result is a prompt Recoil-SOE coincident TOF resonanceas shown in Figure(3.5) centered at -6 ns. The second broad TOF resonance cen-tered at 14 ns relative to the prompt events is of unknown origin, but may resultfrom the rMCP firing later after the ejected recoil impacts the rMCP. We definethe Recoil-SOE TOF as(rMCP - eMCP) TOF = (rMCP[0] - eMCP[0]) - (∆rMCP - ∆eMCP) - δeMCPrMCPChapter 3. Experiment 19where the timing delays are listed in Table(3.1). Note there are differences be-tween the measured delays in Figure(A.1) and those implemented in this anal-ysis in Table(3.1).(rMCP - eMCP) TOF [ns]15− 10− 5− 0 5 10 15 20 25 30 35Counts/0.5 ns02000400060008000100001200014000Figure 3.4: Recoil-SOE coincident TOF spectrum.Table 3.1: Timing OffsetsMeasured Offsets [ns] Added Offset [ns] [ns]∆eMCP 54.5 δeMCPrMCP 2.0∆rMCP 83.7 δrMCPUPMT -36.0∆UPMT 184.0 δrMCPLPMT -38.5∆LPMT 188.6 δeMCPUPMT -34.0δeMCPLPMT -37.0δUPMTLPMT 0.0Similar to the Recoil-SOE coincidence, a prompt coincidence exists in Recoil-SOE-SCINT. For these the beta incident on the scintillator fire the PMT, scat-tering or producing bremsstrahlung photons, which may subsequently fire therMCP. Figure(3.5a,b) displays triple coincident (eMCP - PMT) TOF vs. (rMCP -PMT) TOF using the upper, and lower scintillator, respectively where(eMCP - PMT) TOF = (eMCP[0] - PMT[0]) - (∆eMCP - ∆PMT) - δeMCPPMT(rMCP - PMT) TOF = (rMCP[0] - PMT[0]) - (∆rMCP - ∆PMT) - δrMCPPMT .The subset of events triggered by beta’s and photons correlated with true decaySOE are shown at rMCP-PMT = 0 and eMCP-PMT = 0 ns. The diagonal line ofevents are TOF random false coincidences between the four event channels withthe resonance centered at (1400,1400) ns corresponding to false coincidenceswith the dominant +1 recoil charge state where there is no SOE. The line ofChapter 3. Experiment 20events along eMCP-PMT = 0 ns correspond to true coincidences from decaysoriginating at the trap where the mono-energetic SOE produce a constant SOE-SCINT TOF.a) b)c) d)Figure 3.5: Recoil-SOE-SCINT coincident TOF spectrum with the a,c) upper, and b,d) lowerscintillators, respectively. The expanded region of a,b) in c,d) show the centering in TOFabout (eMCP - PMT, rMCP - PMT) = (0,0) [ns].3.3.3 rMCP Spatial CalibrationPreliminary xz-calibration is performed in Figure(3.6a) with xz-correlated Fourpt4-point mapping algorithm. Given four defined mask calibration points (X1, Z1,X2, Z2, X3, Z3, X4, Z4), and corresponding measured points (x1, z1, x2, z2, x3, z3,x4, z4) a mapping function F can be parameterized for any point (x,z) containedtherein such that F : (x, z)→ (X, Z)x → X = c1 + c2x+ c3y+ c4x · zz→ Z = d1 + d2x+ d3z+ d4x · z(3.1)where the constants c1, c2, c3, c4 and d1, d2, d3, d4 are uniquely determined. Theinversion function F was computed in Mathematica and not shown here forChapter 3. Experiment 21brevity.rMCP x-Position [mm]40− 30− 20− 10− 0 10 20 30 40Counts050100150200250300350400rMCP z-Position [mm]40− 30− 20− 10− 0 10 20 30 40Counts050100150200250300350400a) b)c) d)Figure 3.6: a) Rough recoil hit position calibration of the rMCP from delay-line-anode(DLA) timing signals. b) Hit position following correlated XZ-mapping via Fourpt map-ping algorithm. Projections of b) are shown in c) x, and d) z with overlaid fits to localminima to determine piecewise-linear spatial scaling in x/z required for final calibration.Preliminary calibration was applied to 4-points over the MCP surface in Fig-ure(3.6a) and applied to all recoil events over the detector surface with resultsshown in Figure(3.6b). Projections of Figure(3.6b) in the x, and z planes areshown in Figure(3.6c,d), respectively. This procedure was necessary to removethe xz-linear correlation; however, non-linearities remained as can be seen inthe x and z-projections where the vertical lines indicate true mask position cen-troids. Gaussian fits convoluted with a linear function were applied to each ofthese minima in projected recoil counts with the results overlaid on the pro-jections. The remaining non-linear deviation in the x and z coordinates overChapter 3. Experiment 22the MCP surface was taken to be approximately piecewise linear between adja-cent Gaussian-fit minima in x and z, with corrected recoil 2D event distributionshown in Figure(3.7). The point-like regions with event deficits (roughly 6 vis-ible) likely resulted from HV-discharges within the vacuum during HV condi-tioning and experiment, and locally damaged the MCP channels.Two sets of event types will be analyzed below, the first being recoil shake-off-electron coincidence (rMCP-SOE) representing the largest dataset since eventsare collected in 4pi. The second event type will be triple coincident Recoil-SOE-Scintillator (rMCP-eMCP-PMT) events, which has the advantage of better sig-nal isolation against backgrounds through the narrow timing gate that can beapplied between the beta trigger and arriving SOE, but is limited by the solidangle of our scintillator/collimator. We employ the same recoil summing crite-rion above on the rMCP event in both event types.Figure 3.7: Corrected recoil hit position after uncorrelated piecewise-linear scaling in x, andz from fits performed in Figure(3.6c,d) respectively.3.4 Non-Uniform ~E(~r) FieldThe ~E-field oriented along the +y-axis is applied to separate the the recoilingdaughter charge states in TOF. The proximity of the beta detectors to the trapwas necessary to maximize beta collection efficiency, without interfering withcollection of the recoiling ions. The requirement that the anti-Helmholtz coilsand grounded beta-collimator be so close to the HV electrostatic hoops in ourChapter 3. Experiment 23geometry introduces unavoidable ~E-field nonlinearities, and complicates theotherwise trivial exercise of reconstructing the recoil momentum in a uniformfield.Given the known geometry, the electric potential applied to each electrode wasoptimized using COMSOL 3D finite-element simulations to ensure near-uniform~E-field in the drift volume of the experimental chamber, particularly along thecentral drift axis (y-axis). Due to the proximity of the effectively groundedmagnetic-field coils needed for trapping and the HV electrostatic hoops, fieldnon-linearities are predominantly furthest from the trap region nearest to thecoils as shown in Figure(3.8a) for ~E(0, y, 38mm). Conversely, further from thecoils the field ~E(38mm, y, 0) is reasonably uniform with maximal deviationsO(20V/cm) over the bulk of the drift axis. Non-linearities near hoop #1 (y =75.5 cm) and the rMCP (y = 97 mm) resulted from non-optimal voltage settingsin the experiment which are shown in Figure(3.8b). Naturally, the field non-linearities along the y-axis are further suppressed where ~E(0, y, 0) deviationsare maximally O(10V/cm) over the entire ion drift length.Due to the ~E-field non-linearities, corrections must be applied event-by-eventto map the observed hit position X′ = (x′, z′, t′) to the ideal hit position X =(x, z, t) had the field been perfectly uniform at Eo = 998.5 V/cm along the +y-axis. COMSOL was used to simulate the recoiling 92Sr daughter both in thetrue non-uniform ~E-field and the uniform field, thus comparing the hit positionX’ and X, respectively given identical initial recoil momentum. In this way wemap a given recoil ion event from the data X′ in the non-uniform field into thecoordinate space of the same event X in a uniform field to easily extract the truerecoil kinematics of the decay.In β−-decay, nearly 80% of recoils are in the +1 charge state, having emitted noextra atomic SOE. The abrupt change in the parent nuclei and atomic orbitals,however, can kick out valance electrons yielding +2 and higher recoil chargestates. The COMSOL-generated arrays for each charge state consist of squarehit positions {Xij = (xi,j, zi,j, ti,j}i with position index i = 1, 2, 3, 4 (spaced by4 mm) in a uniform field at time index ti,j = tj (every 4 ns) and correspond-ing hit positions {X′i,j = (x′i,j, z′i,j, t′i,j)}i in non-uniform field as shown in Fig-ure(3.9). Recoil events X′ = (x′, z′, t′) contained within the volume defined byChapter 3. Experiment 24a)b)Figure 3.8: a) COMSOL simulated ~E-field component values as a function of displacementfrom the trap center in the y-axis (rMCP at y = +97 mm) for given translational offsets inx, and z.〈Ey〉is the simulated mean field strength of 998.5 V/cm along the y-axis. b)Expanded view of minimally deviating field component profiles.{X′i,j, X′i,j+1}i passing our selection criteria are then used to compute the set ofcoordinates {X′i = (x′i, z′i, t′)}i at time t′ where for α′i,j = (t′ − t′i,j)/(t′i,j+1 − t′i,j)it follows thatx′i = x′i,j + α′i,j · (x′i,j+1 − x′i,j), z′i = z′i,j + α′i,j · (z′i,j+1 − z′i,j). (3.2)The Fourpt algorithm above was then implemented to map (x′, z′)→ (x, z) andt′ → t where for〈α′i,j〉i= ∑ 4i=1 α′i,j/4t ' tj +〈α′i,j〉i(tj+1 − tj). (3.3)The recoil event coordinate transformation will be demonstrated later whendiscussing the experimental results. It should be noted that in both a uniformand nonuniform ~E-field the transformation from hit coordinate X, X′ to theirrespective recoil momentum coordinates is one-to-one.Chapter 3. Experiment 25Figure 3.9: Mapping recoil hit coordinates in non-uniform ~E-field X′ to coordinates X inuniform field via COMSOL-based mapping of simulated recoil events. a) Simulated re-coil hit coordinate space unit cell in the non-uniform field (S′), and uniform field (S). b)Illustrated Fourpt stretching transformation mapping recoil hit coordinate (x′, z′)→ (x, z).3.5 Recoil Kinetic Energy and MomentaThe recoiling nucleus is non-relativistic, due to its mass Mr compared to thoseof the outgoing beta and neutrino, and thus classical kinematics can be appliedin determining the recoil kinetic energy (Kr). Assuming a uniform collection~E = Eyyˆ field, which is necessarily the case for a recoil hit in the transformedcoordinate space X = (∆x,∆z, TOF), the kinetic energy Kr for charge state q isKr =Mr2[r2TOF2+(lyTOF− qEyTOF2Mr)2]. (3.4)Here, r =√(∆x)2 + (∆z)2 and ly is the distance from the trap to the rMCP frontface (TOF → c · TOF for expression with dimensional consistency). The recoilmomentum ~pr = (prx, pry, prz) is also simply parameterized asprx = Mr · ∆xTOF , prz = Mr ·∆zTOF, and pry =MrlyTOF− 12qEyTOF. (3.5)It is important to note that following the beta decay of 92Rb the recoiling 92Sr isnon-resonant with the MOT trap beams and necessarily is unperturbed by thelaser light of the trap and its associated magneto-optic forces.Chapter 3. Experiment 263.6 Scintillator Energy CalibrationFigure 3.10: Top and bottom scintillator calibration using a combination of gamma-rayCompton edges and β endpoints. A significant non-linear response in the lower scintillatorcan be traced to a defective PMT base. Additionally, the 3.5 cm thick scintillators are thinenough that a small fraction of the most energetic beta’s can leave the volume before de-positing their full energy, or generate bremsstrahlung photons which escape detection. Theresult is less charge deposition at the largest beta energies and this may be another sourceof non-linearity in the detector response.Scintillator media are nominally designed to be linear with the number of pho-tons generated by energetic particles scaling with the energy deposited withinits volume. Similarly, photo-multiplier amplification systems used to collectthese photons and control electronics are also designed to have linear responses.The coupled scintillators and electronic systems, however, require calibration tobuild maps between the integrated charge reported from the photo-multiplierand a calibrant of known energy. Additionally, any physical system will havenon-linearities at some level, which need to be accounted for to obtain an ac-curate calibration. The scintillator calibration performed here utilized a varietyof measurements from γ-ray Compton edges, along with β-endpoint energiesand cosmic muon events. When γ-ray photons of energy Eγ scatter off atomicelectrons in the scintillator its final state energy E′γ depends on the scatteringangle θ,E′γ =Eγ1+ Eγme (1− cos θ). (3.6)Chapter 3. Experiment 27Table 3.2: Scintillator calibration parameters from fits to calibrants in Figure(3.10) using thesaturation equation(3.8).Scintillator I0 [ch.] I1 [ch.] E1 [keV] δE1 [keV]UPMT -2536.89 6138.26 1290.61 4873.30LPMT -6185.41 9241.54 -3782.23 4632.66The maximum energy ECompt deposited in the scintillator occurs for completeback-scatter events (θ = 180◦)ECompt = Eγ − E′γ =2E22E+me(3.7)manifesting as an abrupt edge in the scintillator energy spectra due to energydeposition above this being kinematicly forbidden. Several γ-ray sources wereheld adjacent to the scintillator including 133Ba (356,302,384.0 keV with rel-ative intensity 0.62, 0.18, 0.089), 22Na (511anni, 1274 keV), 137Cs (661.7 keV),92Rb (814.98 keV), 60Co (1173, 1332 keV), 40K (1461 keV), 208Pb (2615 keV),92Sr (1383.93 keV) 92Y (3643 keV) providing calibration points as shown in Fig-ure(3.10) dominantly below 2.5 MeV.Ben Fenker (PhD-Thesis) performed GEANT-4 simulations of cosmic muon in-teracting in our scintillators noting a minimum energy deposition of 5600,5500keV in the top, and bottom detectors with the most probable energy loss of 6300keV. Cosmic muons are identifiable in our geometry by coincident hits in thetop and bottom scintillators in the absence of recoil and SOE events with re-sults shown in Figure(3.10). Additional calibration points were needed above2.5 MeV to better constrain our calibration requiring us to use both the 92Y and92Rb endpoints at 3643 and 8094 keV.An important observation in our calibrations shown in Figure(3.10) was thatboth the upper UPMT (red) and lower LPMT (blue) scintillators appeared tohave notable non-linearities. Although the scintillator and photo-tube in gen-eral have largely linear responses with energy deposition, non-linearities in theresponse can be present in general at some level. Although the upper scintilla-tor (UPMT) appears largely linear, an apparent non-linearity in the detector re-sponse appears around 8 MeV. In the LPMT the non-linearity was significantlyworse. The LPMT was noted to require significantly lower bias to the dynodeChapter 3. Experiment 28stack with only 1350 V (compared to the nominal 2100 V in previous experi-ments) so as not to saturate its response and ensure the 8.1 MeV 92Rb endpointremained on our QDC scale. We have attempted to phenomenologically modelthe non-linear response in both the LPMT and UPMT assuming a saturationresponse, which we parameterize in QDC-channel I vs. EI(E) = I0 +I11+ e−(E−E1)/δE1. (3.8)and fit to the data in Figure(3.10) from the respective detectors to provide ourdetector calibration. E1 represents the saturation energy scale with character-istic width δE1 and saturation value I0 + I1. The calibration parameters fromequation(3.8) are shown in table(3.2).Lower Scint. Energy [MeV]0 1 2 3 4 5 6 7 8 9 10Upper Scint. Energy [MeV]012345678910110210Scint. Energy (Cosmics) [MeV]4 5 6 7 8 9 10Counts/200 keV0102030405060UPMTLPMT TOF [ns]cosmics( UPMT - LPMT )30− 20− 10− 0 10 20 30Counts/1ns050100150200250a) b) c)Figure 3.11: a) UPMT-LPMT coincident events in the absence of recoil’s and SOE’s withcosmic muon in the dashed region having corresponding energy spectrum projections in b)and relative timing difference in c), respectively.With our scintillators calibrated, cosmic muon coincidences are shown in thecorrelated top vs. bottom scintillator energy in Figure(3.11a) with energy pro-jections within the dashed lines in b) and timing difference in c). The timingdifference between the upper and lower scintillators is defined by(UPMT - LPMT) TOF = (UPMT[0] - LPMT[0]) - (∆UPMT - ∆LPMT) - δUPMTLPMTwith measured and imposed timing offsets defined in Table(3.1). The 2 ns leadon the UPMT timing signal compared to the LPMT timing signal is consistentwith the transit time of a relativistic muon punching through both scintillators.Chapter 3. Experiment 293.7 ν¯e Energy Spectrum and Qexp-value ReconstructionSince the 92Rb is nominally highly localized within the trap and cooled to mKtemperatures, the initial state of the parent atom can be considered at rest priorto decay. With the initial recoil momentum ~pr established above and the ~pβmomentum given by√E2β −m2, momentum conservation ~pr + ~pν + ~pβ = 0 isinvoked to reconstruct Eν (assuming negligible rest mass)Eν =√p2r + p2β + 2 · prpβ cos θβr (3.9)where θβr is the β-recoil angle. In the final analysis the DSSD strip detector,along with the absolute trap position and known pr will be used to determineθβr to properly reconstruct Eν. Due to the significant complexity of calibratingthe DSSD strip-by-strip against GEANT4 simulations and building the analysiscode to reject false events, a simplified analysis was performed where we as-sume the beta enters the upper/lower scintillator parallel to the±z-axis, respec-tively. If θr is the angle made between ~pr and the y-axis and φr the azimuthalangle from the x-axis in the xz-plane, then for the initial recoil momenta unitvector (xˆ, zˆ, yˆ) = (sin θr cos φr, sin θr sin φr, cos θr)cos θβr = sin φr ·√1− cos2 θr (3.10)where sin φr = ∆z/r, and cos θr = δ/√r2 + δ2, for δ = ly − 12 ·qEyMr · TOF2.The Q-value of the β-decay transition accounts for the total change in kineticenergy between final and initial states, with rest mass M f and Mi such thatQ = Mi −M f . We define the effective experimental Qexp-value here as the sumover beta, neutrino, and recoil kinetic energyQexp = Eβ + Eν + Kr −m. (3.11)In the GS transition Qexp = Q, while in the decay through excited states Qexp <Q, as the energy carried off by the gamma cascade is not measured. In the caseof 92Rb, the strong GS transition will thus produce a resonance in Qexp at the8.1 MeV Q-value of the decay. In the case of decays to excited states of thedaughter, however, the gamma-cascade carry off a significant fraction of theavailable energy, while also perturbing the recoil momentum and incorrectlyreconstruct Eν, yielding a broad distribution of events with Qexp < Q.Chapter 3. Experiment 303.8 Photo-IonsrMCP x-Pos [mm]5− 4− 3− 2− 1− 0 1 2 3 4 5rMCP z-Pos [mm]5−4−3−2−1−01234505101520253035rMCP x-Position [mm]5− 4− 3−2− 1− 01 2 34 5rMCP z-Position [mm]5−4−3−2−1−012345Counts / 5 min0510152025303540Figure 3.12: Photo-Ion event distribution fit with superposition of three Gaussian ellipsoidsper 5 min of acquisition to check for small time dependent trap drift in the xz-plane.Photo ions are used to monitor the time dependent trap drift, since constrainingthe initial position of the decay is essential to reconstruct the initial recoil mo-menta. A diode pumped 355 nm UV pulsed (0.5 ns) laser is used to singly ionizea subset of the trapped atoms. The rep rate is 10 kHz with 100 µs between pulsesand essentially negligible proportion of coincident beta decay events with PDtrigger. The UV beam enters the vacuum through optical ports at 35◦ to thez-axis in the xz-plane, and is retro-reflected from an external mirror back alongthe incident path. Photo-ion (PI) events are selected from coincident PD-eMCP-rMCP event triggers with 5 min of integration shown in Figure(3.12). ThreeGaussian ellipsoids were fit simultaneously to the xz-distribution to model theinner (I) strongly trapped population , and outer (O1,O2) halo-type populationsdistributed along the x, z-axis, respectivelyNo + ∑k=I,O1,O2Ak · exp[−(x′ − x′k√2wxk)2−(z′ − z′k√2wzk)2](3.12)where (Ak) are the amplitudes, (x′k, z′k) the centroids and (w′k,w′k) the widths ofthe k = I,O1,O2 fit distributions. The inner distribution is defined as that withthe largest amplitude, while the outer halo distributions have the two smallestamplitudes. The three distinct distributions of atoms may be connected withthe recent replacement of the vertical trapping mirrors (along z-axis) with Au-coated pellicle mirrors. Visual observation of the beam profile upon reflectionChapter 3. Experiment 31from the pellicle appeared uniform within 10 cm of the surface, but had no-table intensity non-uniformities at 1m from the surface likely from thicknessnonlinearities of the supporting Kapton membrane. Combined thickness non-linearities were measured by National Photo Color to be 5λ (λ = 770 nm) overa 2 cm beam diameter, while a typical optical mirror on a solid substrate hasλ/10 flatness. Intensity nonlinearities over the trapping region may perturb thetrap population.Cumulative Photo-Ion Run Time [hr]0 2 4 6 8 10 12Amplitude [Counts/5 min]5−051015202530Cumulative Photo-Ion Run Time [hr]0 2 4 6 8 10 12Amplitude [Counts/5 min]02468Cumulative Photo-Ion Run Time [hr]0 2 4 6 8 10 12Amplitude [Counts/5 min]1−01234567Cumulative Photo-Ion Run Time [hr]0 2 4 6 8 10 12rMCP x-Pos [mm]00.20.40.60.811.21.4Cumulative Photo-Ion Run Time [hr]0 2 4 6 8 10 12rMCP x-Pos [mm]00.511.522.53Cumulative Photo-Ion Run Time [hr]0 2 4 6 8 10 12rMCP x-Pos [mm]00.20.40.60.811.21.41.61.822.22.4Cumulative Photo-Ion Run Time [hr]0 2 4 6 8 10 12rMCP z-Pos [mm]1−0.8−0.6−0.4−0.2−00.20.4Cumulative Photo-Ion Run Time [hr]0 2 4 6 8 10 12rMCP z-Pos [mm]1.5−1−0.5−00.511.5Cumulative Photo-Ion Run Time [hr]0 2 4 6 8 10 12rMCP z-Pos [mm]2−1−0123a) b) c)d) e) f)g) h) i)Figure 3.13: Photo-Ion distribution a),b),c) amplitude, d),e),f) x-centroid and width(coloured bounds), and g),h),i) z-centroid and width (coloured bounds) of inner I(red), andouter O1 (blue), O2 (green) elliptical-Gaussian fit parameters, respectively, vs. cumulativerun time of the 1 kV/cm dataset.The cumulative time dependent fit parameters of equation(3.12) to the photo-ion distribution per 5 min exposure are shown in Figure(3.13) with distribu-tion (a,b,c) amplitudes, centroid (d,e,f) x-component (g,h,i) and z-component(widths being the colored bounds) are shown for the I(red), O1(blue), O2(green)distributions, respectively. Measurable time dependent deviation can be seenin all these parameters particularly near the 10 hr mark when we lost the opti-mal laser-lock, and could not be re-established optimally before the end of datataking. Since we cannot distinguish if a decay originates from either the in-ner or outer distributions we establish the trap centroid as a weighted averageof the centroids by their numerically integrated inner (NI), and outer (NOi) fitChapter 3. Experiment 32distribution populations with trap centroid defined as~rIO = [NI~rI + NO1~rO1 + NO2~rO2 ]/[NI + NO1 + NO2 ]. (3.13)Nominally, the I/O1/O2 trap was displaced transversely from the geometriccenter with the width of the inner distribution largely stable with width wxI/wzIof 0.30/0.30 mm, even after the laser miss-tune at the 10 hr mark. The outerpopulations were largely stable in width but appeared to have an invertedresponse following the 10 hr mark with the wxO1/wzO1 expanding/shrinkingwhile wxO2/wzO2 shrinking/expanding. The distribution amplitudes also un-dergo notable reduction following the 10 hr mark.The integrated drift corrected PI event distribution in xz-plane is shown in Fig-ure(3.14). Due to the large sample set, the broad outer halo about the trap ori-ented along the projected MOT beam axes (z, and x-axis) are evident, parame-terized by O1, and O2 distributions, respectively.(eMCP - pd) TOF [ns]20 22 24 26 28 30 32 34 36 38 40 Counts/ 5 min050100150200250300350400Cumulative Photo-Ion Run Time [hr]0 2 4 6 8 10 12eMCP-pd TOF [ns]3030.53131.53232.53333.5(rMCP - eMCP) TOF [ns]1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410 Counts / 5 min050100150200250300350Cumulative Photo-Ion Run Time [hr]0 2 4 6 8 10 12rMCP-eMCP TOF-Pos [ns]13551360136513701375TRAP y-Pos [mm]92 93 94 95 96 97 98 99 100 101 102 Counts / 5 min050100150200250Cumulative Photo-Ion Run Time [hr]0 2 4 6 8 10 12rMCP-eMCP y-Pos [mm]9696.59797.59898.599a)b)c)d)e)f)Figure 3.14: Photo-Ion a) SOE-PD TOF, b) Recoil-SOE TOF, c) y-displacement from trapcentroid examples per 5 min of acquisition with overlaid Gaussian fits. Cumulative runtime dependent Gaussian centroid (black-markers) and width (red-bounds) per 5 min ofacquisition are displayed in d), e), and f), for the respective coincidence.Chapter 3. Experiment 33In the TOF dimension the 1d distributions for the photo-electrons, and photo-ions are shown in Figure(3.14a,b), respectively. The respective cumulative run-time TOF centroids (markers), and widths (red bounds) are shown in Figure(3.14d,e) similarly demonstrate a largely stable trap centroid along the y-axiswith the exception of following loss of an optimal laser lock around the 10 hrmark. The anti-correlation in TOF of the photo-electrons and photo-ions is con-sistent with the trap moving further from the rMCP at the 10 hr mark. Giventhe photo-ions are singularly charged and the ~E-field is known, the ion driftlength ly from the rMCP is determined with projection shown in Figure(3.14c)with nominal Gaussian centroid of 97.5 mm and width of 0.6 mm. Similarly, thecumulative run-time distribution centroid (marker), and width (red bounds)along the y-axis from the rMCP is shown in Figure(3.14f). The trap width andcentroid along the y-axis were again largely stable with the exception of at the 10hr mark, with the trap moving roughly 0.1 mm away from the rMCP. The drift-corrected photo-ion distribution integrated over the entire dataset is shown inFigure(A.2), clearly showing the three distinct populations of atoms in the xz-plane as mentioned above with indicated distribution centroid clearly offsetfrom inner I trap population by the populations in the outer halo region.3.9 Data AcquisitionThe TRINAT DAQ hardware event timing schematic is outlined in Figure(A.1).The ∆E strip detectors are not shown as they have not been used in this analysisas of yet. Timing signals from the MCP’s (eMCP,rMCP) are picked off the frontface of the MCP HV through a preamp and fed into a Constant Fraction (CF)discriminator with the output timing sent to the acquisition. The timing signalsfrom the scintillator PMT’s (UPMT,LPMT) are taken from the first dynode in thephoto tubes and sent to the CF discriminators with outputs sent to the acquisi-tion. Multiple triggers were available to simultaneously fire the DAQ includingA(PMT-singles), C(PMT - eMCP), D(PD - eMCP), E(PD - rMCP), and F(eMCP- rMCP). Trigger A, E, and F were enabled for the 1kV/cm data. Although the+1 charge state in beta decay is the dominant recoil charge state with no atomicSOE, unfortunately the ~E-field was not large enough to collect all recoils in 4pi,and necessarily we have avoided this component of the event stream (triggerA). Since all events of interest in this analysis are above the +1 charge state aSOE trigger from the eMCP is required with double coincident (trigger F) andChapter 3. Experiment 34triple coincident (trigger A and F ) event streams presented in this thesis. Inprinciple, if one was not interested in +1 charge states, enabling trigger C andF (instead of A and F) would cut down the dead time in the DAQ from thedominant beta-singles event type and increase data rates in the higher chargestates.35Chapter 4Experimental Results4.1 Recoil-SOE Coincidence4.1.1 TOF SpectraRecoil-SOE coincident events directly give the recoil momentum from which anaverage value of ξo/ω can be obtained assuming the a1(W) correlation param-eter defined in section 2.4. This represents the main experimental measurementof this thesis. Recoil-SOE coincident events represent an ideal event stream tomaximize statistics, since with sufficiently large ~E-field all recoil charge statescan be collected in 4pi. The distribution of drift corrected impact radius vs TOFis shown in Figure(4.1a) where the +2, 3, 4, and higher charge states can be seenpartially separated in TOF. Gating on events in TOF within the (red), (blue), and(green) regions in Figure(4.1a), we can uniquely separate the +2(red), 3(blue), and4(green) charge states subsets with the indicated 2D rMCP hit position shownin Figure(4.1b). The complication of not having an additional detector coinci-dence, however, leads to non-negligible false coincidence event distributionsthat must be understood to extract physics.Time-random false coincidences can be assessed as nearly uniformly distributedrecoil events in TOF over the rMCP detector surface. Deviations from such uni-formity in TOF, however, can be seen, particularly near 1000 ns in Figure(4.1).This is more obvious in the radially integrated events of Figure(4.1) within the(teal) bounds, with the results shown in Figure(4.2). A resonance in the recoilTOF background indicates decays from a localized planar surface along the y-axis is contributing to the false coincidence event rate, and this is consistentwith an accumulation of 92Rb (and its progeny 92Sr, 92Y) on the HV electrostatichoops, particularly those nearest to the trap. Decays from the central hoops(#5,6 from rMCP) interior surface facing the y-axis can yield a recoil-SOE whichChapter 4. Experimental Results 36a) b)Figure 4.1: Recoil-SOE coincident a) drift corrected ion impact radius vs. TOF, and b) un-corrected recoil hit xz-position for +2(red), 3(blue), and 4(green) charge state events in respec-tively coloured TOF-bounds in (a).may be ejected from the surface and collected in the respective detector, re-sulting in two TOF peaks consistent with those at (975,1025) ns, respectively.Decays from the hoops will preferentially occur from the face nearest transla-tionally to the rMCP, namely the surfaces along the ±z-axis compared to the±x-axis. The rectangular geometry of the HV hoops will necessarily be im-printed on the azimuthally symmetric distribution (about the y-axis) of decaysfrom the trap, and will be discussed later.(rMCP - eMCP) TOF [ns]800 1000 1200 1400 1600 1800 2000 2200Counts051015202530Figure 4.2: Recoil-SOE coincident random background levels vs. TOF integrated over thebounding (teal) lines in Figure(4.1). Fits within the charge state bounds +2(red), 3(blue),4(green) provide a random background rate normalization relative to signal free region(grey) from which time random backgrounds are estimated. The location of the TOF peaksare understood to be real backgrounds and discussed later.Chapter 4. Experimental Results 374.1.2 ~E-Field CorrectionsThe technical details of this important correction were discussed in section 3.6.Mapping the recoil-ion hit coordinates X′ to X was important to correctly recon-struct the recoil Kr across charge states. This is because each charge state sam-ples different physical regions of the non-uniform ~E field, so each one requiresdiffering degrees of correction. With the TOF isolation of the +2(red), 3(blue),4(green) recoil charge states in rMCP-SOE coincidence of Figure(4.1b) a subsetof the recoils which traversed the non-uniform ~E-field have hit coordinates X′plotted in Figure(4.3a,b,c) (black), respectively. Overlaid are the transformed re-coil hit coordinate distributions X had the recoil traversed a uniform ~E-field of998.5 V/cm for the respectively coloured +2(red), 3(blue), 4(green) charge states.The subset of the recoils with largest initial momenta transverse to the y-axissample the fields closer to the hoops, where non-uniformities are larger, andnecessarily require larger corrections in mapping X′ to X, particularly in the +2charge state. The transverse distribution of corrections applied to each chargestate in mapping X′ to X are shown in Figure(4.3d,e,f), respectively. Two polesappear in the corrections in the ±z-axis from the proximity of ground to thecentral region of the hoops along the y-axis (where perturbations the recoil tra-jectory are small) compared to the ±x-axis where ground is much further fromthe central region.4.1.3 rMCP Pulse-Height & Detector EfficiencyIt is well known that MCP have ion impact angle and energy dependent quantum-efficiency for secondary electron emission, with respect to the MCP channelaxis. Generally, the larger the channel angle the smaller these effects can bemade, which is why the large θCH = 20◦ channel pitch angle to normal waschosen for this experiment. Below we detail the qualitative pulse height depen-dencies on our kinematic observables prx, prz,Kr, φr, and the angle to the MCPchannel axis θch. Later we conclude the most practical technique to ascertain therelative efficiency mapped into the recoil Kr observable is through a left/right,and up/down counting rate asymmetry, which had non-negligible Kr depen-dencies, but maximally were on the order 5% in the smallest +2 charge state.Recoil event hardware triggers of the rMCP constant fraction discriminator(CFD) operated at a minimal detection threshold of 10 mV imply that pulsesbelow this would not fire our DAQ. The Z-stack rMCP with fields of 1 kV/mmChapter 4. Experimental Results 38a) b) c)d) e) f)Figure 4.3: Drift corrected recoil xz-coordinate displacement distribution (X) followingmapping above in Figure(3.9) X′ → X (mean field of 〈Ey〉 = 998.5 V/cm) of a) +2(red),b) +3(blue), and c) +4(green) charge states within TOF bounds in Figure(4.1a). The corre-sponding un-corrected recoil event distribution (X′) are shown as black points in the back-ground for each of the respective charge states. The corresponding distribution of recoil hitxz-displacement corrections in mapping from X′ → X are shown in d),e),f), respectively.has the property that their response is largely saturated and independent ofrecoil impact energy, position, or orientation with respect to incident micro-channel axis. Figure(4.4a,b,c) demonstrate the mean recoil MCP pulse height asa function of transverse momentum for charge states +2, 3, 4, respectively. Eachrecoil charge state demonstrate a clear deficit in the mean pulse height for smalltransverse momentum. This may result from the accumulation of 92Rb and theprogeny near the center of the rMCP, which beta decays in 4pi, half of whichmay penetrate multiple MCP microchannels and fire the rMCP continuouslyover the run-time. Continuous firing of the central population of microchan-nels may thus locally degrade the gain over the run-time compared to the pe-ripheral regions of the detector where the progeny is distributed over a largersurface area. In the +2 charge state several locations in transverse momentumreveal pulse height deficits (eg. [-5,7] MeV/c) in the mean pulse height likelyfrom sparks, which have locally damaged the MCP micro-channels reducingChapter 4. Experimental Results 39their amplification. The pulse height distributions as a function of transversemomentum in prx are shown in Figure(4.4d,e,f), and prz in Figure(4.4g,h,i). Athreshold of 50 channel units was imposed to suppress false triggers. No strongpulse height dependencies are immediately obvious in the transverse momen-tum distributions.a) b) c)d) e) f)g) h) i)Figure 4.4: Charge state +2, 3, 4 (left-right) a), b), c) mean rMCP pulse height dependenceon transverse recoil momentum with, d), e), f) pulse height vs. prx, and g), h), i) pulseheight vs. prz distributions, respectively.Pulse height recoil kinetic energy dependencies of the +2, 3, 4 charge statesare shown in Figure(4.5a,b,c), respectively. Our analysis of the pulse heightevent dependency below will be restricted to kinematic regions where the az-imuthal angular distribution φr in transverse recoil momenta are contained in[−45◦, 45◦] and [135◦, 225◦] to suppress hoop backgrounds as discussed above.Overlaid are the mean pulse height values as a function of recoil Kr. The +4charge state has a smaller dependency on Kr, as the impinging ions are closer tonormal incidence than the lower charge states, which nominally sample largerdetector radii. A simple linear fit was made to the mean pulse height distribu-tion of the +4 state and overlaid in red, which was used to normalize the meanpulse heights of the +2, 3, 4 charge states, with results shown in Figure(4.5d,e,f)for the respective charge states. A clear enhancement in the mean pulse heightChapter 4. Experimental Results 40of the +2 charge state is observed, which is roughly linear in Kr up to 240 eVand saturates with a 10% enhancement. Since the +2 charge states sample largerradii, the spread in incident angle to the micro-channel axis is necessarily larger,and so more likely to impact with both smaller (and larger) penetration depthsleading to larger (and smaller) pulse height extremes. We defer discussion ofrelative rMCP detector efficiency to later in this section.a) b) c)d) e) f)Figure 4.5: rMCP pulse height distribution vs. recoil kinetic energy Kr of charge state +2,3, 4 (left-right) a), b), c) charge states with overlaid mean pulse height. Normalized meanpulse height to the linear fit of that of the +4 charge state vs Kr are are shown in d), e), f) forthe respective charge state.Assuming the hexagonal array of MCP channels is on average uniform acrossthe rMCP surface, we can examine dependencies both on azimuthal angle andimpact angle with respect to the channel axis. The impact angle with respectto the channel axis θch is computed from the known recoil impact momentumparameterized as ~pHITr = (pHITrx , pHITry , pHITrz ) where in a uniform ~E-fieldpHITrx = prx , pHITrz = prz , pHITry =MrlyTOF+12qEyTOF , (4.1)it follows thatθch = arccos([pHITrx sin θCH + pHITry cos θCH]/|~pHITr |). (4.2)Chapter 4. Experimental Results 41The pulse height dependence on the incident angle to the rMCP channel axisis better illustrated in Figure(4.6a,b,c) with overlaid mean values for the +2, 3,4 charge states, respectively. The overlaid mean pulse heights are weightedtoward smaller values near 0◦ for ions incident along the channel axis, whilelarger values are nominally obtained off the micro-channel axis. The maximalspread in θch is also clearly reduced with increasing charge state from±9◦, 7◦, 6◦for +2, 3, 4 charge states, respectively, as the recoil ions impinge on the MCPcloser to normal incidence. The azimuthal angular pulse height dependence isshown in Figure(4.6d,e,f) with no strong dependencies over the accepted eventrange. Given the mean micro-channel angle/orientation, nominal diameter andrecoil momentum the ion penetration depth dch can be computed simply asdch = DCH/ tan θch (4.3)with results shown in Figure(4.6g,h,i) for charge states +2, 3, 4, respectively.Due to the orientation of the micro-channels at θCH, the recoils will impinge onthe channels and achieve differing penetration depths for prx > 0 or prx < 0,potentially leading to a pulse height asymmetry and possibly a bias in ourdetection efficiency across the MCP surface. The mean pulse height distribu-tion is overlaid in Figure(4.6g,h,i) for prx < 0 in blue, and prx > 0 in red foreach charge state. Larger pulse heights are correlated with smaller penetra-tion depths accomplished for prx < 0, compared to the more distributed pulseheight distribution for prx > 0, which is evident in each charge state. This isconsistent with the channels being oriented with θCH = −20◦ in the xy-planeof the chosen coordinate system. Additionally, for prx < 0 there is a roughlylinear drop in mean pulse height with penetration depth into the channel upto 65 µm, while for prx > 0 a flatter distribution is achieved. Again, since thelarger charge states nominally impinge on the MCP closer to normal incidence,a smaller maximum penetration depth is achieved, where for +2, 3, 4 the max-imum penetration depths are 130, 110, 100 µm. From the above information itis unlikely that the MCP is fully saturated in its response, although there is noobvious translation of the above into a rMCP detection efficiency that dependson recoil energy Kr.We parameterize the relative MCP detector efficiency by investigating the left/rightand up/down asymmetry in the recoil momentum distribution mapped intothe recoil Kr observable. Plotted in Figure(4.7a,b,c) are the recoil Kr distribu-tion’s for hemispherical cuts in the transverse momentum space for the +2, 3,Chapter 4. Experimental Results 42a) b) c)d) e) f)g) h) i)Figure 4.6: rMCP pulse height dependencies for charge states +2, 3, 4 (left-right) on a),b),c)angle to rMCP channel axis, d),e),f) azimuthal angle in the plane of the rMCP, g),h),i) maxi-mum rMCP channel penetration depth for (prx < 0 black-line), and (prx > 0 red-line).4 charge states, respectively. Recoil Kr spectra were extracted for prx ≥ 0, andprx < 0 in (solid-teal), and (dashed-teal) lines, respectively and similarly for przindicated in (magenta). Asymmetries in the event rate x, and z-axis are com-puted as the ratioseLR(Kr) = 1− dN/dKr(prx ≥ 0,Kr)− dN/dKr(prx < 0,Kr)dN/dKr(prx ≥ 0,Kr) + dN/dKr(prx < 0,Kr)eUD(Kr) = 1− dN/dKr(prz ≥ 0,Kr)− dN/dKr(prz < 0,Kr)dN/dKr(prz ≥ 0,Kr) + dN/dKr(prz < 0,Kr)(4.4)with results displayed in Figure(4.7d,e,f) for each charge states +2, 3, 4, respec-tively. The distributions were fit assuming a linear dependence convolved withan exponential loss in efficiency near the largest recoil Kr. For the +2 chargestate a notable linear reduction in relative efficiency with maximal departuresof order 5% at largest recoil Kr in both the left/right and up/down orientationscan be observed. Such non-linearities are less significant in the larger chargestates consistent with departures from uniform efficiency likely from several ofChapter 4. Experimental Results 43the convolved dependencies mentioned above.Recoil Kinetic Energy [eV]0 100 200 300 400 500Counts20040060080010001200 >= 0rx+2: P <  0rx+2: P >= 0rz+2: P <  0rz+2: PRecoil Kinetic Energy [eV]0 100 200 300 400 500Counts50100150200250300350400 >= 0rx+3: P <  0rx+3: P >= 0rz+3: P <  0rz+3: PRecoil Kinetic Energy [eV]0 100 200 300 400 500Counts50100150200250 >= 0rx+4: P <  0rx+4: P >= 0rz+4: P <  0rz+4: PRecoil Kinetic Energy [eV]0 100 200 300 400 500 [a.u.]rMCP∈00.20.40.60.811.2rx+2: Prz+2: PRecoil Kinetic Energy [eV]0 100 200 300 400 500 [a.u.]rMCP∈00.20.40.60.811.2rx+3: Prz+3: PRecoil Kinetic Energy [eV]0 100 200 300 400 500 [a.u.]rMCP∈00.20.40.60.811.2rx+4: Prz+4: Pa) b) c)d) e) f)Figure 4.7: Recoil kinetic energy Kr spectrum integrated over indicated detector hemi-sphere for charge states +2, 3, 4 in a),b),c), respectively, with (teal) corresponding theleft/right and (magenta) up/down hemispheres. The corresponding Left/Right (LR) andUp/Down (UD) counting rate asymmetries parameterize the relative rMCP detection effi-ciency vs. recoil Kr dependence is shown in d),e),f) with overlaid fit functions parameteriz-ing the relative efficiency eLR(Kr), eUD(Kr).4.1.4 rMCP Detector Response FunctionThe presence of the calibration mask on the rMCP mentioned above in section3.5 introduces significant non-linearities in the recoil Kr spectrum comparedto the theoretical predictions from equation(2.8) above, and this is parameter-ized by a mask transmission function em(Kr) shown in Figure(4.8), via a simpleMonte-Carlo simulation. Additionally, the finite counting and event densityat small recoil radius, and thus small Kr, further introduces non-linearities inthis distribution and is parameterized by e f d(Kr). These detector effects will beused to scale the theoretical predictions of equation(2.8) as will be shown belowChapter 4. Experimental Results 44when fitting experimental recoil Kr spectra in section 4.1.6.a) b)c) d)Figure 4.8: Simple MC of recoil ion in recoil-SOE coincidence assuming recoils with uniformrandomly distributed momenta which traverse a uniform ~E-field of 998.5yˆ V/cm withinthe given charge state TOF bounds in Figure(4.1a). The recoil +2 charge state passing thesecriteria and within the active area of the rMCP have transverse momentum distributionshown in a), and impact hit position distribution shown in b). Azimuthally integratedradial mask transmission efficiency (markers) and finite-density relative event rate effects(lines) of the +2(red) charge state are shown in c). Integrated mask transmission efficiencyem(Kr) (markers) and finite-density event rate efficiency e f d(Kr) (lines) vs. recoil Kr of the+2(red), 3(blue), 4(green) charge states, respectively are shown in d).To quantify em(Kr) and e f d(Kr), simulated recoils were generated from a point-like trap centered on the experimentally determined photo-ion (PI) distributioncentroid from Section 3.10, with uniform-random momentum in 4pi up to 10MeV/c for each charge state. The simulated recoils traverse a uniform ~E-fieldChapter 4. Experimental Results 45of 998.5 V/cm and are incident on a simulated mask/rMCP detector with ac-cepted events in the requisite charge state TOF bounds. The integrated trans-verse momentum distribution, and rMCP xz-displacement distributions cen-tered on the PI distribution for the +2 charge state are shown in Figure(4.8a,b),respectively. As the trap drifts slightly in the xz-plane over the course of theexperimental run-time (Section 3.10) the region of the mask kinematically ac-cessible to the recoils correspondingly changes. The trap drift in effect smearsout the mask pattern imprinted on the integrated 2D momenta, and displace-ment distributions. The simulated mask transmission vs. impact radius of the+2(red) charge state is shown in Figure(4.8c) (markers) determined from the ra-tio of events incident on the mask/rMCP, and the rMCP alone with clear non-linearities from the mask geometry present above 5 mm in radius at nominaltransmission of ∼80%. The trap drift has the effect of smoothing out the ex-trema of the integrated mask transmission function vs. impact radius. Thefinite event density vs. impact radius is also overlaid (line), determined bynormalizing the radial recoil event distribution to the local detector circumfer-ence and clearly demonstrates the finite event density at small impact radius.The corresponding mask transmission function dependence on recoil kinetic en-ergy em(Kr) is similarly computed for +2(red), 3(blue), 4(green) charge states andshown in Figure(4.8d) (markers). Notable non-linearities appear above 20 eVin recoil Kr, which are naturally charge state dependent, as each samples differ-ent regions of the mask surface. Overlaid is the finite event density dependentdetector efficiency e f d(Kr) function (line) computed by normalizing the simu-lated recoil Kr distribution to the steady-state value above 300 eV. Given theform of equation(3.4) it follows that the recoil energy resolution per unit radiusδKr/δr ∝ (r/TOF2) is linear in hit radius. The inverse proportion to TOF2 im-plies the larger charge states with smaller TOF will have a stronger dependencythan a smaller charge state with a longer TOF as clearly demonstrated in thesimulated e f d(Kr).4.1.5 Background EstimationAs discussed above there were two dominant background processes in the recoil-SOE coincidence data stream: the first being from decays which occur uni-formly over the apparatus surfaces and throughout the vacuum along the y-axis so as not to produce TOF resonances IBckRnd(Kr); the second from decaysoriginating on localized surfaces along the y-axis producing TOF resonancesChapter 4. Experimental Results 46 [deg.]rφrMCP ion azimuthal angle 0 50 100 150 200 250 300 350Counts01002003004005006007008009001000+2+3+4Figure 4.9: Recoil ion azimuthal angular distribution of +2(red), 3(blue), 4(green) chargestates. Each charge state was fit with the superposition A+ B cos φr + C cos2 φr.as observed in Figure(4.2), namely from the electrostatic hoops IBckHoops(Kr). Bothof these distributions have characteristic dependence on the recoil Kr and mustbe modeled to match theory with the experimental distributions.The time-random background distribution IBckRnd(Kr) is modeled by selectingevents well outside of the signal region in TOF window ∆TOF, mapping themuniform randomly into the respective charge state TOF bounds (of width δTOF),then computing the recoil Kr spectrum from its hit coordinates X′ = (x′, z′, t′)as above. We then scale this distribution by the ratio of the total integrated in-tensity of time-random fit backgrounds in Figure(4.2) over δTOF, to that over∆TOF. In Figure(4.2) the fits chosen for charge states +3(blue), 4(green) werelinear, as the contribution from the hoops were indistinguishable from time-random distribution. Conversely, the +2(red) charge state had notable TOF res-onances within δTOF from decays originating on the central hoops #4,5 andmust be separated from the time-random background. To do this, two Gaus-sian’s and a step function (dashed) were fit simultaneously to the backgroundin Figure(4.2)(red-line) with the step function defining our time-random eventpopulation within the δTOF of the +2 charge state. With the time-random eventdistribution constrained for the +2 charge state the background recoil Kr wassimilarly obtained.Background events from the HV hoops are produced when the recoil and SOEare ejected from the surface of the electrode facing the trap and necessarily passChapter 4. Experimental Results 47our event triggers. The proximity of the interior hoop surface along the ±z-axis compared to the surfaces along the ±x-axis with the projected boundaryof the rMCP along these axes makes the former the dominant contributor tobackground events from the hoops. The consequence is an event excess in therecoil angular distribution (about the y-axis) at 90◦ and 270◦ as shown in Fig-ure(4.9) for the +2(red), 3(blue), 4(green) charge states, respectively. Althoughdecays from the hoops will be suppressed at smaller detector radii occupiedby the larger charge states, the smaller δTOF bounds near the peak in radius vs.TOF required to uniquely identify the charge state makes the hoop backgroundsmore significant, as can be seen in the angular distributions of Figure(4.9).The transverse recoil momentum distributions are shown in Figure(4.10a,b,c)for each of the respective charge states +2, 3, 4. The integrated recoil Kr spec-tra are shown in Figure(4.10d,e,f) over the azimuthally constrained momentaspace φr within [45◦, 135◦] and [225◦, 315◦] defining the convolved signal plushoop background spectra (magenta), while spectra from the remaining mo-menta space define our signal dominated region (teal) with their difference(black) estimating the hoop background intensity IBckHoops(Kr) for the respectivecharge state. In performing this difference to estimate IBckHoops(Kr) we are also bydefault removing the random background dependence IBckRnd(Kr). Overlaid arethe smoothed hoop background distributions for the +2(red), 3(blue), 4(green)charge states which are used in our analysis below. It will be shown below thatsuch estimates for both IBckRnd(Kr) and IBckHoops(Kr) provide consistent solutions forξo/ω, independently for each analyzed charge state.The theoretical recoil kinetic energy distribution dN/dKr in equation(2.8) mustthus be modified as dN˜/dKr to account for the above mentioned detection effi-ciency effects and our backgrounds in order to compare with the experimentallyobserved distribution, with dN˜/dKr satisfyingem(Kr)e f d(Kr)eLR(Kr)eUD(Kr)NORM· dNdKr+[IBckRnd(Kr) + IBckHoops(Kr)]. (4.5)It should be noted that when fitting equation(4.5) to the data without IBckRnd(Kr)there was considerable disagreement between reported values for ξo/ω be-tween charge states; well outside the 90% statistical C.L.. Only after includ-ing IBckRnd(Kr) for each charge state, were the centroids in ξo/ω brought backChapter 4. Experimental Results 48into agreement, as will be shown below. This lends support that our estimateson IBckRnd(Kr) and their kinematic dependencies have been correctly modeled foreach charge state.a) b) c)d) e) f)Figure 4.10: a),b),c) Transverse recoil momentum distribution of +2, 3, 4 charge states (left-right), respectively. d),e),f) Recoil kinetic energy Kr spectra for respective charge statesextracted from ±x quadrants (teal), ±z quadrants (magenta) with difference (black) an es-timate of background levels dominantly from decays from hoop surfaces in the ±z ori-entation relative to the ±x orientation. The overlaid black diagonal lines are the chosenboundaries for such quadrants.4.1.6 Recoil-SOE Kinematic Spectra and Averaged ξo/ωIntegrated rMCP-SOE coincident Kr spectra are shown in Figure(4.11a,b) withspectra obtained without corrections from the non-uniform ~E-field, and withcorrections, respectively for each of the +2(red), 3(blue), 4(green) charge states.Notably, the uncorrected spectra are systematically skewed to smaller Kr withthe distribution peaking near 380 eV, particularly in the +2 charge state where itis known that the maximal recoil Kr is 430 eV. Overlaid are the respective back-grounds IBckRnd(Kr) as well as IBckHoops(Kr) in the case of the fully corrected spectraChapter 4. Experimental Results 49Recoil Kinetic Energy [eV]0 100 200 300 400 500Counts050010001500200025003000+2 Rnd-bck Theory+3 Rnd-bck Theory+4 Rnd-bck TheoryRecoil Kinetic Energy [eV]0 100 200 300 400 500Thr ] / IThr[ I-I0.15−0.1−0.05−00.050.10.15Recoil Kinetic Energy [eV]0 100 200 300 400 500Counts05001000150020002500+2 Rnd-bck Hoop-bck Theory+3 Rnd-bck Hoop-bck Theory+4 Rnd-bck Hoop-bck TheoryRecoil Kinetic Energy [eV]0 100 200 300 400 500Thr ] / IThr[ I-I0.15−0.1−0.05−00.050.10.15a) b)Figure 4.11: Recoil kinetic energy Kr spectrum for +2(red), 3(blue), 4(green) charge states a)before non-uniform ~E-field ion impact coordinate X′ correction (assuming uniform field ofEy = 998.5 V/cm), and b) following correction in mapping X′ → X where recoils wouldtraverse such a uniform ~E-field. Included are expected random backgrounds and projectedhoop backgrounds necessary for ξo/ω < 0 model fits. (See Appendix A for ξo/ω > 0model fits.)for each of the charge states. In the corrected spectra each charge state peaks asexpected near 430 eV.Each of the recoil charge state Kr spectra in Figure(4.11) were fit assuming themodified dependence in equation(4.5). Since the combined branching to excitedstates is of order 9%, we assume the GS transition is the dominant contributor tothe observed recoil Kr spectrum, with the first-forbidden correlation parameterdefined in equation(2.22). Two parameter χ2 minimization was performed byfloating the nuclear matrix element ratio ξo/ω of the GS transition and arbitrarynormalization NORM with χ2/do f distributions plotted in Figure(4.12a,b,c) forthe uncorrected Kr spectra (setting IBckHoops(Kr) = 0), and in Figure(4.12d,e,f) forthe fully corrected Kr spectra with boundaries defining the 90% statistical C.L.The fit output is overlaid in Figure(4.11a,b) with residuals displayed at the bot-tom of each plot for +2(red), 3(blue), 4(green) charge states. In the corrected spec-tra fit residuals show an event excess at low recoil Kr which might result fromChapter 4. Experimental Results 50NORM3600038000400004200044000460004800050000ω/oξ0.8−0.75−0.7−0.65−0.6−0.55−0.5−0.45−0.4−0.511.522.53/dof (+2)2χUncorrected: NORM110 120 130 140 150 160 170310×ω/oξ0.8−0.75−0.7−0.65−0.6−0.55−0.5−0.45−0.4−0.511.522.53/dof (+3)2χUncorrected: NORM160 170 180 190 200 210 220 230 240 250 260310×ω/oξ0.8−0.75−0.7−0.65−0.6−0.55−0.5−0.45−0.4−0.511.522.53/dof (+4)2χUncorrected: NORM45000 50000 55000 60000 65000 70000 75000ω/oξ0.8−0.75−0.7−0.65−0.6−0.55−0.5−0.45−0.4−0.511.522.53/dof (+2)2χCorrected: NORM150 160 170 180 190 200 210 220 230 240 250310×ω/oξ0.8−0.75−0.7−0.65−0.6−0.55−0.5−0.45−0.4−0.511.522.53/dof (+3)2χCorrected: NORM180 200 220 240 260 280 300 320 340 360310×ω/oξ0.8−0.75−0.7−0.65−0.6−0.55−0.5−0.45−0.4−0.511.522.53/dof (+4)2χCorrected: a) b) c)d) e) f)Figure 4.12: χ2/do f distribution of equation(4.5) fit to recoil Kr spectra in Figure(4.11) afterfloating the nuclear matrix element ratio ξo/ω < 0 and NORM parameters for +2, 3, 4charge states a),b),c) without non-uniform ~E-field corrections, and d),e),f) with non-uniform~E-field corrections in mapping X′ → X. Boundaries of these distributions represent 90 %statistical C.L. for the respective charge state. From the large first-forbidden GS branchingwe assume the beta-energy dependence in the correlation a1 parameter of equation(2.22).The range in ξo/ω was chosen for comparison of the Recoil-SOE, and the GS isolated Recoil-SOE-SCINT coincident data to see the large change in ξo/ω between these channels.transitions to excited states. Transitions to excited states would produce lowerenergy beta’s and necessarily recoils distributed towards lower Kr. It shouldbe noted that two solutions exist for the nuclear matrix element ratio ξo/ω (seesupplementary section for ξo/ω > 0 solutions) with results at 90% C.L. shownin Table(4.1).Results from the Recoil-SOE coincident data reveal consistent values for theξo/ω across each charge state at 90% statistical C.L. in Figure(4.12d,e,f) listed inTable(4.1) with mean value (ξ/ωo)RSOE for positive and negative solutions(ξo/ω)RSOE+ = +0.437± (0.005)stat ± (0.01)sys (4.6)(ξo/ω)RSOE− = −0.512± (0.008)stat ± (0.01)sys. (4.7)As above, this method does not account for transitions to excited states as theycan’t be experimentally distinguished from those decaying through the GS. AChapter 4. Experimental Results 51Table 4.1: Nuclear Matrix Element ratio ξo/ω at 90% C.L. from fits of the recoil Kr spectrumof the specified charge state (CS) using equation(4.5) assuming the dominant transition isthe First-forbidden GS branch in the Recoil-SOE coincidence. See Appendix A for fit resultsfrom ξo/ω > 0. ( ∗Non-linear ~E-field corrected recoil hit coordinate.)CS ξo/ω < 0 do fχ2do f ξo/ω > 0 do fχ2do f+2 −0.569+ (+0.006−0.006)stat ± (0.01)sys 48− 2 0.91 +0.465+ (+0.003−0.003)stat ± (0.01)sys 48− 2 0.90+3 −0.569+ (+0.008−0.008)stat ± (0.01)sys 48− 2 1.26 +0.465+ (+0.005−0.004)stat ± (0.01)sys 48− 2 1.24+4 −0.579+ (+0.010−0.008)stat ± (0.01)sys 49− 2 1.18 +0.471+ (+0.004−0.005)stat ± (0.01)sys 49− 2 1.28+2∗ −0.511+ (+0.006−0.006)stat ± (0.01)sys 51− 2 1.48 +0.435+ (+0.003−0.004)stat ± (0.01)sys 51− 2 1.39+3∗ −0.505+ (+0.008−0.010)stat ± (0.01)sys 51− 2 2.41 +0.434+ (+0.005−0.005)stat ± (0.01)sys 51− 2 2.45+4∗ −0.519+ (+0.010−0.010)stat ± (0.01)sys 51− 2 1.88 +0.441+ (+0.005−0.005)stat ± (0.01)sys 51− 2 1.86similar result was found in 134Sb 0− to 0+ first-forbidden rank-0 decay whereω ∼= 1.8 · ξo or (ξo/ω ∼= 0.56) [26], which would have been a surprising resultfor light nuclei in which |ξo/ω|  1/3 with correlation parameter in equa-tion(2.22) (assuming γ1 ∼= 1) of a1 ∼= 1 [17]. One explanation proposed by Siegl& Scielzo was the feeding of a continuum of highly excited states amounting to17.2(5)% in beta decay strength to obtain a a1 ∼= 1 [26]. We attempt to addressthis question experimentally in the next section, with a preliminary analysis ofthe correlation parameter a1 in Recoil-SOE-SCINT coincidence, where we canisolate the decays through the GS.4.2 Recoil-SOE-SCINT Coincidence4.2.1 Kinematic Observables and ConstraintsAlthough the recoil-SOE coincident events have the advantage of accepting allrecoils in 4pi for charge states +2 and higher in our case, drawbacks exist fromthe large coincident random backgrounds, and the hoop backgrounds as dis-cussed above. Future position sensitivity of the eMCP/WSA coupled with thehighly localized SOE distribution (from their low eV scale energy) will pro-vide additional selection criteria to reject both of these backgrounds, but thisis still being incorporated into our DAQ. Recoil-SOE-SCINT triple coincidentevent channels also provide additional triggers to suppress both backgrounds,though at the expense of restricting the solid angle of accepted events from thesolid angle restrictions of the scintillator/collimator, and introducing additionalβ-detector backgrounds.Chapter 4. Experimental Results 52a) b) c)d) e) f)Figure 4.13: Recoil-SOE-SCINT triple coincident event a),c) drift corrected impact radius vs.TOF (Recoil-PMT) spectrum of +2, 3, 4, and higher charge states for coincidence with upper,and lower SCINT respectively. Overlaid are the respective TOF bounds for the +2(red),3(blue), 4(green) events. (+2) Recoil momentum vs. scintillator energy are shown in b),e)with overlaid kinematic boundaries for Q = 8.1 MeV. Scintillator energy spectra are shownin c),f) with (dashed-line) and without (line) kinematic boundaries applied for indicatedcharge states.Triple coincident Recoil-SOE-SCINT events are shown in Figure(4.13) where wehave imposed recoil hit selection cuts as discussed above in Section 3.3. The trapdrift corrected recoil hit radius vs Recoil-SCINT TOF is shown in Figure(4.13a,d)for coincidence with the upper, and lower scintillator detectors, respectively.Overlaid are the TOF bounds imposed on the +2(red), 3(blue), 4(green) chargestates in this analysis. The addition of the SOE-SCINT timing gate discussedabove was crucial in suppressing random coincident background events. Scin-tillator event thresholds were set at 200 keV to suppress false scintillator triggersfrom electronic noise and events which deposit the majority of their energy else-where in the apparatus before scattering into the scintillator volume. Again, nocut is made using the DSSD detector in this analysis due to time constraints,but this will provide additional event selection constraint in the final analysis.The non-uniform ~E-field corrected recoil momentum plotted as a function ofscintillator event energy for the +2 charge state is shown in Figure(4.13b,e) forChapter 4. Experimental Results 53coincidence with the upper, and lower scintillator, respectively. It should benoted that the recoil momentum ~pr and beta-energy Eβ are kinematically de-pendent from energy and momentum conservation. Assuming Kr  Eβ, Eνand from momentum conservation ~pr + ~pβ + ~pν = 0 it followsEβ + Eν = Eo = Q+m,p2r = p2β + p2ν + 2Pβpν cos θβν(4.8)where since the recoil momenta are bounded by |pβ − pν| < pr < |pβ + pν| itcan be shown[15] the constraints on the accessible beta energy Eβ are(pr − Eo)2 +m22(pr − Eo) < Eβ <(pr + Eo)2 +m22(pr + Eo). (4.9)Overlaid in Figure(4.13b,e) are the kinematic boundaries of equation(4.9) as-suming a Q = 8.1 MeV endpoint for the dominant GS transition. Clearly, asignificant fraction of events particularly at small Eβ and recoil momenta lieoutside the kinematic boundaries for the GS transition. The event populationoutside this region is populated both by decays to excited states, as well as in-correct evaluation of Eβ from bremsstrahlung and betas scattering depositingnon-negligible amounts of energy within the DSSD, or other regions of the ap-paratus volume.Scintillator spectra for the upper, and lower detectors are shown in Figure(4.13c,f)for the +2(red), 3(blue), 4(green) charge states both with (dashed-lines) and with-out (solid-lines) the kinematic boundaries applied to the data. The kinematicbounds demonstrate a significant suppression of pedestal type events wherelow energy beta’s deposit the majority of their energy in the DSSD and othervolumes, with the remaining beta energy sufficient to fire the scintillator PMT.The kinematic bounds, however, assume infinite detector precision and effi-ciency, which is impractical where clearly a small subset of events with largerecoil momentum fall just outside the boundaries from finite recoil momentumresolution. Additionally, a systematic deviation in the maximum SCINT energyfrom the expected kinematic bounds is seen for coincidences in both upper andlower scintillator, suggesting that their calibration is not fully optimized.Chapter 4. Experimental Results 54a) b) c)d) e) f)Figure 4.14: Recoil-SOE-SCINT coincident Qexp-value vs. reconstructed anti-electron neu-trino energy Eν spectrum utilizing the a),b),c) Upper and d),e),f) Lower beta scintilla-tor events within the respective +2, 3, 4 recoil charge state TOF bounds shown in Fig-ure(4.13a,d).4.2.2 Reconstructed Eν and Qexp-value DistributionsWith the known initial recoil momentum ~pr and taking the beta momentum tobe entirely along either the ±z-axis, the neutrino momentum and thus its en-ergy Eν (assuming it is massless for our purposes) can be reconstructed as perequation(3.9), along with the Qexp-value in equation(3.11). The distribution ofQexp-value vs. Eν are shown for +2, 3, 4 charge states using the upper scintillatorin Figure(4.14a,b,c), and lower scintillator in Figure(4.14d,e,f), respectively. Theresonance like feature centered on a Q-value of 8.1 MeV correspond to nuclearβ-decay transitioning through the GS branch. The sparse event distributionwith smaller Qexp-values, which are correlated with smaller Eν, correspond toβ-decays through the excited states since there is less energy available to theneutrino. Events along the Qexp = Eν line result from incorrectly reconstructedEν and thus Qexp from incorrect reconstruction of the beta’s energy, and incor-rect selection of the recoil event. Incorrect reconstruction of the beta’s energyChapter 4. Experimental Results 55can result from energy deposition within the DSSD, scattering from the colli-mator or other non-active volumes, or through bremsstrahlung photons whicheasily escape the low Z plastic scintillators. Similarly, our event selection inthis analysis accepts only the first recoil into the DAQ and thus could lead tomisidentification of the recoil and its momentum. The result is a unitary corre-lated random spread of events along the line Qexp = Eν. Raising the scintillatorthresholds greatly suppress these events uniformly along Qexp = Eν, althoughat the expense of losing sensitivity to the neutrinos with largest Eν.+K-m [MeV]ν+EβE0 2 4 6 8 10 12 14 16 18 20Counts/200 keV110210310UPMT: +2 Rnd-bck OUTUPMT: +3 Rnd-bck OUTUPMT: +4 Rnd-bck OUT [MeV]νE0 2 4 6 8 10 12 14Counts/200 keV0100200300400500600700UPMT: +2 Rnd-bckUPMT: +3 Rnd-bckUPMT: +4 Rnd-bckUPMT: +2(Qwin) Rnd-bckUPMT: +3(Qwin) Rnd-bckUPMT: +4(Qwin) Rnd-bck+K-m [MeV]ν+EβE0 2 4 6 8 10 12 14 16 18 20Counts/200 keV110210310LPMT: +2 Rnd-bck OUTLPMT: +3 Rnd-bck OUTLPMT: +4 Rnd-bck OUT [MeV]νE0 2 4 6 8 10 12 14Counts/200 keV0100200300400500600700800 LPMT: +2 Rnd-bckLPMT: +3 Rnd-bckLPMT: +4 Rnd-bckLPMT: +2(Qwin) Rnd-bckLPMT: +3(Qwin) Rnd-bckLPMT: +4(Qwin) Rnd-bcka) b)c) d)Figure 4.15: Recoil-SOE-SCINT triple coincident events with transition Qexp-value distri-butions of the +2(red), 3(blue), 4(green) charge states with beta’s coincident in the a) Upper,and c) Lower scintillators, respectively (See Appendix A for linear scaled plots). Overlaidare the time random coincident backgrounds (dashed-lines), and kinematically forbiddenevents assuming Q = 8.1 MeV (solid-lines) for the respective charge states. Correspond-ing anti-electron neutrino energy spectra Eν are shown in b), and d) both with kinemati-cally constrained domain and events within 1.5 MeV of Q = 8.1 MeV GS transition (bold-markers), and without such bounds (markers) are shown for each charge state. Expectedtime random coincident background event distribution without the above bounds are over-laid for each charge state (dashed-lines).Chapter 4. Experimental Results 56Projections in Qexp-value of the 2D histograms in Figure(4.14) are shown in Fig-ure(4.15a,c) for +2(red), 3(blue), 4(green) charge states (markers) and time ran-dom events (dashed-lines) for coincidences with the upper, and lower scintil-lator, respectively. Again a sharp resonance at the Q = 8.1 MeV correspond todecays through the GS transition with low energy tail corresponding to tran-sitions to excited states and otherwise incorrectly reconstructed events. Thebreadth of the GS resonance is in part due to the energy resolution of our plas-tic scintillators, but also the assumption that the beta’s have their momenta en-tirely along the z-axis, which is only approximately true. The beta hit position(and energy deposition) in the DSSD will aid in improving the beta momen-tum resolution and thus the spread in the GS Qexp-value distributions in thefinal analysis. Overlaid are the Qexp-value distributions for events that fall out-side of our kinematic boundaries assuming Q = 8.1 MeV for the GS transition,which amount to significant fractions of the low energy Qexp-value tail for eachcharge state. The remaining events in the tail of the Qexp-value distribution thatare within our kinematic boundaries are dominantly populated by transitionsto lower energy excited states.Similarly, projections in Eν of the 2D histograms in Figure(4.14) for each chargestate are shown in Figure(4.15b,d) (solid-lines) with time-random coincidentbackgrounds (dashed-lines) for coincidence with the upper, and lower scintil-lators, respectively. A broad distribution in Eν is observed out to ∼ 8.7 MeVfrom the 3-body phase space of the weak decay kinematically constrained toEν < Qexp (when Eβ ∼ m) with deviations largely due to detector resolutioneffects. Applying a Qexp-value gate on events within 1.5 MeV of the Q = 8.1MeV GS transition, along with our kinematic boundaries defined above, wecan isolate the respective population in Eν with results overlaid for each chargestate (bold-lines) for coincidence with respective scintillator. Notably, in isolat-ing the neutrinos from GS beta transition, the remaining events are correlatedwith lower energy Eν as one would expect from feeding excited states, wherethere is less energy available to the beta’s, and necessarily the neutrinos.The transverse recoil momentum distribution of decays within our Qexp-valuegate and kinematic boundaries for coincidence with the upper, and lower scin-tillator are shown in Figure(4.16a,c), respectively. The small opening angle ofthe scintillator/collimator constrain, naturally, the accessible momentum spaceof the recoil which is dominantly ejected opposite to the respective β-scintillator.Chapter 4. Experimental Results 57a) b)c) d)Figure 4.16: a),d) Recoil-SOE-SCINT coincident transverse recoil momentum distributionof +2 charge state within TOF bounds shown in Figure(4.13a,d) for event subset within±1.5MeV of the GS transition Qexp-value centered at Q = 8.1 MeV, and within the defined kine-matic boundaries. The transverse momentum distribution of events outside the kinematicboundaries are shown in b),d) for coincident events with the Upper, and Lower scintillator,respectively.Particularly, note the absence of events with recoil momenta in the directionof the respective β-scintillators. Beta’s from decays originating on the centralelectrostatic hoop surfaces along the ±z-axis still have line-of-sight access toboth the upper and lower scintillators, though slightly different solid angles.If decays from the hoops were present in this Recoil-SOE-SCINT triple coinci-dent channel there should be a symmetric event excess at large transverse mo-mentum along the ±z-axis as we inferred above in the Recoil-SOE coincidentchannel. The absence of events with large transverse momenta aligned withthe respective β-scintillator indicates that such a background resulting from de-cays off the hoops is greatly suppressed in Recoil-SOE-SCINT triple coincidentchannel with a timing gate applied to the SOE-SCINT TOF as above. Recall thatin the Recoil-SOE-SCINT triple coincident channel the recoil TOF was takenChapter 4. Experimental Results 58between the β-scintillator and the recoil, so naturally hoop backgrounds willbe suppressed in this event type due to the limited solid angle of the scintilla-tor/collimator compared to the Recoil-SOE coincidence which accepts events in4pi. The transverse recoil momentum distribution outside the kinematic boundsare shown in Figure(4.16b,d), and dominantly populate lower transverse recoilmomentum (smaller Kr) resulting from feeding excited states.a) b)c) d)Figure 4.17: Simulated transverse recoil momentum distribution from a point-like sourcewith encoded trap drift, within the TOF bounds, and with conical restrictions placed on theβ-solid angle (22◦) defined by our collimator geometry for coincidence with the a) upper,and c) lower scintillators, respectively. Corresponding mask transmission functions em(Kr)(markers) and finite event density e f d(Kr) are shown for +2(red), 3(blue), 4(green) chargestates for coincidence with the b) upper, d) lower scintillator, respectively.4.2.3 GS Transition ξo/ωWith the GS transition isolated above we can extract the first-forbidden nuclearmatrix element ratio ξo/ω from the recoil Kr spectrum of each charge state.In order to fit the experimental recoil Kr spectrum we similarly employ equa-tion(4.5), but for this channel justifiably assume the background contributionChapter 4. Experimental Results 59from the hoops is small. Since the recoils sample a restricted subset of mo-mentum space due to the restrictions on the beta momentum space, we mustincorporate this into the evaluation of em(Kr), and e f d(Kr). To do this we per-form a crude simulation of a 3-body decay from a point-like source, with theoutgoing beta restricted to uniformly populate the conical momentum spaceaccessible assuming the 22◦ opening angle restriction by the β-collimator. The’neutrino’ energy is kinematically restricted to Eν = Eo− Eβ, but with momentaallowed to occupy 4pi in this simple simulation. The recoil momentum is thengenerated simply from non-relativistic momentum conservation. Again, thetrap drift present in the experiment is encoded in the simulated decay alongwith appropriate recoil TOF bounds in Figure(4.13a,d) with the transverse mo-mentum distribution shown in Figure(4.17a,c) for coincidence with the upper,and lower scintillator, respectively. Similarly, the trap drift smears out the maskpattern in the transverse momentum space with the recoils dominantly pop-ulating the phase space opposite to the respective detector and similar to thedistributions seen in our data in Figure(4.16a,c), respectively.Recoil Kinetic Energy [eV]0 100 200 300 400 500Counts050100150200250300350400450+2 UPMT Rnd-bck Theory+3 UPMT Rnd-bck Theory+4 UPMT Rnd-bck TheoryRecoil Kinetic Energy [eV]0 100 200 300 400 500Thr ] / IThr[ I-I0.6−0.4−0.2−00.20.40.6Recoil Kinetic Energy [eV]0 100 200 300 400 500Counts050100150200250300350400450+2 LPMT Rnd-bck Theory+3 LPMT Rnd-bck Theory+4 LPMT Rnd-bck TheoryRecoil Kinetic Energy [eV]0 100 200 300 400 500Thr ] / IThr[ I-I0.6−0.4−0.2−00.20.40.6a) b)Figure 4.18: Recoil Kr spectra for GS Recoil-SOE-SCINT coincident events in charge state+2(red), 3(blue), 4(green) with beta’s coincident with the a) upper, and b) lower scintillator,respectively. GS events were isolated after applying kinematic bounds, TOF bounds, andgating in Qexp-value on the events within 1.5 MeV of the 8.1 MeV grounds state transition.Overlaid are the time random coincident background’s (dashed-lines) for the respectivecharge states along with model fits taking ξo/ω < 0.Chapter 4. Experimental Results 60The recoil transmission efficiency em(Kr) was similarly computed as the ratioof simulated recoil Kr spectrum of events incident on the rMCP/mask to thatincident rMCP alone for charge states +2(red), 3(blue), 4(green) (markers) shownin Figure(4.17b,d) for coincidence with the upper and lower scintillator, respec-tively. The trap drift requires em(Kr) to be independently determined for coinci-dences with the upper and lower scintillator from the now broken symmetry ofthe imprinted mask pattern observed by each recoil population. The restrictionson the recoil phase space also result in extremely low event rates with trans-verse momenta oriented with the respective scintillator detector, which we pa-rameterize as e f d(Kr) by scaling the simulated Kr spectra incident on the rMCPalone by the value obtained at maximal recoil Kr of 430 eV. The absolute scalingof e f d(Kr) is not relevant given that the normalization NORM in equation(4.5)is a floating parameter. The non-linear drop in efficiency towards zero recoilKr, similar to those in Recoil-SOE coincidence above, again results from the fi-nite event density at low recoil radius for each of the respective charge states.The near-linear dependence in e f d(Kr) out to large Kr results from the restrictedphase space on the beta, and necessarily the recoil.Given the limited statistics in the triple coincidence and the recoil solid an-gle restrictions above, we assume the relative rMCP efficiency functions aboveeLR(Kr) and eUD(Kr) determined from the Recoil-SOE coincidences to be largelyintrinsic to the rMCP detector, and apply them in our evaluation of equation(4.5)for the recoil Kr spectra in Recoil-SOE-SCINT coincidence. Finally, for Recoil-SOE-SCINT coincidences within the +2(red), 3(blue), 4(green) TOF bounds, weapply the above kinematic boundaries and gate on events within 1.5 MeV ofthe 8.1 MeV Q-value, producing the recoil Kr spectra (markers) shown in Fig-ure(4.18a,b) for coincidences with the upper, and lower scintillator, respectively.Random coincidence backgrounds IBckRnd(Kr) were similarly computed as abovefor each charge state and overlaid, which were significantly reduced follow-ing the mentioned SCINT-SOE TOF cuts. Assuming negligible contributionsfrom IBckHoops(Kr) as argued above, the spectra were fit with equation(4.5) float-ing the nuclear matrix element ratio ξo/ω and arbitrary normalization NORMfor charge states +2, 3, 4 with resulting χ2/do f shown in Figure(4.19a,b,c) for co-incidence with the upper scintillator, and lower scintillator in Figure(4.19d,e,f),respectively. Again, the boundary of the χ2/do f distribution defines the 90 %statistical C.L. The solutions for ξo/ω are tabulated in Table(4.2) with distribu-tions and residuals shown in Figure(4.18). (See appendix for similar plots aboveChapter 4. Experimental Results 61with positive solutions of ξo/ω).NORM150 200 250 300 350 400310×ω/oξ0.8−0.75−0.7−0.65−0.6−0.55−0.5−0.45−0.4−00.20.40.60.811.2NORM600 700 800 900 1000 1100 1200310×ω/oξ0.8−0.75−0.7−0.65−0.6−0.55−0.5−0.45−0.4−00.20.40.60.811.21.41.61.822.2NORM800 850 900 950 1000 1050 1100 1150 1200310×ω/oξ0.8−0.75−0.7−0.65−0.6−0.55−0.5−0.45−0.4−00.511.522.533.54NORM150 200 250 300 350 400310×ω/oξ0.8−0.75−0.7−0.65−0.6−0.55−0.5−0.45−0.4−00.20.40.60.811.21.41.61.82NORM600 700 800 900 1000 1100 1200310×ω/oξ0.8−0.75−0.7−0.65−0.6−0.55−0.5−0.45−0.4−02468101214NORM800 850 900 950 1000 1050 1100 1150 1200310×ω/oξ0.8−0.75−0.7−0.65−0.6−0.55−0.5−0.45−0.4−00.511.522.5a) b) c)d) e) f)Figure 4.19: χ2/do f distribution of equation(4.5) fit to recoil GS Kr spectra in Figure(4.18)after floating the nuclear matrix element ratio ξo/ω < 0 and NORM parameters for +2, 3,4 charge states with beta coincident with the a),b),c) Upper, and d),e),f) Lower, scintillators,respectively. Boundaries of these distributions represent 90 % statistical C.L. for the respec-tive charge state. The range in ξo/ω was chosen for comparison of the Recoil-SOE, andthe GS isolated Recoil-SOE-SCINT coincident data to see the large change in ξo/ω betweenthese channels.Consistent values for the GS isolate, first-forbidden rank-0 nuclear matrix ele-ment ratio ξo/ω, across each charge state and each scintillator are seen in ta-ble(4.2) with mean value using the upper scintillator (UPMT)(ξo/ω)BRSOE+ = +0.540± (0.007)stat ± (0.01)sys (4.10)(ξo/ω)BRSOE− = −0.736± (0.016)stat ± (0.01)sys, (4.11)and lower scintillator (LPMT)(ξo/ω)BRSOE+ = +0.542± (0.008)stat ± (0.01)sys (4.12)(ξo/ω)BRSOE− = −0.742± (0.020)stat ± (0.01)sys. (4.13)We note that even though the χ2/do f > 1 the 90% statistical C.L. bounds onξo/ω > 0 are more then half those for ξo/ω < 0, which may suggest the trueChapter 4. Experimental Results 62Table 4.2: Nuclear Matrix Element ratio ξo/ω at 90% C.L. from fits of the recoil Kr spectrumof the specified charge state using equation(4.5) from decays through the First-forbidden GSbranch in the Recoil-SOE-SCINT coincidence. See Appendix A for fit results from ξo/ω > 0.( ∗Non-linear ~E-field corrected recoil hit coordinate.)C.S. SCINT ξo/ω < 0 do fχ2do f ξo/ω > 0 do fχ2do f+2∗ UPMT −0.699+ (+0.012−0.014)stat ± (0.01)sys 49− 2 1.15 +0.525+ (+0.005−0.006)stat ± (0.01)sys 49− 2 1.16+3∗ UPMT −0.743+ (+0.020−0.022)stat ± (0.01)sys 49− 2 2.28 +0.544+ (+0.009−0.008)stat ± (0.01)sys 49− 2 2.36+4∗ UPMT −0.765+ (+0.014−0.016)stat ± (0.01)sys 49− 2 4.38 +0.553+ (+0.006−0.006)stat ± (0.01)sys 49− 2 4.20+2∗ LPMT −0.725+ (+0.012−0.014)stat ± (0.01)sys 49− 2 2.07 +0.535+ (+0.006−0.005)stat ± (0.01)sys 49− 2 2.00+3∗ LPMT −0.755+ (+0.024−0.028)stat ± (0.01)sys 49− 2 14.2 +0.548+ (+0.012−0.011)stat ± (0.01)sys 49− 2 14.2+4∗ LPMT −0.745+ (+0.020−0.022)stat ± (0.01)sys 49− 2 2.58 +0.544+ (+0.008−0.008)stat ± (0.01)sys 49− 2 2.42value of ξo/ω > 0. This follows from the form of the correlation parameterin equation(2.22) having an interference term between the two nuclear matrixelements, which is sensitive to their relative signs in principle. We note thatthis is surprisingly/coincidently consistent with the 134Sb (ξo/ω ∼= 0.56) mea-sured at ANL [26], though this experiment could not rule out contaminationfrom transitions to excited states of the daughter. Evidently, even with the GSisolated events in 92Rb, the hypothesis that |ξo/ω|  1/3 [17] is inconsistentwith our results. It should be noted that even when gating on Qexp-value forthe GS transition, as shown in Figure(4.15a,c), there remains a population oftransitions to excited states, albeit small, within such bounds with recoil Kr nec-essarily skewed towards small values. Such an excess of low energy recoilsabove an already small population from the GS events (with necessarily smalluncertainties), may skew the fit to smaller magnitudes of ξo/ω.4.3 Recoil-SOE-SCINT Coincidence GS Transition ξo/ωThe results for the nuclear matrix element ratio ξo/ω in the Recoil-SOE-SCINTcoincidence from decays through the GS presented in Table(4.2) are notablydifferent from the those in the Recoil-SOE coincidence in Table(4.1) assumingthe first-forbidden rank-0 form of the correlation parameter a1(W) in equa-tion(2.22). The correlation parameter a1(W) however diverges as W → 1. Thescaling coefficient a1(W) · (v/c) of the cos θ dependency in the convolutionfunction equation(2.18) is bounded by ±1, and seems more appropriate to in-vestigate here. Assuming the values for nuclear matrix element ratio ξo/ω at 90% statistical C.L. in Table(4.1) for the Recoil-SOE coincident events a1(W) · (v/c)was plotted in Figure(4.20) in a) for ξo/ω > 0, and c) for ξo/ω < 0, respectively.Chapter 4. Experimental Results 63Asymptotic limits on the correlation parameter show a1(W →Wo)RSOE+RSOE− →0.300.36,respectively, over multiple charge states. The bounds on a1(W) · (v/c) aftergating on events decaying through the GS in Recoil-SOE-SCINT coincidencesare shown in Figure(4.20) in b) for ξo/ω > 0, and d) for ξo/ω < 0, respectively.Asymptotic limits on the correlation parameter show a1(W →Wo)BRSOE+BRSOE− →0.500.58,respectively, over multiple charge states. Considerable deviations in the cor-relation a1 are thus evident in the Recoil-SOE, and Recoil-SOE-SCINT coinci-dent event streams. Without being able to isolate decays through the GS in theRecoil-SOE coincidence, decays to the continuum of highly excited beta transi-tions necessarily reduces the energy available to the recoil and populate smallrecoil Kr on average. In populating small values of Kr from feeding excitedstates in the Recoil-SOE coincidence the result will be a reduction in the aver-age correlation parameter magnitude compared to that from decays through theGS in the Recoil-SOE-SCINT coincidence as observed here. Additionally, the ex-cited states have unconstrained and differing correlation parameters from thoseof the GS transition, which will also distort the recoil Kr spectrum.4.4 Summary of ResultsWe have shown consistent results for the rank-0 nuclear matrix element ra-tio ξo/ω across the +2, 3, 4 charge states in both the Recoil-SOE coincidentdataset in Table(4.1), and triple coincident Recoil-SOE-SCINT dataset in Ta-ble(4.2). The corresponding first-forbidden rank-0 Eβ dependent correlationparameter bounds at 90% statistical C.L. are shown in Figure(4.20) assumingthe beta energy dependence a1(W) of equation(2.22). A significant deviationbetween the mean nuclear matrix element ratio ξo/ω in Recoil-SOE coincidence(ξo/ω)RSOE+ = +0.437± (0.005)stat ± (0.01)sys (4.14)(ξo/ω)RSOE− = −0.512± (0.008)stat ± (0.01)sys. (4.15)and that of the Recoil-SOE-SCINT coincidence after gating on Qexp-value within1.5 MeV of the 8.1 MeV Q-value(ξo/ω)BRSOE+ = +0.541± (0.008)stat ± (0.01)sys (4.16)(ξo/ω)BRSOE− = −0.739± (0.018)stat ± (0.01)sys (4.17)Chapter 4. Experimental Results 64is observed. Interestingly, the bounds on the positive solutions for ξo/ω at 90%statistical C.L. appear to be a factor of 2x smaller than the negative solutions,suggesting ξo/ω > 0, as we have sensitivity to the relative sign of the nuclearmatrix elements from the interference term in a1(W) of equation(2.22).βE1 2 3 4 5 6 7 8( v/c )νβa00.10.20.30.40.50.60.70.80.91+2+3+4βE1 2 3 4 5 6 7 8( v/c )νβa00.10.20.30.40.50.60.70.80.91+2: UPMT LPMT+3: UPMT LPMT+4: UPMT LPMTβE1 2 3 4 5 6 7 8( v/c )νβa00.10.20.30.40.50.60.70.80.91+2+3+4βE1 2 3 4 5 6 7 8( v/c )νβa00.10.20.30.40.50.60.70.80.91+2: UPMT LPMT+3: UPMT LPMT+4: UPMT LPMTa) b)c) d)Recoil-SOEξo/ω > 0Recoil-SOEξo/ω < 0Recoil-SOE-SCINTξo/ω > 0Recoil-SOE-SCINTξo/ω < 0Figure 4.20: First-forbidden (rank-0) beta-neutrino correlation function product a1(Eβ) ·(v/c) in equation(2.22) vs. Eβ from (90 % C.L.) bounds set in Table(4.1) for Recoil-SOEevents in a,c) and GS isolated Recoil-SOE-SCINT coincident events from limits set in Ta-ble(4.2) in b,d) for ξo/ω > 0, and ξo/ω < 0, respectively. In (b,d) the solid and dashed linebounds are from event streams coincident with the Upper and Lower scintillators.65Chapter 5Conclusion and Outlook5.1 Recoil-SOE Coincidence5.1.1 ConclusionThe theoretical neutrino energy spectrum for any 0− to 0+ decay depends onξ0/ω [21], and our determination of this ratio for the GS transition might endup better defining the neutrino energy spectrum than our direct measurement.Our result would be a unique nonzero measurement of ω in these high-Z fissionproducts, a quantity very difficult to calculate because of its dependence on thespatial tails of the wave-function. If ω is finite in general in 0− to 0+ decays, theneutrino energy spectrum would change for all such transitions.In the β−decay of 92Rb it is shown that in Recoil-SOE coincidence, the pos-itive/ negative solutions of the nuclear matrix element ratio (ξo/ω)RSOE+RSOE− =+0.437−0.512± (0.0050.008)stat± (0.010.01)sys and correlation parameter a1(W →Wo)→0.300.36. With|ξo/ω| < 1, and |a1| < 1, our results are inconsistent with theoretical pre-dictions for rank-0 first-forbidden transitions in which |ξo/ω|  1/3, and|a1(W → Wo)| = 1 [17]. These results are similar to those obtained in the134Sb dominant first-forbidden 0− → 0+ rank-0 GS decay, in which they foundξo/ω ∼= 0.56 and a1 = 0.47, but without isolating decays through the GS [26].The authors of [26] attributed this difference to the feeding of excited states.They cite a shell model calculation in which ξo >> ω, and cite a version ofSiegerts theorem applied to forbidden decays. In the case of 92Rb, the dis-agreement in ξo/ω of the Recoil-SOE coincident data with theory would be un-derstandable, as the excited state transitions are indistinguishable from decaysthrough the GS. Interestingly, however, even in the Recoil-SOE-SCINT coinci-dence in which the GS events are isolated in Qexp-value, we find (ξo/ω)BRSOE+BRSOE−=+0.541−0.739 ±(0.0080.018)stat± (0.010.01)sys with a1(W →Wo)→0.500.58, where similarly |ξo/ω| <Chapter 5. Conclusion and Outlook 661, and |a1(W → Wo)| < 1. It should be noted there remains a small popula-tion of excited state events which are within the Qexp-value gate, which mayskew the fits in the Recoil-SOE-SCINT dataset to smaller magnitudes in ξo/ωand a1. In the Recoil-SOE-SCINT coincident dataset we also reconstruct theanti-electron neutrino spectrum Eν following gating on decays through the GS,which will be compared with simulations in the future.5.1.2 OutlookIn the future analysis, the addition of the DSSD detectors will further improveour 3-momentum resolution of the betas, and correspondingly improve our en-ergy resolution of the Qexp-value and Eν. In the analysis above we have utilizeda coarse spatial calibration of the rMCP, and we will perform high-resolutionspatial calibrations in the final analysis to further improve our recoil momen-tum resolution, and corresponding resolution of Qexp-value and Eν. GEANT-4simulations will be necessary to model the Qexp-value of decays through the GS,and expected background levels from inner, and outer bremsstrahlung, and oth-erwise miss-identified events. Once this is complete the population of decaysto excited states can be separated from the data, allowing us to constrain the GSbranching ratio.67Appendix AAppendicesFigure A.1: TRINAT DAQ timing setup with indicated delays into the acquisition. Strip de-tectors are not shown nor are the QDC data-streams from the rMCP, Wedge and Strip Anode(WSA) backing the eMCP, or scintillators. Only relative timing differences are needed forxz-recoil hit position reconstruction of the delay-line-anode (DLA) so delay-line lengths arenot important. The 355 nm UV laser incident on the trap was split off to a photo-diode (PD)to trigger our acquisition.Appendix A. Appendices 68rMCP x-Position [mm]5− 4− 3− 2− 1− 0 1 2 3 4 5rMCP z-Position [mm]5−4−3−2−1−012345050010001500200025003000rMCP x-Position [mm]5− 4− 3−2− 1− 01 2 34 5rMCP z-Position [mm]5−4−3−2−1−012345Counts050010001500200025003000Figure A.2: Drift corrected photo-ion distribution in xz-plane over entire dataset fit withthree independent elliptical-Gaussian’s parameterizing populations of strongly boundatoms and those in the process of being collected in the trap along the beam-line axis.Recoil Kinetic Energy [eV]0 100 200 300 400 500Counts050010001500200025003000+2 Rnd-bck Theory+3 Rnd-bck Theory+4 Rnd-bck TheoryRecoil Kinetic Energy [eV]0 100 200 300 400 500Thr ] / IThr[ I-I0.1−0.05−00.050.1 Recoil Kinetic Energy [eV]0 100 200 300 400 500Counts050010001500200025003000+2 Rnd-bck Hoop-bck Theory+3 Rnd-bck Hoop-bck Theory+4 Rnd-bck Hoop-bck TheoryRecoil Kinetic Energy [eV]0 100 200 300 400 500Thr ] / IThr[ I-I0.1−0.05−00.050.1a) b)Figure A.3: Recoil Kr spectra for +2(red), 3(blue), 4(green) charge states a) before correctionfrom non-uniform ~E-field, and b) following ion impact coordinate correction in a uniform~E-field of 998 V/cm. Included are expected random backgrounds and projected hoop back-grounds necessary for ξo/ω > 0 model comparison.Appendix A. Appendices 69NORM3600038000400004200044000460004800050000ω/oξ0.40.420.440.460.480.50.520.540.560.580.511.522.53/dof (+2)2χUncorrected: NORM110 120 130 140 150 160 170310×ω/oξ0.40.420.440.460.480.50.520.540.560.580.511.522.53/dof (+3)2χUncorrected: NORM160 170 180 190 200 210 220 230 240 250 260310×ω/oξ0.40.420.440.460.480.50.520.540.560.580.511.522.53/dof (+4)2χUncorrected: NORM45000 50000 55000 60000 65000 70000 75000ω/oξ0.40.420.440.460.480.50.520.540.560.580.511.522.53/dof (+2)2χCorrected: NORM150 160 170 180 190 200 210 220 230 240 250310×ω/oξ0.40.420.440.460.480.50.520.540.560.580.511.522.53/dof (+3)2χCorrected: NORM180 200 220 240 260 280 300 320 340 360310×ω/oξ0.40.420.440.460.480.50.520.540.560.580.511.522.53/dof (+4)2χCorrected: a) b) c)d) e) f)Figure A.4: χ2/do f distribution of equation(4.5) fit to recoil Kr spectra in Figure(4.11) afterfloating the nuclear matrix element ratio ξo/ω > 0 and NORM parameters for +2, 3, 4charge states a),b),c) without non-uniform ~E-field corrections, and d),e),f) with non-uniform~E-field corrections in mapping X′ → X. Boundaries of these distributions represent 90 %statistical C.L. for the respective charge state. From the large first-forbidden GS branchingwe assume the beta-energy dependence in the correlation a1 parameter of equation(2.22).Appendix A. Appendices 70Recoil Kinetic Energy [eV]0 100 200 300 400 500Counts0100200300400500+2 UPMT Rnd-bck Theory+3 UPMT Rnd-bck Theory+4 UPMT Rnd-bck TheoryRecoil Kinetic Energy [eV]0 100 200 300 400 500Thr ] / IThr[ I-I0.6−0.4−0.2−00.20.40.6Recoil Kinetic Energy [eV]0 100 200 300 400 500Counts0100200300400500+2 LPMT Rnd-bck Theory+3 LPMT Rnd-bck Theory+4 LPMT Rnd-bck TheoryRecoil Kinetic Energy [eV]0 100 200 300 400 500Thr ] / IThr[ I-I0.6−0.4−0.2−00.20.40.6a) b)Figure A.5: Recoil Kr spectra for GS Recoil-SOE-SCINT coincident events in charge state+2(red), 3(blue), 4(green) with beta’s coincident with the a) upper, and b) lower scintillator.GS events were isolated after applying kinematic bounds, TOF bounds, and gating in Q-value on the events within 1.5 MeV of the 8.1 MeV grounds state transition.Appendix A. Appendices 71NORM150 200 250 300 350 400310×ω/oξ0.40.420.440.460.480.50.520.540.560.5800.20.40.60.811.2NORM600 700 800 900 1000 1100 1200310×ω/oξ0.40.420.440.460.480.50.520.540.560.5800.20.40.60.811.21.41.61.822.22.4NORM800 850 900 950 1000 1050 1100 1150 1200310×ω/oξ0.40.420.440.460.480.50.520.540.560.5800.511.522.533.54NORM150 200 250 300 350 400310×ω/oξ0.40.420.440.460.480.50.520.540.560.5800.20.40.60.811.21.41.61.82NORM600 700 800 900 1000 1100 1200310×ω/oξ0.40.420.440.460.480.50.520.540.560.5802468101214NORM800 850 900 950 1000 1050 1100 1150 1200310×ω/oξ0.40.420.440.460.480.50.520.540.560.5800.511.522.5a) b) c)d) e) f)Figure A.6: χ2/do f distribution of equation(4.5) fit to recoil GS Kr spectra in Figure(4.18)after floating the nuclear matrix element ratio ξo/ω > 0 and NORM parameters for +2,3, 4 charge states with beta coincident with the a),b),c) Upper, and d),e),f) Lower, scintilla-tors, respectively. Boundaries of these distributions represent 90 % statistical C.L. for therespective charge state. The range in ξo/ω was chosen for comparison of results betweenRecoil-SOE, and Recoil-SOE-SCINT coincidence. The range in ξo/ω was chosen for com-parison of the Recoil-SOE, and the GS isolated Recoil-SOE-SCINT coincident data to see thelarge change in ξo/ω between these channels.Appendix A. Appendices 72+K-m [MeV]ν+EβE0 2 4 6 8 10 12 14 16 18 20Counts/200 keV020040060080010001200140016001800UPMT: +2 Rnd-bck OUTUPMT: +3 Rnd-bck OUTUPMT: +4 Rnd-bck OUT [MeV]νE0 2 4 6 8 10 12 14Counts/200 keV0100200300400500600700UPMT: +2 Rnd-bckUPMT: +3 Rnd-bckUPMT: +4 Rnd-bckUPMT: +2(Qwin) Rnd-bckUPMT: +3(Qwin) Rnd-bckUPMT: +4(Qwin) Rnd-bck+K-m [MeV]ν+EβE0 2 4 6 8 10 12 14 16 18 20Counts/200 keV02004006008001000120014001600180020002200 LPMT: +2 Rnd-bck OUTLPMT: +3 Rnd-bck OUTLPMT: +4 Rnd-bck OUT [MeV]νE0 2 4 6 8 10 12 14Counts/200 keV0100200300400500600700LPMT: +2 Rnd-bckLPMT: +3 Rnd-bckLPMT: +4 Rnd-bckLPMT: +2(Qwin) Rnd-bckLPMT: +3(Qwin) Rnd-bckLPMT: +4(Qwin) Rnd-bcka) b)c) d)Figure A.7: Recoil-SOE-SCINT triple coincident events with transition Q-value distribu-tions of the +2(red), 3(blue), 4(green) charge states with beta’s coincident in the a) Upper, andc) Lower scintillators, respectively. 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