Fast-Timing Measurements with a Spatially-Distributed Source

Transcription

1 Fast-Timing Measurements with a Spatially-Distributed Source A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2017 Michael James Mallaburn School of Physics and Astronomy University of Manchester

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3 Contents List of Figures 7 List of Tables 11 Abbreviations 13 Abstract 15 Declaration of Authorship 17 Copyright Statement 19 Acknowledgements 23 1 Introduction 25 2 Nuclear Theory Nuclear Structure and Deformation The Shell Model Collective Vibration Collective Rotation The Nilsson Model Selection Rules Reduced Transition Probabilities Isomeric Nuclear States Monte Carlo Calculations Experimental Background Heavy-Ion Fusion-Evaporation Reactions Direct Electronic Timing Contributions to Timing Uncertainty Mirror Symmetric Centroid Difference Method Recoil Distance Method

4 4 Contents 4 Radiation Detectors Interaction of Electromagnetic Radiation with Matter Sodium Iodide Detectors Lanthanum Bromide Detectors Semiconductor Detectors The Experimental Configuration at the University of Jyväskylä A New Fast-Timing LaBr 3 Array LaBr 3 Array Setup LaBr 3 Array Resolution LaBr 3 Array Intrinsic Photopeak Efficiency The Prompt Response of the LaBr 3 Array Chapter Summary Investigation into the Effect of Spatial Distribution on Fast-Timing Measurements Introduction to Timing of a Spatially Distributed Source Details of the RaPToF Technique An Experimental Trial of the RaPToF Correction Implementation of the RaPToF Technique using Source Data Experimental Details Experimental Results Experimental Discussion A Method to Quantitatively Link Spatial Distribution to an Increase in Lifetime Uncertainty Parameterisation of the Spatial Diffuseness of a Source and the Surrounding Detector Geometry Evolution of the Lifetime Uncertainty as a Function of Distribution Uncertainty Future Experimental Considerations Conclusion A New Fitting Procedure for Obtaining the Individual Time- Walk Parameters with Reduced Complexity for Use with a Multi- Detector Array Existing Prompt Response Correction Method The New Prompt Response Correction Method Motivation for a New Prompt Response Correction Method Details of the New Prompt Response Correction Method Experimental Test of the New Prompt Response Correction Method Chapter Summary

5 Contents 5 7 A New Analysis Technique Providing an Improved Accuracy in Fast-Timing Measurements The Symmetrised Convolution Lifetime Measurement (SCLM) Technique Applying the SCLM Technique to Simulated Data Limitations of Both MSCD and SCLM Analysis Techniques Limitations of the MSCD Method Limitations of the SCLM Method Chapter Summary Summary and Outlook Summary Outlook A An Effective Variance to account for Uncertainties in Two Independent and One Dependent Variable(s) in a Least-Squares Minimisation Routine applied to a Surface Function utilising Existing Algorithms 197 B More Information on the Time-Measurement Walk (TMW) Parameters 203 Bibliography 213 Final word count: 40122

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7 List of Figures 1.1 Nuclidic chart showing known bound nuclei and their associated half-lives A chart of the different measurement techniques and associated ranges of mean lifetime in which they are implemented A schematic of a two-detector experimental configuration used to measure the time between consecutively emitted γ rays from both point-like, and spatially-distributed radioactive sources Two-neutron separation energies of isotones showing sharp changes at the magic shell closures Calculated energy levels for single-particle nucleon states using a modified simple harmonic oscillator potential Illustration of vibrational spherical deformation in nuclei Image of nuclear quadrupole deformation showing oblate and prolate deformed shapes Nuclear deformation parameterised by quadrupole deformation, β, and deviation from axial symmetry, γ A Nilsson diagram showing the single-particle energies of neutron states as a function of deformation, ɛ A single-particle orbit about a nuclear core displaying axiallysymmetric quadrupole deformation Nuclear excitation energy as a function of deformation, difference in spin and difference in the spin projection onto the symmetry axis between an initial and final state Partial level schemes of 138 Gd and 136 Sm A schematic of the de-excitation methods of a compound nucleus formed via a fusion-evaporation reaction A schematic drawing of a two-detector experimental arrangement used to measure the time difference between two coincidentally emitted γ rays Simulated data of typical γ-ray timing coincidences Circuit representation of a photomultiplier-tube anode output voltage as an RC circuit Current and voltage diagrams of a photomultiplier anode as a function of time

8 8 List of Figures 3.6 Example of a detector pulse crossing a voltage threshold, demonstrating the statistical deviation in timing pick-off, (jitter) resultant from input voltage noise Plots of typical signal voltage pulses at the input of a discriminator, highlighting the uncertainty in timing pick-off Circuit diagram of the photomultiplier tube voltage divider used in the present work Simulated data following a Gaussian-exponential convolution for various nuclear-state lifetimes in both gating conditions used in the MSCD-method data analysis A typical experimental configuration for performing a recoil-distance Doppler-shift method measurement A diagram of the relative intensities and energy separation of the Doppler-shifted and unshifted γ-ray energies A schematic energy-level diagram Energy-dependent cross sections of several interaction mechanisms between γ rays and NaI The intensity per unit energy of a NaI(Tl) detector to a monochromatic 137 Cs γ-ray source Schematic of the electron band structure within a scintillator material Circuit diagram of an off-axis photomultiplier tube voltage divider, highlighting the path traversed by the accelerated electrons A measurement of the 1333-keV and 1173-keV γ-ray energies in 60 Co using NaI, LaBr 3 and HPGe detectors Schematic of the experimental equipment available at the University of Jyväskylä featuring JurogamII, RITU and GREAT A technical drawing of the LaBr 3 detectors used in the present work A scale drawing of the LaBr 3 frame as viewed looking upstream of the recoil axis A scale drawing of the frame with the LaBr 3 detectors inserted as viewed looking upstream of the recoil axis, demonstrating the number convention used throughout this thesis A scale drawing of the array with detectors mounted showing the angle of inclination with respect to the recoil axis as viewed from the side A scale drawing of the LaBr 3 array frame showing the angles of the inner- and outer- four detectors with respect to the recoil axis in the horizontal plane Photographs of the LaBr 3 -detector array Diagram of the electronics used for the LaBr 3 array in the present experiment An energy spectrum of the 1333-keV γ ray emitted during the decay of 60 Co

9 List of Figures Time spectrum corresponding to the 2 + state in 60 Ni from a single detector pair, obtained by collecting data started by the 1173-keV γ ray from the decay of 60 Co and stopped by the 1333-keV γ ray Sum of the 28 detector-pair time spectra corresponding to the 910(20)-fs half-life 2 + state in 60 Ni A LaBr 3 -energy spectrum collected using a source of 152 Eu Measured timing uncertainty (jitter) as a function of energy The calculated timing uncertainty surface of the LaBr 3 detectors in the array for all combinations of start and stop energies between 100 kev and 900 kev The intrinsic photo-peak efficiency of the eight LaBr 3 detectors in the array Partial level schemes of 152 Gd and 152 Sm Plot of the prompt response difference function A simple drawing of a two-detector experimental configuration used to measure the time between consecutively emitted γ rays from both point-like, and spatially-distributed radioactive sources A scale drawing showing the different flight times for photons emitted from a given DSSSD pixel position towards a particular detector pair The time difference between the 867-keV and 244-keV internaltransition γ rays in 152 Sm between LaBr 3 Detectors 1 and 8, collected with a 152 Eu source positioned at the bottom-left of the vacuum chamber and at the bottom-right of the vacuum chamber when viewed upstream of the recoil direction axis A schematic illustration of the experimental configuration viewed from above LaBr 3 -energy spectra showing the total projection of the γ-ray coincidence matrix observed in a 30-ns coincidence window, and those in coincidence with the populating and depopulating γ rays corresponding to the 2 + state in 138 Gd The distribution of recoils implanted over the two side-by-side DSSSDs Time spectra of the 2 + states in 138 Gd and 136 Sm, both with and without the RaPToF correction being applied Time spectra showing the effect on timing measurements from the detector geometry used in the present work Time spectra showing the effect on timing measurements from the detector geometry of FATIMA at Super-FRS An example of simulated data showing the natural and reverse time spectra with N = 1000 counts in each spectrum and a distribution uncertainty of 0.11 ns A histogram of the results of lifetime calculations using the MSCD method with N = 1000 counts in each spectrum and a distribution uncertainty of 0.11 ns

10 10 List of Figures 5.12 The evolution in lifetime uncertainty as the distribution uncertainty changes, and for various levels of statistics, N The eight individual w C k CFD time walk components for the LaBr 3 array used in the present work Surface plots of the prompt response centroid for each detector pair as a function of the γ-ray energies used in starting and stopping a TAC Heatmaps showing the time measurement walks (TMWs) for each detector pair as a function of the start- and stop-signal γ-ray energies The individual time walk components used to describe the average time walk for all eight CFDs used to provide start or stop signals to a TAC Surface plots of the mean TMW for the LaBr 3 detector array as a function of the γ-ray energies used in starting and stopping a TAC Spectra of natural and reverse time data corresponding to the 2 + state in 138 Gd, comparing existing and new prompt response correction methods Simulated data representing the natural and reverse time spectra observed when measuring a nuclear state with a lifetime of 0.3 ns and a timing resolution parameterised by a standard deviation 0.27 ns with 1000 counts in each spectrum The lifetime uncertainty provided by the fitting program using the SCLM method and propagated from centroid uncertainties using the MSCD method. Also shown are the lifetime spectra using both methods Deviation of measured lifetimes from the known value, of 300 ps, as a function of N for both SCLM and MSCD methods Deviation of the measured mean lifetime from the known value, of 10 ps, as a function of counts, N, utilising the MSCD method B.1 Surface plots of the prompt response centroid for each detector pair as a function of the γ-ray energies used in starting and stopping a TAC

11 List of Tables 2.1 Weisskopf single particle transition probabilities for the first four valid multipoles in nuclear transitions Configuration for the Lyrtech ADC cards used in processing the energy signals from the LaBr 3 detector array Optimised P/Z values for the LaBr 3 detector array Configuration for the Lyrtech ADC cards used in processing the timing signals from the LaBr 3 detector array LaBr 3 -array detector efficiencies and peak-to-total measurements Parameter values describing the prompt response difference function, resulting from the fit to the data as described in the text Covariance matrix for the prompt response difference parameters listed in Table Lifetime results obtained from the data shown in Figure 5.7 using the MSCD method Parameter values used in describing the time-walk contribution of each CFD to a timing measurement using the LaBr 3 array Parameter values used in describing the variation in time measurement walk with respect to start and stop signals due to the detection of γ rays of various energies Covariance matrix for the eight parameters defining the average time walk contribution from both the start signal energy and stop signal energy used in the present work Lifetime results obtained from the uncorrected data, shown in Figure 5.7, and the corrected data, shown in Figure 6.6, using the MSCD method B.1 Covariance matrix for the 32 parameters defining the time walk contributions from the eight CFDs used in the present work

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13 Abbreviations ADC AIDA CFD DDCM DESPEC DSAM DSSSD FAIR FATIMA FIFO FWHM GCD method GO box GREAT HPGe MARA MSCD MWPC P/Z PMT PRD PRF RaPToF RC circuit RDDS RDM RIT RITU SCLM TAC TDR TFA TMW ToF Analogue-to-Digital Converter Advanced Implantation Detector Array Constant Fraction Discriminator Differential Decay-Curve Method Decay Spectroscopy Doppler-Shift Attenuation Method Double-Sided Silicon Strip Detector Facility for Antiproton and Ion Research Fast Timing Array Fan-In-Fan-Out Full Width at Half Maximum Generalised Centroid Difference method Gain-Offset box Gamma Recoil Electron Alpha Tagging High-Purity Germanium Mass Analysing Recoil Apparatus Mirror Symmetric Centroid Difference Multi-Wire Proportional Counter Pole/Zero Photomultiplier Tube Prompt-Response Difference Prompt-Response Function Relative Photon Time of Flight Resistor-Capacitor circuit Recoil Distance Doppler-Shift Recoil Distance Method Recoil-Isomer Tagging Recoil-Ion Transport Unit Symmetrised Convolution Lifetime Measurement Time-to-Amplitude Converter Total Data Readout Timing Filter Amplifier Time-Measurement Walk Time of Flight 13

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15 Abstract The effect of the spatial distribution of a radioactive source on timing measurements has been investigated with particular consideration toward the focal plane of recoil separators. The work conducted during this thesis is a precursor to understand the magnitude of such effects for the upcoming fast timing array (FATIMA) at FAIR. An experiment was undertaken at the University of Jyväskylä using the K130 cyclotron to accelerate a 36 Ar beam to 190 MeV, directed onto a 106 Cd target, to produce recoils of 138 Gd and 136 Sm via fusion-evaporation reactions. Recoils directed using RITU to the focal-plane DSSSD of GREAT were distributed over the majority of the 124-mm 40-mm extension of the DSSSD. A new array consisting of eight LaBr 3 detectors was used to measure the time between coincident promptγ rays emitted following the de-excitation of isomeric recoil states implanted into the DSSSD. Lifetimes were measured to be 213(20) ps and 200(100) ps for the first-excited 2 + states in 138 Gd and 136 Sm respectively. Positional information, extracted from the DSSSD, was used to correct for the difference in the timeof-flight of γ rays as they travelled from the implantation position to the LaBr 3 detectors. When accounted for, the lifetimes were remeasured to be 217(20) ps and 210(90) ps, respectively, showing no significant change in value or error. A method of quantifying the increase in uncertainty of a lifetime measurement due to the spatial distribution of the source and the position of the surrounding detectors, supported by simulation, has been provided to explain these observations. A new technique for extracting the time-walk from each of the CFDs in a multidetector array has been presented. The new technique offers a reduced complexity in calculations by accounting for the correlated time-walks present in time measurements from different detector-pairs sharing a common CFD. Work towards a technique for extracting lifetimes from time data has been presented. Dubbed the Symmetrised-Convolution Lifetime Measurement (SCLM) method, this technique essentially applies a model-dependent convolution of the prompt-response with nuclear exponential decay on both time spectra, obtained by inverting the start and stop conditions of a TAC, simultaneously and draws parallels to the Mirror Symmetric Centroid Difference method. 15

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17 Declaration of Authorship I confirm that no portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 17

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19 Copyright Statement i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the Copyright ) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intellectual property (the Intellectual Property ) and any reproductions of copyright works in the thesis, for example graphs and tables ( Reproductions ), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see DocuInfo.aspx?DocID=24420), in any relevant Thesis restriction declarations deposited in the University Library, The University Library s regulations (see and in The University s policy on Presentation of Theses 19

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21 [And] radiation is one of the things that s guaranteed to kill you. Professor Mark Blaxter 1 1 As reported by BBC News, URL accessed:

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23 Acknowledgements Firstly, I would like to acknowledge my supervisor, Dr David Cullen, for the role he has played in helping me throughout my studies and the opportunities I have had as a Ph.D. student. I also acknowledge the group at the accelerator laboratory in Jyväskylä for the experiences both offered and gained during my long-term attachment there. In particular, I would like to thank Dr Catherine Scholey for always making herself available to answer my questions whilst away from Manchester, and also for lending me a much-needed bicycle. I would like to thank the nuclear physics group at Manchester for their part in making my Ph.D. experience enjoyable, particularly Dr Paul Campbell, who has always made himself available to help with random physics questions, and the drongos in Room 4.19, who have offered their support and made the office a great place to work. I would also like to thank Dr B. S. Nara Singh for always being available to discuss ideas and concepts, and for the lightening-fast proof reading of anything I have given him. He also made sure that I took a break every now and then during the writing of this thesis that no doubt helped to keep my sanity in check. Lastly, I would like to thank my parents and girlfriend for their support during my Ph.D. studies and also for the patience required to deal with my being mostly unavailable throughout the period of writing of this document. 23

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25 Chapter 1 Introduction Measurements of the lifetimes of excited nuclear states are a crucial tool in the study of nuclear structure. It is well known that nuclear decay is a random process, with a probability per unit time described by an exponential function. The mean time value of the decay probability distribution defines the lifetime of a nuclear state. Equivalently, the concept of half-life may be used to describe the time period in which a population of radioactive nuclei is halved. The lifetime, τ, and half-life, T 1/2, are related by a constant, ln 2, in the standard and well-known equation, τ = T 1/2 ln 2. (1.1) Lifetime measurements of nuclear states are typically used to compute electromagnetic matrix elements that govern the transition between nuclear states. These matrix elements can be compared to those derived from theoretical consideration to allow a model-dependent probe into the nuclear structure. Lifetimes of nuclei far from stability have recently become available [1 4] and will be useful in testing and understanding various systematics close to the limits of bound nuclear structure. The structure of nuclei is reasonably-well described by the independent particle model [5], discussed in Section 2.1.1, in which nuclei are treated as spherical shells of protons and neutrons. Shells are considered to be completely filled when the 25

26 26 Chapter 1 number of protons or neutrons is equal to one of; 2, 8, 20, 28, 50, 82 or 126 [6]. Calculations of the excited nuclear-state energies, based on the independent particle model, tend to agree with experimental measurements when the proton and neutron numbers are close to a filled shell value. Figure 1.1 shows the nuclidic chart, that displays the known bound nuclei as a function of neutron (N) and proton (Z) numbers. Z, number of protons 137 Gd 136 Eu 135 Sm 138 Gd 139 Gd 137 Eu 138 Eu 136 Sm Sm Z=82 N=Z N=126 Z=20 Z=28 Z=50 N=50 N=82 Half-life (s) > <10-15 unknown Z=8 N=8 N=28 N=20 N, number of neutrons Figure 1.1: Nuclidic chart showing known bound nuclei and their associated half-lives. The colour used to represent each nucleus contains the half-life information, given to the nearest power of 10. The shell closures are indicated for protons and neutrons with horizontal (blue) and vertical (black) rectangles, respectively. The magnified section shows the region of nuclei studied in this thesis, with the specific nuclei, 138 Gd and 136 Sm, highlighted with a pink outline. Image adapted from Reference [7] according to Reference [8]. Non-deformed, spherical nuclei are typically observed in regions of the nuclidic chart where both the proton and neutron numbers are close to a shell closure, shown by the overlapping of the vertical and horizontal rectangles in Figure 1.1. Often, the spherical constraint of the independent particle model is lifted, and nuclei are considered to be deformed symmetrically about a single axis, or axially deformed, to form prolate or oblate shapes. Such deformation allows collective

27 Chapter 1 27 excitation of the nucleus via quantum mechanical phonon interactions, to form vibrating structures or via a permanent deformation of the nucleon probability distribution that allows collective rotating structures. The rotational excitations cannot be built upon a spherical nucleus due to symmetry, as a sphere under any given rotation about its centre is indistinguishable from a sphere that has not been rotated. The excitation energies of nuclear states for nuclei with a half-filled shell usually have better agreement with models that include deformation [9]. Theoretical models exist, some of which are discussed later, that are built upon features of nuclear structure and aim to correctly predict excitation energies and transition rates of various nuclear states. As the focus of the present work is on lifetime measurements, the examples of nuclear structure given in this thesis are restricted to those cases that can be tested using lifetimes or electromagnetic matrix elements derived from lifetimes. The examples of nuclear structure information include: Clustering of smaller groups of nucleons within a larger nucleus [10]. Nuclear decay modes that emit, for example, a 12 C or 16 O ejectile may have a transition probability that competes with other decay modes, such as alpha decay. These nuclei may then be explained as having clusters within their structure, increasing the probability of those clusters being ejected as a whole [10]. By calculating the expected lifetime from the interactions between such a cluster and the rest of the nuclear material, a direct comparison to measured lifetimes can be performed. Deviations in the shell gaps, where the ratio between the T = 1 and T = 0 isospin components of the nuclear force differs to that predicted using the independent particle model [11]. This difference can alter the single-particle excitation energies to the extent that the shell gaps at neutron values of 20 and 28 disappear in some neutron-rich nuclei. Reduced transition probabilities, calculated from measured lifetime values of nuclear states, typically agree with values derived from models describing single particle excitation at closed shells and are larger at the mid-shell nucleon numbers. The reduced transition probabilities can be compared to calculations based on specific excited configurations to deduce the most probable configuration.

28 28 Chapter 1 Shape coexistence, in which the collective rotational band structures may show evidence of a transition from an oblate to a prolate shape, or vice-versa, with increasing spin via a mixed configuration of both shapes [12]. Changes in the transition quadrupole moments, that can be calculated from lifetime measurements, indicate a change of the nuclear shape and thus shape changes can be observed. Isospin symmetry breaking, in which the transition matrix elements measured for nuclei with the same isospin may differ [13]. Such measurements aim to test the charge independence of the interaction between nucleons. The interaction between neutrons and protons, where reduced transition probabilities, derived from measured lifetimes, can be used to test possible interaction terms between the two types of nuclei [14, 15]. In this thesis, the lifetimes of the first excited states of both 138 Gd and 136 Sm are measured. Figure 1.1 shows these nuclei as existing in a typically deformed region of the nuclidic chart - where the proton numbers, of 64 and 62, respectively, are far from the nearest shell closures. In this region, the proton structure within the nucleus tends to favour collectivity, resulting in deformed ground state bands. The neutron structure, having a number closer to a complete shell than the protons, tends to be more spherical than the proton structure, allowing single-particle excitations. The two shapes compete within these nuclei, allowing mixing to occur between deformed and spherical configurations. Like other neutron deficient, N = 74 nuclei with even proton and neutron numbers, 138 Gd and 136 Sm exhibit two-quasineutron K π = 8 single-particle isomeric (> 1 ns half-life) configurations that decay to the collective yrast bands [16, 17]. In both nuclei, the assigned configuration for the K π = 8 isomeric state is ν{ 7 + [404]} ν{ 9 [514]}. The 2 2 half-life of the K π = 8 isomeric state has been established to be 6.2(2) µs in 138 Gd and 15(1) µs in 136 Sm [17 19]. The half-life provides information on the change between the single-particle and collective behaviours within these nuclei. Further details of this K-isomerism is discussed in Section The main focus of this thesis is on the measurement of nuclear-state lifetimes. Lifetimes are typically observed in a range extending from s and longer, as

29 Chapter 1 29 shown in Figure 1.1, and various experimental techniques are used to measure them in these different time ranges [20 23]. Figure 1.2 displays the different measurement techniques and the range of lifetimes which they are used to measure. Blocking DSAM RDM X-ray coincidences Electronic timing Lifetime (s) Figure 1.2: A chart of the different measurement techniques and associated ranges of mean lifetime in which they are implemented. The electronic timing has been extended (dotted line) to account for more recent fast-timing capabilities. Image adapted from Reference [20]. In the shortest time region, < 1 fs, blocking and X-ray coincidence techniques are used to measure lifetimes [20]. The Doppler-shift attenuation method (DSAM) is used to measure lifetimes between 1 fs and 1 ps [20]. The recoil distance method (RDM) is used to measure lifetimes between 1 ps and 1 ns, and the electronic timing method is used to measure lifetimes above 1 ps [20]. An extensive review of the different techniques available, produced by Nolan and Sharpey-Schafer, covers each of these techniques in detail [20]. As seen in Figure 1.2 some overlap exists between many of the lifetime measurement techniques, which is significant for the RDM and fast-timing electronic measurement techniques between s and 10 8 s. In the case of the first excited 2 + states in 138 Gd and 136 Sm, the lifetimes are of the order of s [24, 25], and therefore both the RDM and the electronic timing method are applicable to measure them. Of particular relevance to the present work, lifetime measurements using the RDM have been performed [24] on the ground-state band of 138 Gd. This measurement was performed at the target position, where an accelerated

30 30 Chapter 1 beam impacted a stationary target to induce a fusion-evaporation reaction at the University of Jyväskylä [26]. The RDM was implemented using the Köln plunger device [27], consisting of a target foil and, further downstream of the beam, a degrader foil to retard the recoiling reaction products and create two velocity regions. The two velocity regions give rise to two discrete Doppler shifts in the energy of emitted γ rays, manifest as two peaks in an energy spectrum whose relative intensity at various foil distances allows the measurement of nuclear state lifetimes. A potential issue with the RDM is the presence of systematic uncertainties arising from the unobserved statistical γ rays that populate a state of interest, often termed unobserved side-feeding [22]. This effect can be mitigated by performing a coincidence analysis, in which only those γ rays observed in coincidence with the γ ray that populates a state of interest are considered. Measurements made using the RDM are compared to measurements performed using fast-timing in the present work. As both techniques have independent systematic uncertainties, a comparison is useful in the identification of potential bias present in either technique. The RDM is discussed in detail in Section 3.3. In comparison with the RDM, the fast-timing technique is not restricted to being implemented only at the target position to detect prompt γ rays. The lifetimes of the K-isomeric states in both 138 Gd and 136 Sm, mentioned above, are large enough (order of 400 ns or more) for recoiling products of the fusion-evaporation reactions, used to produce them, to be separated from the beam particles and directed toward a focal plane of a separator. This results in an increased selectivity of the nuclei of interest and a corresponding reduction in the proportion of other events being observed, as many of the produced nuclei do not have isomeric states and thus decay before reaching the focal plane. In addition, the Recoil-Isomer-Tagging (RIT) [19, 28 34] technique may be used at the focal plane of a separator to select events of interest in which a γ ray was observed within approximately three times the half-life of the isomeric state. Experiments have been performed that demonstrate the high selectivity of the tagging technique; not just in RIT [35], that is implemented in the present work, but also tagging using other particles e.g. alpha [1, 36], proton [2 4, 37] and beta [38] decays. By positioning the fast-timing

31 Chapter 1 31 detectors at the focal plane of a separator, fast-timing can be used to measure short-lived state-lifetimes populated by the internal decays of these longer-lived isomeric states. In this thesis, the direct electronic timing technique is used to measure the lifetime of fast-decaying states, populated by isomeric states, of nuclei that are separated from beam particles and transported to a focal plane for use with RIT, discussed above. The electronic timing method of lifetime measurement is performed by measuring the time difference between a γ ray populating a nuclear level of interest and a γ ray depopulating the state. A minimum of two γ-ray detectors are required to measure this time difference, each outputting a signal pulse upon the detection of a γ ray. The time difference is then measured using the electronics components detailed in Section 3.2. A time spectrum of many measured time differences from a sample of nuclei will follow the known exponential decay law, ( N (t) = N 0 exp t t ) 0, (1.2) τ where N (t) is the population of a nuclear state at time, t, N 0 is the initial population of the state at time t 0 and τ is the mean lifetime. The lifetime of the state may be extracted from the time spectrum using one of several techniques [20]. In the case where the lifetime is larger than the timing resolution of the detection system, measurement of the gradient of the logarithm of the time spectrum allows the extraction of the lifetime. By taking the logarithm of Equation 1.2, the following relation is obtained; ln N (t) = ln N 0 t t 0. (1.3) τ It is evident from Equation 1.3, that the lifetime is the negative of the inverse of the gradient of the logarithmic time spectrum. If the lifetime is short compared to the detection system timing resolution, it may be observed as a centroid shift when compared with the time spectrum obtained for a pair of instantaneously detected, or prompt, γ rays i.e. with very short delay relative to the lifetime of

32 32 Chapter 1 the state and the timing resolution. This latter spectrum is referred to as the prompt response function (PRF). In typical fast-timing coincidence measurements, the γ-ray source is assumed to be point-like and the source-to-detector distances are fixed for all detectors, such as with measurements performed at the target position of a detector array [39 41]. Figure 1.3 (top) shows a simple picture demonstrating this for a source placed at the centre of two detectors. The distance between the point-source and each of the detectors is equal and, more importantly, remains constant over many observed γ decays. This ensures the specific detector geometry does not affect the measured time differences of the emitted γ rays, or, in the case of unequal distances, can be corrected for by the addition of a constant.

33 Chapter 1 33 γ rays Radioactive Detector 1 Point-Source Detector 2 Radioactive Source Distribution Possible Decay Points Figure 1.3: A schematic of a two-detector experimental configuration used to measure the time between consecutively emitted γ rays from a point-like radioactive source (top). As the distance each γ ray (pictured) has to travel to reach the detector is equal, there is a zero net relative time difference between measurements. The bottom diagram shows how uncertainty in source position can introduce a relative difference in the time of flight of each γ ray in a pair which may travel different distances to reach the surrounding detectors. However, at the focal plane of a spectrometer, recoils are implanted into a detector over an area that may be significant when compared to the distance to the surrounding fast-timing detectors, creating a distributed source of γ rays. Figure 1.3 (bottom) represents this source distribution, in which two possible points of γ-ray emission have been highlighted, demonstrating the differing distances each photon must travel to reach the detector. Multiple observations of the time difference from consecutively emitted γ rays is no longer relative to a single

34 34 Chapter 1 constant time-offset, as in the point-like case, and must be considered as a function of the source distribution and specific detector geometry. The inverse of the speed of light is 3.3 ps / mm, showing how sensitive picosecond-scale time measurements are to a small deviation in the γ-ray flight-path. As the emission point moves closer to one of the detectors it may also move away from the other, depending on the specific detector geometry, so time measurements may shift by upto twice this value. If the emission point of each measurement is known, along with the detector positions, the effect may be corrected for by subtracting the difference in time-of-flights (ToFs) of the γ-ray pair from the time difference observed from that pair of γ rays. Plans for experiments that utilise large-area focal planes, such as with Super-FRS at FAIR, having a potential focal-plane geometry of 28 cm 8 cm, could be problematic when attempting to perform picosecond-scale fast-timing measurements using an array of LaBr 3 detectors due to the difference in ToFs observed. In preparation for such experiments, the work presented in this thesis investigates the effect on lifetime measurements of the spatial distribution of recoil products at the focal plane of the Recoil-Ion-Transport-Unit (RITU) [42] at the University of Jyväskylä. An array of fast-timing LaBr 3 detectors was designed and installed at the Gamma-Recoil-Electron-Alpha-Timing (GREAT) focal plane spectrometer [43] to measure the 2 + states in the ground state bands of 138 Gd and 136 Sm that are populated from the decay of isomeric excited states. The method of identifying the implantation position, over the two 60 mm 40 mm Double-Sided-Silicon-Strip-Detectors (DSSSDs), and the results from this experiment are discussed in Chapter 5. In addition, a new method for calibrating a fast timing detector array, consisting of more than two detectors, to account for the γ-ray energy dependence of the timing of the γ ray is presented in Chapter 6. Chapter 7 demonstrates a potential lifetime extraction technique based on the mirror-symmetric centroid difference (MSCD) method, a variation of the centroid shift method.

35 Chapter 2 Nuclear Theory 2.1 Nuclear Structure and Deformation Nuclear structure refers to the current understanding of how nuclei are bound together. A comparison of the observed properties of nuclei can be made to those derived from theoretical predictions in order to gain information on the structure of those nuclei. An example of such an estimated structure, referred to as a model, was provided by Rutherford when he suggested that the positive and negative charge distributions within the atom had very different radii [44]. Before that, it was widely thought that nuclei consisted of overlapping spherical distributions of positive and negative charge. The motivation for creating a new model was provided by experimental data showing alpha particles scattering at large angles when incident on a thin gold foil. If the positive and negative electric charges were equally distributed, the alpha particles would be assumed to scatter by displacement of the positive and negative charges within the atom upon approach. This displacement would then create an electric field to deflect the incoming particles. This model, however, was insufficient to explain the large scattering angles observed. Since then, many models have been developed to describe observations of nuclear excitations and level structures, and various properties of nuclei such as state lifetimes. 35

36 36 Chapter 2 A discussion of nuclear structure will begin with the shell model. Following this, various other models will be discussed that have been developed to deal with shortcomings of the shell model by modelling nuclei as deformed spheres The Shell Model The shell model, or independent particle model [5], describes the energy of a single nucleon within a nucleus. The shell model was designed in an attempt to explain various observations of abrupt changes in some nuclear properties, discussed below. Such behaviour was consistently observed for particular numbers of either protons or neutrons within different nuclei. These nucleon numbers are 2, 8, 20, 28, 50, 82 and 126, and are now known as the magic numbers [6]. Figure 2.1 demonstrates the effects of the magic numbers as a sudden localised decrease in the two-neutron separation energies of several isotonic chains as the neutron number increases above a magic value. These energies are relative to those calculated using the liquid drop model [8], a model that does not account for magic numbers.

37 Chapter Two-neutron separation energy (MeV) O Ca Ni Kr Cd Number of neutrons Dy Pb U Figure 2.1: Two-neutron separation energies of even-even isotopes showing sharp changes at the magic shell closures. The energy scale is the difference between experimentally determined two-neutron separation energies and those calculated using the liquid drop model, that does not account for magic numbers. Data for image taken from Reference [45]. Other examples showing abrupt changes at magic numbers include an increase in the energy of alpha particles emitted from radon isotopes with over 126 neutrons, neutron capture cross sections of many nuclei showing a dip of around two orders of magnitude at magic numbers of neutrons or protons and the nuclear charge radii having sudden systematic discontinuities at magic neutron numbers [8]. The shell model attempts to account for the magic numbers in a way that is analogous to the electron orbital solutions in atomic physics [5]. To describe the quantum mechanical nucleus, a mean potential is used as a simplification of the interactions between many nucleons in order to construct a Hamiltonian. This mean potential is considered to depend only on radial distance from a central point and so is invariant under rotational transformations. Several potentials can be used to approximate the nuclear potential. A potential that provides good

38 38 Chapter 2 agreement with observations is the Woods-Saxon potential that takes the form, V (r) = V e r R a, (2.1) where V 0 indicates the depth of the potential, R is the mean nuclear radius and a describes the diffuseness of the potential [46]. The Woods-Saxon potential does not lead to analytically solvable solutions for nuclear energy states so a simple harmonic oscillator potential is used instead for discussion. The simple harmonic oscillator potential takes the form, V (r) = 1 2 Mω2 r 2, (2.2) where ω is the oscillator frequency and M corresponds to the nucleon mass. The time-independent non-relativistic Schrödinger equation is used to find the energy values (eigenvalues) of the standing waves (eigenfunctions) that describe the states of particles in a given potential. The Schrödinger equation is applied to the potential, V(r), to give ( ) p 2 Hψ = 2M + V (r) ψ nlm (r) = E nlm ψ nlm (r), (2.3) where H is the Hamiltonian operator, p is the momentum operator and n, l, and m are quantum numbers used to denote the eigenfunctions of ψ. The n quantum number is the principle quantum number and can take the value of 1 and integers above 1. The l quantum number is the orbital quantum number and can take integer values from 0 to n 1. The m quantum number is the magnetic quantum number and can take integer values from l to l. The energy eigenvalues of ψ nlm are given by E nl = ( 2n + l 1 ) ω. (2.4) 2 The m quantum number in Equation 2.4 has been dropped due to degeneracy in energy. The energy levels calculated with the simple harmonic oscillator potential are shown in Figure 2.2 (left).

39 Chapter 2 39 Figure 2.2: Calculated energy levels for single-particle nucleon states using a simple harmonic oscillator potential (left). Corrections to the potential are shown that account for nucleon screening (centre, right) and the interaction between nucleon spin and orbital angular momentum (right). With both corrections, the experimentally verified magic numbers are correctly reproduced. Image taken from Reference [5].

40 40 Chapter 2 The simple harmonic oscillator produces energy states dependent on the n quantum number. Due to degeneracy, multiple states have similar energies and the numbers on the left of Figure 2.2 give the total number of states below that energy. These numbers are seen to match the magic numbers at first, but deviate as the excitation energy increases. An improvement on the simple harmonic oscillator potential is made by the addition of an l 2 term. This term has the effect of flattening the potential at very small distances, and is added to account for the nucleons inside a nucleus experiencing approximately the same force as they are completely surrounded by other nucleons. The strong nuclear force, responsible for attracting the nucleons to each other, is short ranged, justifying the approximately constant potential within the nuclear radius. The l 2 correction term is not required when using the Woods-Saxon potential. To correctly recreate the magic numbers, an additional l s, term is added to the potential. This correction is required to account for the coupling of nucleon spin to the orbital angular momentum of the nucleus. With the l 2 and l s correction terms included, the potential given by Equation 2.2 then becomes V (r) = 1 2 Mω2 r 2 + Cl 2 + Dl s, (2.5) where C and D are constants to be empirically deduced. The population of nuclear states is subject to the Pauli principle, in which no two fermion particles may occupy the same quantum state. Nucleons are classed as fermions due to having a spin value of 1. The result of the Pauli principle is 2 that it is not possible for all nucleons to exist at the lowest possible energy. In general, nucleons arrange such that protons and neutrons each fill up the energy levels shown in Figure 2.2, favouring the least energetic, or most tightly bound, states. Due to the proton having a positive charge, the energy levels are increased in comparison to the neutron energy levels due to Coulomb repulsion competing with the attractive nuclear force.

41 Chapter Collective Vibration In regions far from a shell closure, some nuclei exhibit excitation due to collective behaviour. An example of collectivity is isoscalar vibrations, in which the nuclear volume is modelled as being an instantaneous deformation about a spherical equilibrium shape. Nuclei are excited, by means of phonons, into quantised vibrational modes [8] described by a set of equations known as the spherical harmonics. A point on the nuclear surface has a coordinate given by R (θ, φ) = R av + λ α λµ Y λµ (θ, φ), (2.6) λ 1 µ= λ where R av is the average nuclear radius, taken to be R 0 A 1/3 with R 0 describing the radius of a single nucleon and A the number of nucleons [47]. The angular momentum of the phonon, λ, and its projection onto the symmetry axis, µ, are quantum numbers that describe a particular spherical harmonic. The parameters, α λµ are coefficients of the spherical harmonics. States with low angular momenta generally dominate over those with higher angular momenta, however, certain properties of nuclei exclude contributions from those states with λ < 2. With λ = 0, Y 0µ describes a monopole excitation of the spherical nuclear surface, however, this excitation-mode is disallowed due nuclear matter being modelled as incompressible. Dipole excitations are described by λ = 1, and amount to a displacement of the nucleus. Structural information may be inferred from dipole excitations by observing the strength of giant dipole resonances or pygmy dipole resonances. The giant dipole resonance results from an oscillation of the proton structure in a manner that is out of phase with that of the neutron structure [5]. Pygmy dipole resonances are widely interpreted to be a result of excess neutrons oscillating out of phase with an isospin-saturated (N Z) core [48, 49]. The next mode of excitation considered is the quadrupole deformation, given by λ = 2. The λ quantum number can be any positive integer above and including 2 whilst the µ quantum number is limited to integers in the range λ to λ. Figure 2.3 shows different vibrational modes of excitation for λ = 2 to λ = 4.

42 42 Chapter 2 Figure 2.3: Illustration of vibrational spherical deformation in nuclei. Quadrupole (λ = 2), octupole (λ = 3) and hexadecapole (λ = 4) excitations are pictured. Image taken from Reference [47] Collective Rotation A nucleus with permanent deformation in its ground-state shape has its spherical symmetry broken. Therefore, a component of rotation, perpendicular to an axis of symmetry, causes a time-dependent variation in the nucleon probability distribution. Deformed nuclei that exhibit rotation are generally found in the ranges, A 80, 150 < A < 190 and A > 220, where A is the atomic mass number [8, 50]. The deformed nuclear surface is described by the equation R (θ, φ) = R av [1 + α λµ Y λµ (θ, φ)], (2.7)

43 Chapter 2 43 where R av, α λµ and Y λµ (θ, φ) are defined as discussed in Section [47]. As before, the quadrupole (λ = 2) rotational mode is usually the dominant one. A quadrupole deformed nuclear surface can be described by the coefficients of the spherical harmonics in which λ = 2, i.e. α 2µ, with µ ranging from 2 to 2. In the nuclear quantum mechanical system, two internal degrees of freedom exist and are labelled β and γ [47]. The β degree of freedom is the quadrupole deformation parameter and the γ degree of freedom is the triaxiality parameter, given in Equation 2.8. Symmetry considerations in transforming the coordinate system between the intrinsic frame of reference and the lab frame of reference require only two independent variables in the α 2µ quadrupole coefficients [47]. It is therefore given that α 2 2 = 1 2 β sin γ, α 2 1 = 0, α 20 = β cos γ, (2.8) α 21 = 0, and α 22 = 1 2 β sin γ. Values of γ in the range 0 to 60 describe all collective behaviours in rotational nuclei. For γ = 0, a prolate collective shape is formed. For γ = 60, an oblate collective shape is formed. For values between this range, the nucleus is deformed in a way that is not symmetrical about a single axis. This is known as triaxial deformation. Prolate deformed nuclei can be described as rugby-ball shaped while oblate nuclei can be described as the shape of the Smarties R confectionery. Figure 2.4 shows a projection of the oblate (left) and prolate (right) shapes.

44 44 Chapter 2 z z x O a c a y x O a c a y Figure 2.4: Image of nuclear quadrupole deformation showing oblate (left) and prolate (right) deformed shapes. The z-axis is the axis of symmetry in each case. Image taken from Reference [51]. The value of β is positive in the case of a prolate shape and negative in the case of an oblate shape. It is noted that, for K π = 8 isomers (discussed in Section 2.1.7), rotation of single-particle excited states of the prolate deformed (β > 0) nucleus occurs about the symmetry axis, with γ = 120 [52]. Figure 2.5 shows the nuclear deformation at extreme points on the surface defined by β and γ.

45 Chapter oblate β γ 0 prolate single-particle rotation spherical collective rotation oblate prolate Figure 2.5: Nuclear deformation parameterised by quadrupole deformation, β, and deviation from axial symmetry, γ. It can be seen that single-particle configurations (γ = 60, 120 ) rotate about the symmetry axis and the collective configurations (γ = 0, 60 ) rotate about an axis perpendicular to the symmetry axis. Image adapted from Reference [53]. In the case of an axially symmetric rotor (γ = 0 ), the collective rotational excitations of the nucleus depend on angular momentum, I, and the energy is given by E rot = 2 I (I + 1), (2.9) 2I where is the reduced Plank constant, I is the moment of inertia and I is the angular momentum quantum number [8]. From Equation 2.9, the energy ratio of the states with spin I = 4 and I = 2 is calculated to be which is characteristic of rotational excitation. E (4) E (2) = 10 3 = 3. 3, (2.10)

46 46 Chapter The Nilsson Model The Nilsson model implements the single-particle nuclear shell model under the consideration of nuclei being deformed with axial symmetry. A deformed Hamiltonian [5] is used, based on the harmonic oscillator potential introduced in Section 2.1.1, that takes the form, H = p2 2m mω2 0r 2 mω0r π 3 5 δy 20 (θ, φ) + Cl s + Dl 2, (2.11) where ω 0 is a spherical oscillator frequency, l and s are the shell model quantum numbers for orbital angular momentum and spin respectively, Y 20 (θ, φ) is the spherical harmonic corresponding to a quadrupole deformation along the symmetry axis and the deformation parameter, δ, is related to both the difference between the semi-major and semi-minor nuclear ellipsoidal axes, R, and the root-mean-square radius, R r.m.s., by [54] δ = R R r.m.s.. (2.12) The quadrupole deformation parameter, β, is related to δ by β = 4 3 and to another deformation parameter, ɛ 2, by [54] π δ, (2.13) 5 16π ɛ 2 β =, n N. (2.14) 5 3n n In the case of a small deformation, the Y 20 (θ, φ) term can be considered as a perturbation to a spherical shape, resulting in a small energy shift of E (Nljm) = 2 ( 3 ω 0 N + 3 ) [ 3K 2 j (j + 1) ] [ 3 j (j + 1)] 4 δ, (2.15) 2 (2j 1) j (j + 1) (2j + 3) where N, l, and m are shell model quantum numbers, the total angular momentum of a single nucleon is j = l + s and K is the projection of the total angular momentum onto the symmetry axis, given by the sum of the m single-particle

47 Chapter 2 47 contributions. The predicted single-neutron excitation energies for deformed nuclear states are shown in Figure /2[651] 7/2[514] 1/2[660] 5/2[642] 3/2[402] 1/2[400] 1/2[770] 1/2[640] 3/2[761] 13/2[606] 3/2[521] 5/2[402] 1/2[521] 7/2[633] 9/2[505] 82 5/2[523] Single-particle energy (ħω) /2[411] 1/2[420] 3/2[422] 3/2[541] 7/2[523] 7/2[404] 1/2[550] 5/2[413] 5/2[532] 3/2[402] 9/2[514] 1/2[400] 1/2[431] 5/2[402] 2d 1h 3/2 11/2 3s 1/2 1g 7/2 2d 5/2 7/2[523] 11/2[505] 11/2[505] 3/2[402] 3/2[411] 1/2[431] 1/2[400] 9/2[514] 3/2[422] 7/2[404] 5/2[402] 1/2[411] 1/2[550] 5/2[413] 3/2[532] 3/2[651] 1/2[530] 1/2[660] 5/2[532] 5/2[642] 1/2[541] 1/2[651] 1/2[660] 1/2[411] 5.0 1/2[301] 3/2[431] 5/2[422] 7/2[413] 1/2[440] 50 1g 9/2 3/2[431] 5/2[422] 9/2[404] 9/2[404] 7/2[413] 1/2[301] 1/2[420] 3/2[541] 3/2[301] 1/2[301] 1/2[550] 3/2[541] 1/2[301] 5/2[303] 5/2[532] 1/2[541] 3/2[301] Deformation ( ) 2 Figure 2.6: A Nilsson diagram showing the single-particle energies of neutron states as a function of deformation, ɛ 2. The highlighted states are those that correspond to the isomeric K π = 8 configuration for the N = 74 isotones with deformation, ɛ Image adapted from Reference [54]. Deformed nuclei are no longer isotropic and thus have a definable orientation

48 48 Chapter 2 in space, specified by the projection of the total angular momentum on the symmetry axis, K. A set of Nilsson quantum numbers are constructed that are the symmetry-axis projections of the shell model quantum numbers. The Nilsson quantum numbers, used to define a nuclear eigenstate, are usually given as Ω π [N n z Λ], (2.16) where n z, Λ and Ω are the symmetry-axis projections of N, l and j respectively and π defines the parity of the state [47]. The projection of the total angular momentum of a nucleus onto the symmetry axis, K, is equal to the sum of the individual contributions from the constituent nuclei, Ω i, where the index i is used to denote a given nucleon. A visual representation of the Nilsson quantum numbers is offered by Figure 2.7. Figure 2.7: A single-particle orbit about a nuclear core displaying axiallysymmetric quadrupole deformation. The mapping of the shell model quantum numbers (l, s, j) to the Nilsson quantum numbers (Λ, Σ, Ω) onto the axis of symmetry (z) is shown. Image taken from Reference [55].

49 Chapter Selection Rules The selection rules describe which transitions between nuclear states are allowed under the consideration of conserved angular momentum and parity [8]. For a nucleus in an initial state with spin I i and parity π i, decaying to a final state with spin I f and parity π f, the following rules must be obeyed: I i =λ + I f π i =π γ π f, (2.17) where λ and π γ are the multipolarity and the parity of the photon transferred during the transition, respectively. The spin values in Equation 2.17 are shown to be vector quantities, allowing λ to take a range of values given by I i I f λ I i + I f. (2.18) As no intrinsic nuclear monopole interaction is allowed, λ = 0 is forbidden but higher orders of λ are valid. The parity values can be either 1 (even) or 1 (odd). This means that the parity of the photon must be even if the initial and final nuclear states have the same parity, or odd if those states are of different parity. Emitted photons can be of either electric or magnetic character, each having a different parity for a given multipolarity. The parity of the photon, as a function of its multipolarity, is stated as Electric : π γ = ( 1) λ Magnetic : π γ = ( 1) λ+1. (2.19) Given these rules, it is possible to describe certain transitions as being allowed or not. Section builds on these rules in terms of the probability of such decays. Considering the case of a nucleus with an even proton number and an even neutron number, usually referred to as an even-even nucleus, the constituent nucleons are all paired and so the total nuclear spin is zero and the parity is even. The nucleus must then have a ground state of 0 +. To excite this nucleus into a state of 2 +

50 50 Chapter 2 requires a photon with multipolarity of two and even parity, according to the rules in Equation Using Equation 2.19, it is clear that the interaction is of electric character, thus the only allowed transition is that of an E2, or electric quadrupole. It may be more useful to consider the possible modes of decay from an excited state, without knowing exactly what the final state is. In this case, still using the example of a 2 + excited state, it is clear that λ = 1 is a valid multipolarity according to Equation 2.18 and thus, to maintain even parity, the transition is magnetic in nature. The state then has two possible modes of decay, either an electric quadrupole or a magnetic dipole, M1. Both of these decay modes are in competition and the intensities of each can be used to infer information about the structure of particular nuclei, in a similar manner to that discussed in the next section Reduced Transition Probabilities The probability per unit time, for a nucleus in an initial excited state of spin I i, ψ i, to decay to a final less-excited state of spin I f, ψ f, is given by T (σλ; I i I f ) = 8π (λ + 1) λ [(2λ + 1)!!] 2 ( ) 2λ+1 Eγ B (σλ; I i I f ), (2.20) c where λ is the multipolarity of the transition, σ is used to denote the nature of the transition as either electric (E) or magnetic (M), E γ is the energy of the photon emitted as a result of the transition and B (σλ; I i I f ) is the reduced transition probability [56]. The reduced transition probability, in terms of the reduced matrix elements of the multipole interaction tensor, M (σl), is defined as B (σλ; I i I f ) = 1 2I i + 1 ψ i M (σl) ψ f 2. (2.21) For nuclear states that decay predominantly to a single less-excited state, T (σλ; I i I f ) can be calculated from the mean lifetime, τ as τ = 1 T (σλ; I i I f ). (2.22)

51 Chapter 2 51 Equation 2.22 assumes a negligible component of internal conversion, in which the wavefunction of an atomic electron overlaps with the nuclear wavefunction and allows nuclear transitions to occur without the involvement of a γ ray. As mean lifetimes are experimentally observable, Equations 2.20, 2.21 and 2.22 allow the transition matrix elements to be determined. An approximation of the wavefunctions for the magnetic and electric multipole cases, under the assumption of a single proton transition in a spherical nucleus, gives rise to the Weisskopf estimates for single particle transition probabilities [8]. For the electric transitions, σ = E, the Weisskopf transition probability is calculated using T (Eλ) 8π (λ + 1) λ [(2λ + 1)!!] 2 e 2 4πɛ 0 c ( ) 2λ+1 ( ) 2 Eγ 3 cr 2λ 0 A 2λ 3, (2.23) c λ + 3 where ɛ 0 is the vacuum permittivity, R 0 is the radius of a nucleon and A is the nucleon number. The Weisskopf single particle transition probability for magnetic transitions is ( T (Mλ) T (Eλ) µ p 1 ) 2 ( ) 2 ( ) 2 λ + 3 R 2 0 A 2 3, (2.24) λ + 1 m p c λ + 2 where m p is the mass of a proton and µ p is its free nuclear magnetic moment. The Weisskopf estimates for the first four multipoles are shown in Table 2.1, where the probabilities are in units of s 1 and γ-ray energies are in units of MeV. Multipolarity (σλ) Transition probability (s 1 ) E A E γ E A E γ E A 2 7 E γ E A E γ M E γ M A E γ M A E γ M A 2 9 E γ Table 2.1: Weisskopf single particle transition probabilities for the first four valid multipoles in nuclear transitions. The γ-ray energy, E γ, is in units of MeV.

52 52 Chapter 2 As the Weisskopf estimates are for single particle excitations, the observation of significant deviations from the values presented in Table 2.1 suggest a collective mode of excitation. In some highly collective nuclei, the E2 transition probability is observed to be higher than that of the M1 transition probability despite being several orders of magnitude lower in the Weisskopf estimates [8]. For axially-symmetric rotationally-deformed nuclei, the intrinsic electromagnetic quadrupole moment is given [47], to first order, as Q 0 = 3 5π ZeR 0 2 β, (2.25) where Z is the number of protons and β is the axially-symmetric quadrupole deformation parameter as given in Section Equation 2.25 relates nuclear deformation to an intrinsic quadrupole moment. The reduced transition probability for an E2 transition in a quadrupole deformed rotational band, between an initial state with spin J + 2 and a final state with spin J, is given by B (E2; J + 2 J) = 15Q π (J + 1) (J + 2) (2J + 3) (2J + 5). (2.26) A measurement of a nuclear state lifetime can be used to infer information about the collective deformation of that state using Equations 2.20, 2.25 and Isomeric Nuclear States Isomeric nuclear states are excited configurations in which the process of deexcitation is hindered such that the lifetime of the state exceeds 1 ns [57]. Isomers exist largely due to differences in either configuration (cf. Equation 2.21) or spin between a given state and available states in which decay can occur. In such cases, the overlap between wavefunctions through the electromagnetic interaction is small, hindering the transition rate. Figure 2.8 (left) shows excitation energy as a function of shape elongation. A secondary minima is demonstrated at large

53 Chapter 2 53 deformation, forming a metastable state in which the nucleus cannot decay without additional energy or quantum-mechanical tunnelling though the energy barrier. Energy shape isomer Energy spin trap Energy K-trap Shape elongation Spin Spin projection Figure 2.8: Nuclear excitation energy as a function of deformation (left), difference in spin (centre) and difference in the spin projection onto the symmetry axis between an initial and final state (right). Also depicted (top) are sketches of the types of motion each region describes. Image taken from Reference [57]. In the case of spin isomerism, both the magnitude of the spin itself and its projection onto the symmetry axis, in the case of prolate deformed nuclei, is considered. Large differences in spin between an initial and final allowed state require a high multipolarity transition. As discussed in Section 2.1.6, the probability of a transition occurring per unit time decreases rapidly as multipolarity increases, hence there is an increase in the lifetime of the excited state. For states where a local minimum energy is obtained with a large projection of the total angular momentum along the symmetry axis, K, isomeric states can be formed. To de-excite such nuclear states to a state of lower energy, a large change in the value of K is required. Such a large change in K may be naively assumed to require an interaction with a high multipolarity, obeying the following soft selection rule for allowed transitions: K λ, (2.27)

54 54 Chapter 2 where λ is the multipolarity of the radiated photon. This rule is observed to be broken in several nuclei and has lead to the adoption of the degree of forbiddenness, ν = K λ, as a means of parameterising this behaviour. Each increase in the degree of forbiddenness systematically, with some anomalies, reduces the transition probability by a factor between 30 and 200 [58, 59]. Known as the hindrance per degree of K-forbiddenness (or reduced hindrance), this phenomenon is described by the equation, ( τ f ν = τ w ) 1 ν, (2.28) where τ w is the lifetime according the Weisskopf transition probability estimates and τ is the empirically observed isomeric state lifetime. Some nuclei have very low reduced hindrances, suggesting K may not be a good quantum number. One explanation to account for this behaviour suggests a transient breaking in the axial symmetry of the prolate deformed shape, allowing transitions to occur by tunnelling through triaxial configurations [57]. Another explanation is that there is a mixing between the high-energy isomeric state and a lower-energy rotational state with the same spin. Systematics show a relationship between the energy difference of these two states and the reduced hindrance [57]. Other explanations include high spin K-isomers exhibiting Coriolis mixing [59] and the mixing of states in regions of high level density. The K π = 8 isomeric states in 138 Gd and 136 Sm are examples of K-forbidden transitions in the N = 74 isotones [16]. Figure 2.9 shows the partial level schemes for both 138 Gd (left) and 136 Sm (right), highlighting the K π = 8 isomeric states in each case.

55 138 Gd 136 Sm Figure 2.9: Partial level schemes of 138 Gd and 136 Sm showing the energies of states in both nuclei. The arrows represent possible γ ray decay paths and the number shown is the energy of that γ ray. The width of the arrows indicates the relative intensities of the γ rays and the white component of the arrow indicates the proportion of the intensity that is due to internal conversion. The K π = 8 isomeric states have been highlighted. Image adapted from References [17] and [19]. Chapter 2 55

56 56 Chapter 2 The decay of the above isomers to the K π = 0 + ground state bands, via 7-times K-forbidden E1 transitions, is observed with a reduced hindrance of f ν = 24.1 and f ν = 25 for 138 Gd and 136 Sm, respectively [16, 18]. These isomeric states have been assigned to the 7 + [404] 9 [514] two-neutron quasiparticle configuration. 2 2 The bands seen to populate the K π = 8 isomeric states in Figure 2.9 are known four-quasiparticle (two-neutron, two-proton) configurations [17, 19] When the nucleus is excited above 8, competition exists between the collective rotational ground state band configuration and the two-quasineutron isomeric configuration. Due to the charge independence of the nuclear force, the K π = 8 isomeric states have also been observed in regions of the nuclear chart with Z 74, corresponding to a region with atomic mass of A 170 (cf. Figure 1.1) [17]. As an example, the Z = 72 Hf isotopes have been observed [60] to contain the same 7 + [404] 9 [514] 2 2 two quasi-particle configuration in the proton structure. The study of systematics in this region allows the matrix elements of the nuclear force that governs the overlapping of nuclear states with different deformation to be determined. 2.2 Monte Carlo Calculations Whilst not unique to the field of nuclear physics, Monte Carlo calculations can be employed to solve problems that may be difficult to solve analytically. Monte Carlo calculations are used to solve problems statistically, rather than analytically, by employing the use of random numbers. An example of solving a problem statistically, that is related to nuclear physics, is to find the percentage of radioactive nuclei that remain after a decay period of three half-lives. Note that this example can be solved analytically to verify the Monte Carlo solution, but not all problems may be so easily solved. As nuclear decay is a statistical process with a probability distribution described by an exponential function, the solution is obtainable simply by generating many random numbers following this distribution, and calculating the percentage of those numbers that happen to be above three half-lives.

57 Chapter 2 57 Computer algorithms are deterministic and so true randomness does not exist. True random numbers must be derived by observing properties of naturally occurring randomness, such as nuclear decay or statistically-described thermal fluctuations. A pseudorandom number generator, by contrast, is deterministic, however, it may be considered to be good if the generated number sequence has a long period (i.e. it does not repeat for any practical use) and passes randomness tests such as the spectral test or k-distribution tests [61, 62]. The reproducibility of pseudorandom number generators may also be considered an advantage as it allows for the repeatability of results obtained from the numbers. A pseudorandom number generator is used to produce random numbers following a uniform probability distribution over a range of possible values [62]. To generate numbers that follow arbitrary probability distributions, the uniform distribution must be transformed into the desired distribution. If the distribution is analytically definable, such as with exponential nuclear decay, the cumulative integral form of the normalised probability distribution function can be used to define a continuous transform of the random uniform distribution into the analytical form. This process is equivalent to the mapping of an interval of the uniform probability distribution to a bounded area of the analytical distribution with equal magnitude to the uniform value. It is possible to discretise the mapping of uniformly distributed random numbers to an analytically definable probability distribution function by choosing discrete intervals in which to compute the probability. The cumulative integral of this set of probabilities can be approximated as the sum of all probability values upto a given point, ensuring that the total cumulative integral is normalised to one.

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59 Chapter 3 Experimental Background 3.1 Heavy-Ion Fusion-Evaporation Reactions Nuclear fusion defines the process in which two initially separate nuclei join together to form a single nucleus. Fusion is inhibited by the electrostatic potential, or Coulomb barrier, between the nuclei involved and therefore require a large kinetic energy, of order 1-10 MeV per nucleon, to overcome this barrier and form a compound nucleus [8]. Atoms are ionised and accelerated to form a beam with large linear momenta. When a beam of nuclei, with sufficient energy to fuse, is incident on a stationary target, compound nuclei with large angular momenta, of order , are formed [5, 8]. The angular momentum of a compound nucleus, l, is stated in terms of the linear momenta of a beam particle, p, and the distance between the two nuclei, r, as [55] l = r p. (3.1) It is implicit from Equation 3.1 that more massive, therefore larger, nuclei can interact at larger values of r, still undergoing fusion, to produce large angular momenta. 59

60 60 Chapter 3 Compound nuclei have large angular momenta and are generally neutron-deficient, meaning they have less neutrons than would be typically observed for stable nuclei with the same proton number. The fused product is therefore closer to the N = Z line shown in Figure 1.1. An increasing number of neutrons is required to overcome the Coulomb repulsion exhibited by protons, via the strong force, hence nuclei in this region are generally unstable. In order to lower their energy and spin, compound nuclei may radiate neutrons, protons and alpha particles, although the latter two cases are inhibited by the Coulomb barrier of the compound nucleus [8]. Figure 3.1 indicates where the various stages of fusion-evaporation reactions occur in terms of excitation energy and spin, highlighting the particle evaporation with blue arrows. Figure 3.1: A schematic of the de-excitation methods of a compound nucleus formed via a fusion-evaporation reaction. The large energy and spin obtained from its formation is lost through particle radiation and subsequently γ-ray emission. Image taken from Reference [55].

61 Chapter 3 61 This evaporation of particles continues until the compound nucleus no longer has enough energy to overcome the binding potential by means of further evaporation [5]. Once this threshold is passed, the compound nucleus lowers its energy by means of γ-ray emission until it reaches the lowest energy state for a given spin, or the yrast state as labelled in Figure 3.1. The compound nucleus continues to lower spin by de-excitation of γ rays, closely trailing the yrast line until transitions to the ground state occur and a minimum energy state is achieved. A rotating nucleus is necessarily deformed from sphericity with a minimum multipolarity of λ = 2, or quadrupole deformation - see Section for more details. According to the Weisskopf single particle transition rates, discussed in Section 2.1.6, M1 and E2 are the most probable transition multipolarities between closely-packed states of the same parity. The large collectivity of a high-spin nucleus allows a stronger E2 transition rate relative to the M1 transition rate, contrary to the single particle case, due to quadrupole transitions in the rotational-band decay. 3.2 Direct Electronic Timing Direct electronic timing refers to the timing of electronic pulses generated by detectors when radiation is incident. By detecting two events with a pair of detectors, the difference in time between those events can be measured. Figure 3.2 shows a schematic experimental configuration using two detectors to measure the time difference between two coincidentally-emitted γ-ray photons.

62 62 Chapter 3 γ rays γ A γ B τ γ B γ A Detector Radioactive Point-Source Detector Figure 3.2: A schematic drawing of a two-detector experimental arrangement used to measure the time difference between two coincidentally emitted γ rays, γ A and γ B. A schematic level scheme is shown on the left to indicate how the timing of these γ rays allows the lifetime of the nuclear state, τ, to be measured. Signals from the detectors are processed through a discriminator, discussed later in this chapter, that generates a logic pulse when the input voltage exceeds a specified limit. The logic pulses are then passed to a time-to-amplitude converter (TAC) that generates a pulse with an amplitude that is proportional to the time difference between two input pulses. Figure 3.3 contains generated data of typical timing spectra from the measurement of γ-ray timing coincidences.

63 Chapter 3 63 Counts per unit time (a) (b) (c) Time (arb. units) Figure 3.3: Simulated data of typical γ-ray timing coincidences. The spectra observed when the lifetime is larger than the timing resolution is shown in both linear (a) and logarithmic (b) scales. When the lifetime is smaller than the timing resolution (c), the resultant spectrum (thick line, blue) has a mean that is delayed compared to that of the prompt response function (thin line, red). Measurements of the time difference between γ rays known to populate and depopulate a particular excited state in a nucleus give rise to a time distribution that resembles a standard exponential nuclear decay, as long as the lifetime is large compared to the resolution of the timing system as shown in Figure 3.3 (a). In the limit where the lifetime of the state of interest is much smaller than the timing resolution of the system, two coincidentally emitted γ rays corresponding to the populating and depopulating nuclear transitions are said to be prompt. Timing measurements made with prompt γ rays follow a time distribution that is the PRF, shown by the spectrum with a thin line (red) in Figure 3.3, with a width limited by the resolution of the system [21]. The prompt response function is discussed in detail in Chapter 6. When the γ rays are not prompt, the observed time distribution is a convolution of the prompt-response function and the nuclear

64 64 Chapter 3 exponential decay function [21, 63]. Extracting the lifetime of the nuclear state from such a time distribution can be done in several ways [20]. Perhaps easiest, is in the case with a large time profile that extends much further than the PRF (cf. Figure 3.3 (a) and (b)). In this case, an exponential function fitted to the data far from the PRF will result in extraction of the lifetime as a parameter of the fit (cf. Figure 3.3 (a)). Alternately, the logarithm of the data may be taken and a linear fit may be performed to extract the lifetime (cf. Figure 3.3 (b)). Another technique to extract the lifetime, that works when the lifetime is too short to isolate from the PRF (cf. Figure 3.3 (c)), is the centroid method [20, 63]. This technique calculates the shift in centroid between an experimentally-measured prompt time distribution and one including the lifetime of the state. The difference between the centroids is equal to the lifetime of the state [21]. Detectors that have desirable timing properties that are able to directly measure nuclear state lifetimes in the sub-nanosecond range, such as scintillator materials like LaBr 3, are referred to as being fast and the lifetime measurement technique itself is referred to as fast-timing [64, 65] Contributions to Timing Uncertainty The mean lifetime of excited electronic states within a scintillator crystal affects the achievable resolution on timing pick-off. This is demonstrated by first considering the electronics within the anode output of a photomultiplier tube (PMT), which is detailed in Chapter 4. Figure 3.4 shows a resistor-capacitor (RC) circuit used to represent the PMT anode output.

65 Chapter 3 65 i R i(t) i C + R C V(t) Figure 3.4: Circuit representation of a photomultiplier-tube anode output voltage, V (t), as an RC circuit comprising of a resistor, R, and a capacitor, C. The current, i (t), depends on the decay time of the scintillator material. Image adapted from Reference [66]. The charge collected, Q, at the anode is proportional to the charge emitted by the photo-cathode, and hence the number of scintillations emitted by the crystal in a given pulse. The charge per unit time, or current, i (t), is then defined to be proportional to the decay activity of the crystal, i (t) = Q τ e t/τ, (3.2) where the normalisation factor, Q/τ, has been calculated by integrating the current with respect to time to give the charge, Q [66]. The current in Figure 3.4 is related to the voltage by the equation, dv (t) i (t) = C + V (t) dt R, (3.3) where C is the capacitance and R is the resistance. Equating Equations 3.2 and 3.3, and solving for V (t) results in, V (t) = 1 λq ( e θt e λt), (3.4) λ θ C

66 66 Chapter 3 where λ = 1 and θ = 1. To minimise the error on the amplitude observed at the τ RC anode, the circuit time-constant, RC, is usually chosen to be much larger than the decay time of the scintillator, τ [66]. This effectively allows the voltage to build up over a longer period of time, integrating a larger proportion of the total charge and resulting in a lower statistical error on the maximum pulse height. In this limit, Equation 3.4 becomes V (t) = Q C ( e θt e λt). (3.5) Figure 3.5 shows the forms of the input current (top) and the resultant output pulse (bottom) under the assumption, RC τ. Figure 3.5: Current and voltage diagrams of a photomultiplier anode as a function of time. The voltage shown is in the limit of the circuit decay-time being much larger than the mean lifetime of excitation within a scintillator material. Image adapted from Reference [66].

67 Chapter 3 67 The start of the voltage pulse in Figure 3.5 (bottom) can be described by taking the small-time limit, t RC, of Equation 3.5, V (t) = Q C ( 1 e t/τ ). (3.6) The gradient of the voltage pulse in the small-time limit is then dv (t) dt Q e t/τ =. (3.7) C τ It is apparent from Equations 3.6 and 3.7, that a small value for τ results in a voltage pulse with a larger gradient that approaches its maximum value, Q/C, more quickly. The time in which the pulse takes to increase from 10 % of its amplitude to 90 % of its amplitude is referred to as the rise-time [66, 67]. To realise the effect of the rise-time on the timing uncertainty, the concept of a leading-edge discriminator, where the time of a signal is taken at a point where the detector voltage exceeds a set threshold, is implemented. A voltage threshold is defined such that when it is exceeded by the anode pulse voltage, the time is extracted. Figure 3.6 shows how statistical deviations in the voltage result in statistical deviations in the time extraction.

68 68 Chapter 3 Figure 3.6: Example of a detector pulse crossing a voltage threshold, V T HR demonstrating the statistical deviation in timing pick-off, σ t, (jitter) resultant from input voltage noise, σ n. Image adapted from Reference [68]. The relationship between the statistical error in the time extraction, σ n, is related to that of the voltage-induced noise, σ n, assuming no additional error from electronics components, to be ( ) 1 dv σ t = σ n, (3.8) dt where dv/dt is the gradient of the voltage pulse as is passes the voltage threshold, given in Equation 3.7. It is clear that a small rise-time results in a large gradient and ultimately a lower uncertainty on the pickoff time. The timing uncertainty discussed above is referred to as the time jitter. The jitter of a pair of detectors varies as a function of the energy of detected γ rays. A typical detector output pulse is shown in Figure 3.7, adapted from Reference [69], which uses the concept of a leading edge discriminator to visualise the statistical uncertainty in timing.

69 Chapter (a) 0.0 Voltage (arb. units) (b) (c) Time (arb. units) Figure 3.7: Plots of typical signal voltage pulses at the input of a discriminator. The three signal lines in each plot represent a typical signal (middle) and the uncertainty bands due to statistical fluctuations in the baseline voltage, (a), and the signal amplitude (b). Vertical lines show how much variation there is between the points in which the signal voltage rises above the threshold voltage (horizontal dashed line). Plots (a) and (b) correspond to a leading edge discriminator time pick-off and are included for visualisation only. Plot (c) shows a typical CFD bipolar pulse that reduces the systematic uncertainty from amplitude dependence. The statistical uncertainties at the peak maximum and leading edge will naturally be different so the statistical uncertainty is not reduced as Plot (c) would suggest. A leading-edge discriminator is a device which generates an output logic pulse when an input signal voltage passes a certain threshold voltage. When considering such a signal, two effects contribute to the timing uncertainty. Firstly, the noise on the baseline voltage of the detector output contributes to the jitter as the voltage rises above the discriminator threshold at different times. This is shown in Figure 3.7 (a), with the upper (green) and lower (red) lines showing the positive and negative uncertainty limits of the centre (blue) line. Figure 3.7 (b) shows how the amplitude uncertainty affects the timing uncertainty at the threshold voltage. Qualitatively, it can be seen that as the signal amplitude in Figure 3.7 (b) increases, the larger

70 70 Chapter 3 the gradient of the leading edge of the signal as it crosses the threshold voltage and therefore the smaller the contribution to the timing uncertainty. Figure 3.7 (c) shows the voltage of a typical constant-fraction-discriminator (CFD), in which the input signal is delayed and added to an inverted and attenuated input signal. The output signal of the CFD is then generated when the voltage crosses the baseline. This reduces the systematic timing uncertainty, or time walk, but makes it more difficult to visualise the effects on time jitter. The maximum of the attenuated signal should be aligned to the position in which in inverted signal rises to % of the maximum voltage [70] before the signals are summed to produce the bipolar pulse. The statistical fluctuations in the voltage at these two points will be different and therefore do not cancel as nicely as would be naively assumed from Figure 3.7 (c) and instead, the timing pick-off uncertainty on a CFD is higher than that of a leading-edge discriminator [66]. An additional source of error is introduced due to the variation of electron drifttimes through the multiple dynodes of a PMT (cf. Figure 4.4). The variation of electron drift-times is also referred to as the transit time spread, defined as the full width at half maximum (FWHM) of the time distribution [67]. The transit time spread convolutes with Equation 3.6, causing the rise-time to increase and the resultant voltage to have a smaller gradient upon crossing the trigger threshold, resulting in an increased time jitter. The photomultiplier tube electronics diagram specific to that used in conjunction with the LaBr 3 detectors in the present work is shown in Figure 3.8.

71 Chapter 3 71 K G Acc DY1 DY2 DY3 DY4 DY5 DY6 DY7 DY8 P R11 R12 R13 R14 R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 -HV R1: 390 kω R2: 1.44 MΩ R3, R4: 450 kω R5 to R10: 300 kω R11: 1 MΩ R12 to R14: 51 Ω C: µf C C C GND SIG Figure 3.8: Circuit diagram of the photomultiplier tube voltage divider used in the present work. The series of resistors, R, are used to distribute the single high-voltage input, -HV, to each dynode, DY. The capacitors, C, are used to smooth variations in the dynode voltages [66]. Acc is an accelerating electrode and G is a grid of electrodes used to focus the photoelectrons emitted from the photocathode, K. Image taken from Reference [71]. The experimental section of the present work uses the anode output, P in Figure 3.8, for the purposes of timing extraction and the last dynode output, DY8, for the energy information. This decision has been suggested to be because of the intense light output of the LaBr 3 scintillator crystal that causes the voltage output at the anode to saturate [72]. By taking the energy from the final dynode, a charge with opposite polarity to that at the anode is collected that corresponds to the leftover charge as electrons are liberated in the process of amplification. This collected charge is equal to the amount of electrons lost to the amplification provided by the last dynode only, and thus is smaller than the charge collected at the anode, avoiding the saturation issue Mirror Symmetric Centroid Difference Method The MSCD method is the main fast-timing measurement technique used in the present work [73]. By building upon the centroid method detailed in Section 3.2, the MSCD method takes advantage of the presence of an inverted time spectrum

72 72 Chapter 3 when the start and stop conditions are reversed in a γ ray coincidence timing measurement. A reduction in the statistical uncertainty by a factor of 2 is gained when utilising the MSCD method over a standard centroid shift, assuming an infinite precision on the prompt centroid of the latter. In addition, all of the data used to infer the lifetime using the MSCD method is acquired simultaneously, and any electronic drift affecting calibration during the experiment is cancelled out. This is not the case for the standard centroid shift method, in which the prompt centroid is determined at a separate moment in time to the delayed centroid value, allowing the introduction of a systematic uncertainty to the lifetime measurement. Figure 3.9 shows simulated time spectra for various nuclear state-lifetimes, τ, demonstrating the MSCD method. Counts / 0.05 ns τ = 10 ps τ = 200 ps τ = 1000 ps γ γ stop τ start γ γ start stop τ Time (ns) Figure 3.9: Simulated data following a Gaussian-exponential convolution for various nuclear-state lifetimes in both gating conditions used in the MSCDmethod data analysis. For each line thickness (colour), the solid and broken step patterns indicate time spectra that are measured starting above and below a state of interest, respectively, as indicated by the inset level schemes. The standard deviation of the Gaussian component was 350 ps.

73 Chapter 3 73 The observed time distribution is that of the prompt response function convoluted with an exponential function descriptive of a standard nuclear decay activity, t D (t) = nλ P (x) e λ(t x) dx, λ = 1 τ, (3.9) where D is the delayed time distribution, P is the prompt time distribution, t is time and x is a parameter of the convolution described by the equation [73]. The six plots shown in Figure 3.9 were constructed under the assumption that the PRF is symmetric and Gaussian in form, with a standard deviation, σ, of 350 ps. The different-coloured lines show the time spectra as the lifetime of a nuclear state increases, from the case where τ σ (red), to where τ and σ are of roughly the same magnitude (green) to the case where τ > σ (blue). The inset level diagrams on Figure 3.9 specify the gating conditions required to observe the spectra with centroids shifted toward them. When a timing measurement is made with the γ ray populating a state of interest triggering the start signal of a TAC module, and a depopulating γ ray is used to trigger a stop signal, the centroid of the time distribution is shifted to the right in comparison to the prompt response distribution. This condition is referred to in the present work as the natural condition and describes a natural time spectrum. This phrase was chosen as it describes the time distribution that is similar to a straightforward nuclear-source activity measurement i.e. the probability decreases as time increases. With the gating condition reversed, the TAC is started by a signal triggered by the depopulating γ ray, and stopped by a signal triggered by the populating γ ray, made possible as the stop signal includes an artificial delay such that the instantaneous point is not zero, but rather a constant positive value. This condition is referred to as the reverse condition in the present work and describes a reverse time spectrum, chosen as it is resulting from reversing the gating conditions of the TAC. The thinnest lines (red) in Figure 3.9 approximate the prompt response function due the small lifetime of the time distribution relative to the timing resolution. It can be seen that the natural and reverse time spectra are almost identical from the significant overlap shown. The medium-thick lines (green) are shown to have clearly separate centroids, as the resolution and the lifetime are of

74 74 Chapter 3 similar magnitude. It is noted that the asymmetry of the natural and reverse time plots is very small and is only present in the tails of the distribution, showing the centroid is a much more sensitive probe of any underlying lifetime than small deviation in the shape. The thickest lines (blue) show separate centroids and clearly identifiable exponential tails. To extract the lifetime from natural and reverse time spectra, such as those shown in Figure 3.9, the centroid of each distribution is calculated and the difference taken to give C = C n C r, (3.10) where C n is the centroid of the natural time spectrum and C r is the centroid of the reverse time spectrum. The prompt response function varies as a function of γ-ray energy, for reasons discussed in Section 3.2, and must be accounted for. The full discussion of the PRF correction is deferred until Section 6.1, where it is analysed in-depth in the context of intoducing an analogous procedure, however, it is mentioned here that the correction is defined as the prompt-response difference (PRD). The lifetime, τ, of a nuclear state is calculated according the MSCD method as τ MSCD = C PRD. (3.11) 2 If the PRD correction term is ignored, for purposes of understanding, Equation 3.11 clearly shows that the difference in centroid position of the natural and reverse time spectra is twice the lifetime of the state. 3.3 Recoil Distance Method Whilst the work conducted in this thesis employs only the direct electronic timing method of lifetime measurement (cf. Section:3.2), comparisons are made to previous measurements conducted using the RDM. For lifetimes that are in the order of s, the RDM, or recoil distance Doppler-shift (RDDS) method, allows an indirect means of measurement [22]. This is achieved by making use of the Doppler shift exhibited during the measurement of γ rays emitted from a

75 Chapter 3 75 moving source of radiation. With the use of heavy-ion fusion-evaporation reactions, recoiling fusion products, from a beam impinging on a target foil, can be emitted with a large speed of typically 3-5 % of the speed of light. Figure 3.10 shows this schematically. Thin target d Stopper foil Beam θ E s Eu Doppler-shifted γ ray Unshifted γ ray Figure 3.10: A typical experimental configuration for performing a recoildistance Doppler-shift method measurement, showing a particle beam incoming from the left and hitting a thin target. Recoil products then decay in-flight, emitting a γ ray with an observed energy of E s, as they travel to the stopper foil (right). The remaining recoils are stopped in the foil and subsequently decay, emitting a γ ray with an observed energy of E u. The relative intensities of the resultant Doppler-shifted and so-called unshifted components are used to derive the nuclear state lifetime. Image adapted from Reference [74]. A thin target foil is used to allow the escape of recoils, to ensure that they are not completely retarded within the target itself. The recoils travel toward a thicker foil, at a velocity, v, for a distance, d, where they are slowed down and stopped within the thick foil. Optionally, a type of differential plunger exists [22] which uses another thin foil in place of the stopper foil. This is then referred to as a degrader foil, as it slows the recoils down instead of stopping them completely. Using a degrader foil results in two separate velocity regions that have differing Doppler-shifts but the analysis technique is essentially the same as when using a stopper foil. The energy of γ rays emitted whilst the recoils are moving, or

76 76 Chapter 3 in-flight, are Doppler-shifted according to E s = E u 1 β 2 1 β cos (θ), (3.12) where β = v/c, θ is the angle of observation relative to the direction of recoil velocity, E u is the so-called unshifted (i.e. without a Doppler shift) γ-ray energy to be observed and E s is the Doppler-shifted γ-ray energy [22]. When observed from an angle greater than π/2 rad (cf. Figure 3.10), the Doppler-shifted energy is less than the unshifted energy which results in an observed spectrum similar to that shown in Figure Decreasing target-to-stopper distance E s E u Increasing γ-ray energy Figure 3.11: A diagram of the relative intensities and energy separation of the Doppler-shifted (E s ) and unshifted (E u ) γ-ray energies. These components correspond to the in-flight and stationary recoil-decay γ emissions observed using the RDDS measurement technique, respectively. The intensities are seen to vary as the distance between the target foil and stopper foil is changed. Image adapted from References [53, 75]. Also shown in Figure 3.11 is the intensity distribution of the shifted and unshifted components as a function of the target-to-stopper distance. In the simplest of cases, in which a single excited nuclear state is populated before decaying to emit

77 Chapter 3 77 a γ ray, the intensity of the shifted component is stated as and the intensity of the unshifted component as ( [ I s (t) = I (t) 1 exp d ]), (3.13) vτ [ I u (t) = I (t) exp d ], (3.14) vτ where I s and I u are the intensities of the shifted and unshifted components, respectively and I is the maximum unshifted intensity at zero-separation distance, which depends on time, t = d/v. Due to the complex arrangement of nuclear states, a level of interest, denoted L i, may be populated by transitions from many higher-energy levels, denoted L k, and also depopulate by transitions to many other lower-energy levels, denoted L h. Figure 3.12 uses this notation in demonstrating such an arrangement of energies with arrows to indicate possible decay-transition paths. Figure 3.12: A schematic energy-level diagram showing a level of interest, i, having many states that populate it, k, and many states in which depopulating transitions can occur, h. Image adapted from Reference [22]. The decay curve, R (t), of state L i is calculated from experimental measurements to be R (t) = I u (t) I u (t) + I s (t). (3.15)

78 78 Chapter 3 The population of state L i, denoted n i (t), at a time, t, is described by the Bateman equations d dt n i (t) = λ i n i (t) + N k=i+1 λ k n k (t) b ki, (3.16) where N is the number of the highest feeding level to be considered, λ = 1/τ and b ki is the branching ratio from state k to state i [22]. The solution of the Bateman equation is stated as follows, R i (t) = P i e tλ i + N k=i+1 [ ] λi M ki e tλ k e tλ i, (3.17) λ k where P j is the direct feeding intensity of a level, j, and M ki takes the form, ( ) λi N M ki 1 = b ki P k b ki λ k m=k+1 M mk + k 1 m=i+1 M km b mi λ m λ k. (3.18) Equation 3.17 is then fitted to experimental data to deduce the lifetime of the state. The RDDS method may depend on states that are unobserved, due to statistical limitations. This phenomena is referred to as side-feeding and its magnitude may be estimated from the difference in intensities of γ rays populating and depopulating the state of interest. The lifetime of the unobserved states may be estimated to be the mean of the observed-state lifetimes [22]. A technique involving selection of energies within a single observed decay-chain, that populates the state of interest, in a short period of time is clearly seen to eliminate the vast majority of side-feeding. This technique is referred to as coincidence gating, however, it comes at a statistical cost as each coincidence fold reduces the accepted events by a factor equal to the absolute γ-detection efficiency of the singles events, i.e. with no coincidence condition. An improvement to the RDDS technique is offered by the differential decay curve method (DDCM). With the DDCM, the integration of Equation 3.16 is performed

79 Chapter 3 79 to obtain n i (t) = N i (t) + k b ki N k (t). (3.19) Using Equation 3.19, along with the fact that the lifetime of state i can be defined as d dt N i (t) = λ i n i (t), (3.20) τ i = N i (t) + b ki N k (t) k d N. (3.21) dt i (t) Equation 3.21 is stated, in terms of experimentally measurable quantities R i (t) and R k (t), to be R i (t) + b ki α ki R k (t) k τ i = d R, dt i (t) where α ki = ω k (Θ) ɛ ( ) E γ k ω i (Θ) ɛ ( ). E γ i (3.22) Here, ω (Θ) and ɛ (E γ ) denote the angular distribution of the γ radiation and the efficiency of detecting the γ-ray energy respectively [22]. All of the quantities listed in Equation 3.22 are measurable. The decay curve, R (t), is measured according to Equation 3.15 and its derivative can be inferred by varying the time-of-flight of recoils, i.e. by choosing different distances. A polynomial, usually of degree two, is often used to fit the R (t) measurement data, thus no constraint is imposed on the form of R (t).

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81 Chapter 4 Radiation Detectors 4.1 Interaction of Electromagnetic Radiation with Matter To detect electromagnetic transition γ rays, an understanding of the ways in which they interact with matter is required. Several interactions exist [66] that occur with a probability that is dependent on the energy of the γ ray, and the proton number for a given material. Figure 4.1 shows the cross sections for various means of interaction between γ rays and a typical detector material (NaI). 81

82 82 Chapter 4 Figure 4.1: Energy-dependent cross sections of several interaction mechanisms between γ rays and NaI. Dashed lines represent the individual contribution of the labelled interaction whilst the solid lines represent the total cross section. Photoelectric absorption and Compton scattering are the main interactions observed in the energy range 100 kev kev and are of main relevance to the present work [76]. Image taken from Reference [66]. The energies of the γ rays measured in this thesis lie between 100 kev and

83 Chapter kev. Therefore interactions via the pair production mechanism, which require a minimum energy of MeV to decay into and electron and a positron, each with a rest mass of 511 kev / c 2, are prevented. It can be seen from Figure 4.1 that the largest interaction cross section for sodium iodide in the range 100 kev kev is that of photoelectric absorption. In the range 250 kev kev, the most probable interaction mechanism is Compton scattering. These two interactions are discussed below, however, details of the other interactions shown in Figure 4.1 may be found in Reference [66]. The probability for a γ ray to be absorbed via the photoelectric effect, τ, is given as τ = A Zn Eγ 3.5, (4.1) where A is a constant, Z is the proton number of the interacting material, E γ is the γ-ray energy and n is a number that can vary between 4 and 5 over a typical range of interesting energies [66]. The probability of a γ ray undergoing Compton scattering, σ, is given as a function of solid angle, Ω, by the Klein-Nishina formula: ( ) dσ dω = 1 2 ( 1 + cos 2 ) ( ) θ α 2 (1 cos θ) 2 Zr α (1 cos θ) 2 (1 + cos 2, θ) [1 + α (1 cos θ)] (4.2) where α E γ /m 0 c 2, m 0 is the rest mass of an electron, θ is the scattering angle and r 0 is the classical atomic radius [66]. Equations 4.1 and 4.2 both show an increase in the probability of interaction with an increasing proton number, albeit the effect is much stronger in the case of photoelectric interaction. For radiation shields, which are designed to stop photons, nuclei with a higher proton number are preferred to maximise the interaction cross section. The probability per unit path length, µ, is defined as µ = τ + σ, (4.3) in the limit where E γ < 1 MeV i.e. with no contribution from the pair-production mechanism [66]. The inverse of µ gives the mean free path that a γ ray can traverse through a material. Figure 4.2 demonstrates how photoelectric absorption and Compton scattering

84 84 Chapter 4 are manifest in the measurement of a 137 Cs source of radiation using a thalliumactivated NaI detector, discussed in Section Photopeak Differential Intensity, dn de 10 5 Pb K X-rays Backscatter Peak Compton Edge γ-ray Energy, E (kev) Figure 4.2: The intensity per unit energy of a NaI(Tl) detector to a monochromatic 137 Cs γ-ray source. Shown are the peak due to photoelectric absorption of the total energy of the 662-keV γ ray (right) and the distribution due to Compton scattering between the 662-keV γ ray and electrons in the detector (centre). Also shown is a peak corresponding to the X-ray fluorescence of lead that surrounded the source (left). The photopeak shown in Figure 4.2 results when all energy of an incident photon is collected within the detector material. This happens due to photoelectric absorption, where a γ ray interacts with an atomically-bound electron that is liberated with an energy equal to that of the photon less its binding energy. If the γ ray has high energy (above 250 kev) it is more likely to Compton scatter before photoelectric absorption occurs. The full energy is recorded so long as no scattered γ rays or electrons remove energy from the detector material. The distribution between the backscatter peak and the Compton edge, as labelled in Figure 4.2, is due to Compton scattering. In this case, an incident photon

85 Chapter 4 85 interacts with an electron in the detector material to transfer some of its energy and momenta. The amount of energy transferred depends upon the angle, relative to the initial direction of motion, in which the incident photon is scattered during the interaction according to E E E = E 1 + E m 0 (1 cos θ), (4.4) c 2 where E is the energy of the incident photon, E is the energy of the scattered photon, m 0 is the rest mass of the electron and θ is the angle of scattering. From Equation 4.4, it can be seen that there is little energy transferred to the electron if the scattering angle is small. It can also be seen that there is a maximum energy transfer corresponding to the maximum scattering angle, of π radians, that corresponds to the Compton edge seen in Figure 4.2. The backscatter peak corresponds to the photoelectric absorption of a photon that has initially passed through the detector material, undergone a large-angle Compton scatter in surrounding material, and subsequently re-entered the detector. The Pb K X-rays in Figure 4.2 are due to interactions between the γ ray emitted from the 137 Cs source and the K-shell electrons in the Pb shielding that surrounded the equipment. K-shell electrons are liberated via the interaction and other atomic electrons fill the vacancy, releasing an energetic X-ray. 4.2 Sodium Iodide Detectors Sodium iodide (NaI) detectors are a type of scintillation detector based on an inorganic alkali halide [66, 77, 78]. Scintillation detectors are named as such due to the absorption and subsequent re-emission (fluorescence) of incident photons. The electrons within a scintillation crystal absorb energy from incident photons via one or more of the mechanisms discussed in Section 4.1. Electrons within a scintillator are bound to a band of densely-packed states, the valence band, unless enough energy is provided to promote them into another band, the conduction band, due to the two bands being energetically separated by a so-called band gap.

86 86 Chapter 4 Conduction band Band gap Scintillation photon Activator excited states Activator ground state Valence band Figure 4.3: Schematic of the electron band structure within a scintillator material. Electrons are excited from the valence band to the conduction band when excitated, via γ-ray interaction, with more energy than the band gap. Electrons then populate the activator states where a scintillaition photon is emitted. Image adapted from Reference[66]. Excited electrons in the conduction band are free to move throughout the crystal and can transfer some kinetic energy to promote other electrons into the conduction band, and can thus create many excited electrons. Many photons are re-emitted when these conduction band electrons de-excite. Sodium iodide crystals are activated with thallium to add additional states into the electron energy structure of the crystal that are situated in the band gap. The thallium doping both lowers the frequency of re-emitted photons to be within the visible range and causes the re-emitted photons to be transparent to the crystal structure as the lower energy cannot promote other electrons across the band gap, preventing re-absorption that would negatively affect the efficiency of light collection. Sodium iodide detectors produce photons per MeV of incident photon radiation, although not all of this light is collected due to losses such as the re-absorption of scintillated light. This number is large when compared to some other scintillator materials, such as BaF 2 or CsI each producing 1400 and 2000 photons per MeV, respectively [66]. The photons, or signal carriers, are integrated during the collection of scintillations and the result is a Poisson distribution. The uncertainty on the number of collected photons is therefore the square root of the number itself, so the relative uncertainty

87 Chapter 4 87 decreases as more photons are collected. It is therefore preferable to have a large number of signal carriers to reduce uncertainty. The electrons that are excited to the conduction band in NaI, upon interaction with an incident photon, have a lifetime of 230 ns. This lifetime affects the speed at which the scintillation occurs and therefore the ability to resolve the time at which the photon is detected. To count the scintillations, a PMT is used. Figure 4.4 shows a voltage dividing circuit for a PMT with arrows to indicate the motion of electrons. K LIGHT F Dy1 Dy2 Dy3 P e - e - e - e - PHOTOELECTRONS SECONDARY ELECTRONS ANODE CURRENT Ip A V1 V2 V3 V4 V5 POWER SUPPLIES Figure 4.4: Circuit diagram of an off-axis photomultiplier tube voltage divider, highlighting the path traversed by the accelerated electrons. The series of power supplies, V, provide a high electric field at each dynode, DY. F is a focussing electrode used to focus the electrons emitted from the photocathode, K. Image taken from Reference [67]. The first stage of a PMT contains a photo-cathode, in which a material with a low work function and electron affinity interacts with the scintillated light to liberate electrons via the photoelectric effect. The work function is the minimum energy required to liberate an electron from the surface of a material and electron affinity is the amount of energy required to bind another electron to an atom. A material with a negative electron affinity has a potential barrier at its surface that is lower than the bottom of the conduction band, allowing more time to liberate electrons before recombination with an associated hole occurs [66]. The liberated electrons, of number in the few hundreds, are accelerated in the presence of an electric field toward a dynode. Several electrons within the dynode surface are promoted to the conduction band as a result of interaction with the accelerated initial energetic electron. Some of the newly excited electrons have momenta in the required

88 88 Chapter 4 direction to overcome the work function of the dynode and become liberated, and are subsequently accelerated toward another dynode with another electric field. This multi-stage process amplifies the number of electrons progressively until such a large signal is created, consisting of 10 6 electrons, that the charge can be collected using standard electronics. 4.3 Lanthanum Bromide Detectors Lanthanum bromide (LaBr 3 ) detectors are a relatively new type of scintillation detector that have been developed in more recent years than earlier detectors such as sodium iodide [79]. LaBr 3 detectors are doped with cerium, at a concentration of 0.5 %, that is used as an activator, and so the detector is often denoted as LaBr 3 (Ce 3+ ). Several advantages are offered over NaI detectors, including an improved energy resolution of 2.8 % at 662 kev compared to 5.6 %, owed in part to a larger number ( / MeV) of scintillations per MeV. LaBr 3 is also of higher density than NaI, at 5.29 g cm 3, compared to 3.67 g cm 3, which increases the number of electrons in a given unit volume that are available for interaction with incident radiation, therefore offering improved efficiency for a given crystal size. In addition, the lifetime of electrons that are excited to the conduction band is only 25 ns, compared to 230 ns in NaI. Section 3.2 discusses the effect of the mean lifetime of the scintillator on the timing uncertainty. A lower lifetime results in a improved timing resolution for LaBr 3 detectors, which makes these detectors well suited to fast-timing measurements, as covered in Chapter 3. One feature of LaBr 3 detectors that must be taken into account is the intrinsic radioactivity caused by the decay of 138 La [80]. The decay of 138 La may occur via electron capture or positron-decay, with a probability of 0.664(5), to form an excited state of 138 Ba [54]. A γ ray of energy (10) kev is emitted following the decay of 138 Ba to its ground state. The other method in which 138 La decays, with a probability of 0.336(5), is via beta-decay to form an excited state of 138 Ce. An emitted γ ray with energy (8) kev is observed as the 138 Ce

89 Chapter 4 89 decays into its ground state [54]. The half-life of 138 La is 102(1) Giga-years [81]. Most uses of the LaBr 3 detectors in the present work are for measurements that involve the timing of coincident γ rays. In such cases, any intrinsic radioactivity can be negated as the probability of a decay occurring within a small (< 50 ns) time-interval is low. 4.4 Semiconductor Detectors Semiconductor detectors, usually made using silicon or high-purity germanium (HPGe) crystals, generally achieve much higher resolution than scintillator detectors [66]. Semiconductor detectors function by interacting with incident radiation, via the mechanisms given in Section 4.1, to excite many electrons that are collected at an electrode to form a charge pulse. The detectors are doped with impurities that provide additional free electrons to the crystal lattice structure, in the case of n-type semiconductors, to increase the conductivity of the material. P-type semiconductors are formed when impurities are added that increase the number of unbound valence electrons in the lattice structure The electrons that are usually, in a pure crystal, bound to a neighbouring atom are therefore free to bind to other nearby electrons. When this happens, the electron that becomes bound must necessarily leave behind a state in which it was previously bound, effectively moving a localised positive charge in the opposite direction of the electron. This phenomenon can be described by defining the hole left by an unbound electron as its own quasiparticle. Due to the formation of holes, the conductivity of p-type detectors is also increased, despite having less electrons in the lattice. A junction is formed between the n- and p-type semiconductors to form a depletion region, where the relative abundance of holes in the p-type material diffuse into the n-type material and, equivalently, an abundance of electrons diffuses from n-type to p-type materials. The diffusion forms an equilibrium at the junction where the potential difference, created by the movement of charge, prevents further net diffusion of charge carriers. The depletion region is extended using a large

90 90 Chapter 4 external potential, typically 3 kv to 5 kv in HPGe detectors, to cover the entire crystal volume. In this depleted volume, electron-hole pairs are created when incident photons interact with the valence-band electrons and promote them to the conduction band. Due to the large potential, the pairs of signal carriers are separated and accelerated to opposing electrodes, where the charge is collected to form a measurable pulse. The band gap of germanium is 3 ev, giving a theoretical maximum number of signal carriers per unit energy of / MeV. Due to charge trapping, caused for example by inconsistencies in the crystal lattice structure, and electron-hole pair recombination, not all of this charge is collected but it is this high number, relative to that of scintillator signal carriers, that results in a very good energy resolution in HPGe detectors. HPGe detectors have an energy resolution, in terms of FWHM, between 1 kev and 2 kev at 662 kev. To avoid the thermal excitation of electrons to the conduction band, the detectors are operated at 77 K, achieved by cooling with liquid nitrogen. The timing resolution of HPGe detectors is typically worse than for LaBr 3 detectors. This is due to the time it takes signal carriers to drift to the electrodes, typically 100 ns/cm although the mobility of electrons differs from that of holes, and also due to differences in the pulse shape output for interactions at different positions within the detector material. A comparison of NaI, LaBr 3 and HPGe detectors has been made, in which a source of 60 Co has been measured. The results are displayed in Figure 4.5, showing two peaks corresponding to the γ-ray energies of 1173 kev and 1333 kev [54].

91 Chapter 4 91 Differential Intensity, dn de (a) 10 5 (b) 10 5 (c) 1173 NaI LaBr 3 HPGe γ-ray Energy, E (kev) Figure 4.5: A measurement of the 1333-keV and 1173-keV γ-ray energies in 60 Co using NaI, LaBr 3 and HPGe detectors. The NaI detector has a FWHM of 73 kev at 1333 kev. The LaBr 3 detector has a FWHM of 37 kev at 1333 kev. The HPGe detector has a FWHM of 2.1 kev at 1333 kev. 4.5 The Experimental Configuration at the University of Jyväskylä The experimental section of this thesis, contained within Chapter 5, was undertaken at the Accelerator Laborartory in the University of Jyväskylä. In order to accelerate ions to form a beam, used to perform a fusion evaporation reaction, the K130 cyclotron was utilised [82]. The K130 cyclotron consists of two 78 accelerating plates, or dees, in which a radio frequency of between 10 MHz and 21 MHz is used in applying an oscillating potential of up to 50 kv. The potential of the dees generates an oscillating electric field that is used to accelerate charged ions in a vacuum of 10 7 mbar at a fixed orbital frequency with a bending limit of 130 MeV. A beam accelerated using the K130 cyclotron is directed towards

92 92 Chapter 4 a stationary target located at the centre of the JurogamII HPGe array [83 85] using dipole magnets for steering and quadrupole magnets for planar focussing. A drawing of the detector configuration, containing JurogamII, RITU [42] and GREAT [43], is shown in Figure 4.6. GREAT RITU Beam from K130 cyclotron Dipole magnet Quadrupole magnets JurogamII Figure 4.6: Schematic of the experimental equipment available at the University of Jyväskylä featuring JurogamII, RITU and GREAT (labelled). The four labelled magnets are part of RITU. Image adapted from Reference [86]. Whilst not used for the work presented in this thesis, JurogamII consists of 24 clover detectors - each a cluster of four HPGe crystals - and 15 tapered HPGe detector totalling 111 individual HPGe crystals. Each of the tapered HPGe and clover detectors use bismuth germanate, Bi 4 Ge 3 O 12 (BGO), detectors to form Compton suppression shields to increase the peak-to-total ratio of recorded events. Compton suppression shields work by vetoing those events in which a Compton scatter from within the HPGe detector resulted in a γ ray leaving the crystal without its full energy being absorbed. If a BGO detector is triggered at the same time as a corresponding HPGe detector, the event is vetoed. The detector array is arranged to form two rings of 12 clover detectors, at 75.5 and 104.5, and two

93 Chapter 4 93 rings of tapered detectors at (10 detectors) and (5 detectors), relative to the direction of the incoming beam [84]. The JurogamII array is highly suited to the selection of weakly populated nuclear states following fusion-evaporation reactions, due to the large solid angle, of 40 % of the total, and the ability to select high multiplicity events. Products of the fusion-evaporation reaction at the target position contain linear momentum, transferred from the beam particles, and therefore recoil out of the back of the target towards RITU. RITU is a gas-filled separator (usually He or H 2 at a pressure of 1 mbar) that consists of four main magnets and has a total length of 4.8 m. An initial vertical focussing quadrupole magnet is used to focus the recoiling nuclei (hereafter recoils) into the acceptance window (± 80 mrad vertical, ± 30 mrad horizontal) of the dipole magnet that follows [42]. The dipole magnet bends the recoils by 25 from their initial trajectory upon entrance, separating them from the lighter beam particles. The recoils achieve an average charge state by transferring electrons in collisions with the gas. The average charge increases the recoil transmission through the dipole magnet by the narrowing of the distribution of mass-to-charge ratios for the recoils of interest. The reason for this becomes clear when considering the effect of the mass-to-charge ratio, m/eq, on the radius of curvature, ρ, in the presence of a constant magnetic field, B, given by the equation, ρ = mv Beq, (4.5) where v is the velocity of the recoils [42]. Recoils move from the dipole magnet to two consecutively-positioned quadrupole magnets, and are focussed in both vertical and horizontal planes in preparation for entrance to GREAT. The GREAT spectrometer is used for measuring the decay products of recoils that have been transported from RITU. Several detector systems are implemented to identify and measure the energy of these decay products. Recoils, transported from RITU, first pass through the multi-wire proportional counter (MWPC) [43], a 131-mm 50-mm arrangement of wires, where energy is lost to the detector via ionisation of an isobutane atmosphere. The energy

94 94 Chapter 4 loss, timing between recoils entering and leaving the detector, and positional information derived from the energy absorbed as a function of wire number is recorded and used to distinguish recoil events from the events resulting from the radioactive decay of the recoils at the focal plane. Two Mylar windows are used to separate the isobutane within the MWPC from the He gas in RITU and the vacuum throughout the rest of GREAT. Recoils are then implanted into one of two side-by-side DSSSDs, kept under vacuum, that are used to measure the energies of the recoils and of their subsequent decays [43]. The DSSSDs are separated by 4 mm horizontally and each has an active area of 60 mm 40 mm, with a strip pitch of 1 mm and a thickness of 300 µm. Electronic signals are taken from each strip to provide positional information. An array of mm 28-mm silicon PIN (a semiconductor junction with a p-type layer, an intrinsic layer and an n-type layer) diode detectors is positioned around the DSSSDs, in vacuum, to detect electrons emitted as a result of internal conversion within implanted recoils [43]. The PIN diodes were not used in the present work. Usually positioned 10 mm downstream of the DSSSDs is a planar double-sided HPGe strip detector, used for the measurement of X-rays, low-energy γ rays and high-energy beta particles [43]. The planar detector has an active area of 120 mm 60 mm and has a strip pitch of 5 mm. The planar detector was removed during the measurements taken in the present work due to the attenuation of low-energy (< 500 kev) γ rays based on simulations showing a factor of five increase in absolute efficiency for the LaBr 3 detectors discussed in Section 4.6 [87]. Outside of the vacuum chamber containing the DSSSDs, three clover HPGe detectors were arranged such that one was vertically above the DSSSDs and the other two were positioned horizontally, facing each other and the DSSSDs. The focal plane clover detectors use the same Compton suppression shielding as the JurogamII clover detectors. These clovers were used for the calibration part of the experimental work performed in this thesis.

95 Chapter A New Fast-Timing LaBr 3 Array In addition to the standard experimental configuration, shown in Section 4.5, an array of eight LaBr 3 detectors has been designed and implemented at the GREAT focal plane. The work contained in this thesis presents the first use of this detector array in the measurement of picosecond-scale nuclear-state lifetimes that are the result of the decay of short-lived states populated by a long-lived isomeric state within recoils that are spatially distributed at the focal plane of a separator LaBr 3 Array Setup An array of LaBr 3 detectors was constructed to test the effect of the spatial distribution of recoils at the focal plane of a separator on the lifetime measurement of nuclear states. LaBr 3 detectors were selected due to their excellent timing characteristics, discussed in Section 4.3. The housing for the LaBr 3 detectors used in the present work, of diameter 1.5 inches and length 2 inches, was designed by Smith [88], and is shown in Figure 4.7. Rear End Clamp PMT Hamamatsu H1057MOD Front End Clamp M3 12 Hex SK Cap Screw ±0.5 PMT Housing 41.5±0.5 Crystal Housing O' Ring - BS 226 Internal Spacer Cylinder Casing O' Ring - BS 233 Outer Shielding Detector Assembly B380 Figure 4.7: A technical drawing of the LaBr 3 detectors used in the present work. The LaBr 3 crystal is housed on the right and is connected to a photomultiplier tube (PMT) that has three pins on the left (not all visible in the drawing) for anode output, last dynode output, and high-voltage. Distances are given in millimetres. Image adapted from Reference [88].

96 96 Chapter 4 The PMTs inserted into the housing shown in Figure 4.7 were of model R9779 [71]. A frame was designed to be mounted onto the frame surrounding the GREAT spectrometer and hold eight LaBr 3 detectors [89]. Figure 4.8 shows a technical drawing of the array frame, indicating the positions of the holes for the detectors along the horizontal (x-axis) and vertical (y-axis) axes FROM BEAM CENTRE HORIZONTAL FROM BEAM CENTRE HORIZONTAL DIMENSIONS SHOWN ON THIS VIEW ARE TO THE CENTRE OF THE PLATE HOLES ON THE FRONT FACE OF THE PLATE FROM BEAM CENTRE VERTICAL FROM BEAM CENTRE VERTICAL Figure 4.8: A scale drawing of the LaBr 3 frame as viewed looking upstream of the recoil axis. The height in which the top of the array frame is positioned below the frame surrounding the GREAT DSSSD is given, along with the locations of the holes for the detectors. Distances are given in millimetres. Image adapted from Reference [89]. Figure 4.9 shows the LaBr 3 array frame with the detectors inserted along with the numbering convention used throughout this thesis.

97 Chapter Figure 4.9: A scale drawing of the frame with the LaBr 3 detectors inserted as viewed looking upstream of the recoil axis, demonstrating the number convention used throughout this thesis. Image adapted from Reference [89]. Figure 4.10 shows the LaBr 3 array frame from the side, showing the angles that the detectors face with respect to the recoil axis (z-axis) in the vertical plane (y-z-plane) Figure 4.10: A scale drawing of the array with detectors mounted showing the angle of inclination with respect to the recoil axis as viewed from the side. The right side of the drawing depicts the vacuum box surrounding the DSSSD. Distances are given in millimetres. Image taken from Reference [89].

98 98 Chapter 4 Figure 4.11 shows the LaBr 3 array frame, with the detectors removed, from a top-down perspective. DSSSD DISTANCE FROM DSSSD WHERE THE INNER LaBr WOULD FOCUS WHEN INTERSECTING THE BEAM CENTRE DISTANCE FROM DSSSD WHERE THE OUTER LaBr WOULD FOCUS WHEN INTERSECTING THE BEAM CENTRE ALL DETECTORS ARE 90 mm FROM DSSSD TO LaBr CRYSTAL FRONT FACE Figure 4.11: A scale drawing of the LaBr 3 array frame showing the angles of the inner- and outer-four detectors with respect to the recoil axis in the horizontal plane. The two groups of detectors are seen to have different focal points to maximise the solid angle facing the DSSSDs. Also shown are the clamps used to attach the LaBr 3 detectors to the frame. Distances are given in millimetres. Image adapted from Reference [89]. The angles of the LaBr 3 detectors closest and furthest from the recoil axis are shown to be 8.5 and 34, respectively, to the z-axis in the x-z-plane. To determine the positions of the detectors for the experimental work presented in Chapter 5, the x- and y-positions of the holes were taken from Figure 4.8. The z-positions of the holes were determined by using a ruler to measure the distance from the outside of the vacuum chamber surrounding the GREAT DSSSD to a fixed position on the array frame. The distance between the fixed position and the holes along the z-axis, determined using the scale drawings, was then added to the measured distance. The distance between the detector faces to the front of the frame was

99 Chapter 4 99 chosen to be the same value, l = 158 mm, for all detectors. The directional unit-vector for each detector, ˆd i, was calculated from the angles provided in the above figures. The position vectors of the detector faces, f i, were then calculated from these measurements using vector addition according to f i = h i + l ˆd i, (4.6) where i is the detector index and h i is the position vector of the corresponding hole in the frame. The origin was taken to be the centre of the two DSSSDs to simplify the determination of the DSSSD pixel position vectors. Pictures of the finished detector array, attached to the frame surrounding the GREAT spectrometer, can be seen in Figure 4.12.

100 100 Chapter 4 Figure 4.12: Photographs of the LaBr 3 -detector array. The top-left picture shows the detectors mounted in the frame when pulled back from the GREAT spectrometer, along with the three clover HPGe detectors that were not specifically installed for the present work. The top-right picture shows the array looking upstream of the recoil axis. The bottom-left picture shows the array during the cabling of detector outputs to the electronics units and the bottom-right picture shows the array from a wide angle with respect to the recoil axis. The cabling and electronics are discussed in Section The red cables seen in Figure 4.12 provide high voltage to the detectors whilst the other cables were used to provide energy and timing information via electronic signal processing, discussed in the section below LaBr 3 Array Electronics After the design and installation of the LaBr 3 array, discussed in Section 4.6.1, the detectors were connected to a series of electronics units used to process and store information on the signal pulses. The electronics used to extract energy-related information from the LaBr 3 detectors in the experiment differed to that used to

101 Chapter extract the timing information. Figure 4.13 shows a schematic of signal processing and data acquisition equipment used in the experiment. LaBr 3 Array + 8 _ in + _ _ out + _ +_ in _ +_ + _ Start in out 7 8 out in out CFDs FIFOs Common Stop TACs TFAs in out GO Box in out Attenuating GO Box in out Lyrtech ADC Figure 4.13: Diagram of the electronics used for the LaBr 3 array in the present experiment. The green wires from the dynode ( ) outputs of the detectors carried the energy signals. The purple anode (+) wires were used for the timing branch. Red wires from the CFDs were used to start the seven TACs. The blue CFD wires formed the common stop for the TACs. Different coloured wires represent the different combinations of stop signals propagated through the layers of Fan-In-Fan-Out (FIFO)s, with the combinations through a particular FIFO labelled by stop detector numbers. The voltage of each photomultiplier tube was set to 1300 V after fine tuning was performed to optimise the timing and energy resolutions [90]. The LaBr 3 detectors have two outputs, an anode and a dynode. Signals from the dynode and anode were used to process the energy and timing information of the γ rays respectively.

102 102 Chapter 4 The energy signals were fed into a timing filter amplifier (TFA) to both amplify the signal and stretch out the energy pulse, increasing the decay constant and making it more compatible with the Analogue-to-Digital Converters (ADCs). The Ortec 474 TFAs [91] were configured with an integration time of 500 ns, a course gain of 6, a fine gain of 4, differentiation time OUT and were set to non-inverting mode. The signals from the TFAs were taken through a patch panel to a gain-offset (GO) box. The GO box inverted the signal and allowed the baseline voltage to be set within the ±1 V range accepted by the Lyrtech digitiser. Lyrtech ADC cards were used to digitise signals recorded using the Total Data Readout method [92] to an accuracy of 14-bits. Data were recorded every 10 ns using a 100 MHz metronome. Table 4.1 contains the settings used for the Lyrtech ADC, configured using the MIDAS software package [93]. Parameter Value CFD Threshold 8-15 TFA Shaping Time 200 Shaping Time 2.16 Rise Time Peak Sample Peak Separation Decay Time Constant (µs) 8.1 Table 4.1: Configuration for the Lyrtech ADC cards used in processing the energy signals from the LaBr 3 detector array. Most values have not been assigned units as they were not specified by the MIDAS firmware. The decay constant configured in the Lyrtech parameters was shorter than that of the input pulse. This was due to a limitation in the Lyrtech firmware. As a test, Detector 7 had a pre-amplifier inserted between its output and corresponding TFA. This stretched the signal so the decay time going into the Lyrtech was 1400 ns, up from 600 ns. Elongating the decay time constant using the pre-amplifier brought the input signal s decay time closer to the decay constant setting in the Lyrtech ADC, resulting in an improved resolution. While this improved the energy resolution for Detector 7, see Section 4.6.2, there were insufficient pre-amplifiers available to install for the other detector channels.

103 Chapter Whilst not employed for the experiment in the present work, another test was conducted that involved taking the dynode outputs from the LaBr 3 detectors directly into the Lyrtech ADC and recording signal traces. By performing an integration of each trace to calculate the peak area, the energy resolution was made comparable to that when using both a TFA and a pre-amplifier. It is also noted that the maximum signal amplitude during this test was small compared to the range converted by the ADC, so it may be possible to improve the energy resolution further with some level of amplification. Based on these observations, it has been decided [94] that future fast-timing experiments at the University of Jyväskylä will not use pre-amplifiers or TFAs. Instead, the LaBr 3 PMT output will be connected either directly into the Lyrtech ADC cards or, more likely, into a piece of equipment that can integrate the signals coming from the detector in firmware to avoid having to record traces. The timing signals were taken from the LaBr 3 detector anodes and fed into CFDs. Two quad Ortec 935 CFDs [95] were used with a 10 ns cable delay. The pole-zero (P/Z) was configured to minimise the change in output signal time as a function of input signal amplitude. This was done by recording data from a radioactive source of 60 Co and constructing a matrix of LaBr 3 energy against time with a pre-condition that a different detector had observed the 1332-keV γ ray that is emitted during the decay of the source [54]. By looking at this matrix, a peak corresponding to the 1173-keV γ transition in the decay of 60 Co was observed, along with its Compton background. The P/Z was adjusted until the time difference between the 1333-keV transition and the Compton background was constant over as large a range of energies as possible. While using Compton background may be less than ideal, the time dependence with amplitude is revisited later and done with a larger degree of accuracy in Section The final P/Z voltages found for each CFD used are given in Table 4.2.

104 104 Chapter 4 CFD # P/Z (mv) Table 4.2: Optimised P/Z values for the LaBr 3 detector array. Each CFD had three outputs which were used to provide start and stop signals for the assigned TACs. Figure 4.13 shows the start signal cables in red. The CFD outputs were connected through a series of Fan-In-Fan-Out (FIFO) units to increase the available CFD outputs and to multiplex multiple CFD signals into a single so-called common stop signal [21]. Figure 4.13 shows a detailed wiring diagram for the FIFO configuration. A total of seven TACs were used with this common-stop method. The Ortec 566 TACs [96] were configured to the minimum range of 0-50 ns. Each TAC converted the time between the start and stop electronic signals into a pulse with an amplitude representative of the time difference. The pulse amplitude varied from 0 V to 10 V over the range configured for the TAC. An attenuating GO box was used that reduced the amplitude of its output by a factor of five to fit the range accepted by the Lyrtech ADC. The TAC outputs were passed through the attenuating GO box and subsequently into a Lyrtech ADC, configured for the TAC signals according to Table 4.3.

105 Chapter Parameter Value CFD Threshold 30 TFA Shaping Time 200 Shaping Time 1.8 Rise Time Peak Sample 1.53 Peak Separation 3.6 Decay Time Constant (µs) Table 4.3: Configuration for the Lyrtech ADC cards used in processing the timing signals from the LaBr 3 detector array. Most values have not been assigned units as they were not specified by the MIDAS firmware LaBr 3 Array Resolution To determine the energy resolution of the LaBr 3 detectors, γ rays emitted from a source of 60 Co were measured. A third-order polynomial was used to calibrate the ADC channels of observed peaks to the known γ-ray energies emitted by the source [54]. Calibrated data were subsequently resampled to have an interval of 1 kev / bin and data from all eight detectors were summed. The resultant summed spectrum is shown in Figure 4.14.

106 106 Chapter Counts / 1 kev Co FWHM = 50 kev Energy (kev) Figure 4.14: An energy spectrum of the 1333-keV γ ray emitted during the decay of 60 Co. The spectrum is the sum of all eight detectors in the LaBr 3 array. The peak at 1333 kev was used to measure the energy resolution, as there were a negligible amount of background events recorded at this energy. The standard deviation of the 1333-keV peak shown in Figure 4.14 was 21 kev. The standard deviation, σ, was converted using, FWHM = 2σ 2 ln 2, (4.7) to give a FWHM of 50 kev at 1333 kev. As mentioned in Section , a preamplifier was used to improve the energy resolution for Detector 7. With this preamplifier, the energy resolution, in terms of FWHM, was 37 kev. A measurement of the timing resolution attainable for all pairs of LaBr 3 detectors in the array was made by collecting data from a radioactive source of 60 Co. The data were sorted into a three-dimensional, or 3D, (energy, energy, time) histogram.

107 Chapter A gate was set on the 1173-keV transition in one detector to use as the start signal to the TAC and the 1333-keV transition in another detector to use as the TAC stop signal, where both detectors received a signal within a 30 ns of each other [54]. The resultant time projection of the 3D histogram was a time spectrum, shown in Figure 4.15, that is representative of the first excited 2 + state of 60 Ni Counts / 0.05 ns Time (ns) Figure 4.15: Time spectrum corresponding to the 2 + state in 60 Ni from a single detector pair, obtained by collecting data started by the 1173-keV γ ray from the decay of 60 Co and stopped by the 1333-keV γ ray. Detectors 3 and 8 were used to provide the start signal and stop signal respectively. The resolution, in terms of FWHM of this particular detector pair, was 270 ps. Time spectra were created similarly for all 28 detector-pair combinations and the centroid of each spectrum was calculated. The mean lifetime of the first excited 2 + state in 60 Ni, 1.310(30) ps [97], was subtracted from each of the centroid values to obtain the prompt points. The prompt points are the time values in which two simultaneously emitted γ rays start and stop a TAC with negligible delay from the lifetime of the state between these emissions. By subtracting a given detector-pair s prompt point from each measured value, the data from

108 108 Chapter 4 each detector pair is synchronised to the same timing value of zero. After each detector pair has been synchronised, the 60 Co decay data were re-sorted and all 28 resultant time spectra were summed to produce the time spectra shown in Figure This time spectrum has the appearance of a Gaussian distribution resulting from the fact that it is a convolution of a prompt response function and the exponential nuclear-decay distribution [21] in the limit of a small lifetime. Figure 4.16: Sum of the 28 detector-pair time spectra corresponding to the 910(20)-fs half-life 2 + state in 60 Ni which follows the decay of 60 Co. The TACs were started on the 1173-keV transition and stopped on the 1333-keV transition. The spectrum is the sum of all timing from the 28 LaBr 3 detector combinations in the array, after synchronising the prompt signal position to 0 ns. The prompt response function can be asymmetric, especially when the start and stop γ-ray energies largely differ, but it is generally well described as a Gaussian with a standard deviation that is representative of the intrinsic timing resolution of the detector pair. As mentioned, the mean lifetime of the first excited 2 + state in 60 Ni is 1.310(30) ps [97], which was small compared to 127.0(3) ps, the standard deviation of the time spectra shown in Figure Because of this, it is inferred

109 Chapter that the exponential component of the convoluted spectrum is dominated by the Gaussian component. Therefore, the stated standard deviation corresponds directly to that of the intrinsic timing resolution of the detector pairs used to make the spectrum. The time jitter, discussed in Section 3.2, was calculated for the detectors that constitute the LaBr 3 array in the present work using a radioactive source of 152 Eu. Figure 4.17 shows the energy spectrum for 152 Eu, obtained by summing the spectra from the eight LaBr 3 detectors Eu Counts / 1 kev γ-ray Energy, E (kev) Figure 4.17: A LaBr 3 -energy spectrum collected using a source of 152 Eu. γ rays of various energies, coincident with the 244-keV internal transition γ ray in 152 Sm, were used as start signals and the 244-keV γ ray was used as a stop signal. This γ ray was chosen as it has a large range of possible transition energies in coincidence from states whose decay populate the 4 + state [54]. The resultant spectra were fitted to the form of an exponential-gaussian convolution with the lifetime parameter fixed to 83.2 ps [54], and the resultant σ parameter values for

110 110 Chapter 4 each energy were taken to be the time jitter. Figure 4.18 shows the time jitter at various energies Time (ns) Energy (kev) Figure 4.18: Measured timing uncertainty (jitter) as a function of energy. All data points are plotted at the energy of the γ ray used to start a TAC and were each stopped by a γ ray of energy 244 kev, corresponding to various states within 152 Eu. The lower line indicates the component contribution of the start event and stop event. Adding the component contributions in quadrature for a given pair of energies estimates the jitter for that energy pair. See text for more information. The data in Figure 4.18 have been fitted to the functional form used to describe CFD time-walk [21]. To estimate the jitter for an arbitrary pair of γ-ray energies, it is assumed that each timing measurement can be separated into two components; the arrival time of the start signal, and the arrival time of the stop signal. The time measurements would be resultant of a basic subtraction of start time from stop time, thus error propagation dictates that the observed uncertainty is the quadrature sum of the two component measurements assuming that jitter is the dominant source of statistical error and that the measurements are independent. This latter point is valid as, although the start and stop time signals are correlated,

111 Chapter separate detectors and separate CFDs are used in determining the time pick-off of each event. If σ (E) is the observed uncertainty at energy, E, then it holds that σ (E) 2 = σ (E) 2 + σ (244) 2, (4.8) where σ (E) is the component error due to a single start or stop event, assumed to be the same, and the energy of 244 is in units of kev. By evaluating Equation 4.8 for E = 244, σ (E) can be related to σ (E) as follows, σ (244) 2 = 2σ (244) 2. (4.9) Substituting Equation 4.9 into Equation 4.8, and rearranging to solve for σ (E), results in σ (E) = σ (E) σ (244)2. (4.10) Equation 4.10 was used to produce the line below the data points in Figure Generalising Equation 4.8 to an arbitrary energy pair, independent of 244 kev, the following equation is defined. σ (E 1, E 2 ) = σ (E 1 ) 2 + σ (E 2 ) 2, (4.11) where σ (E 1, E 2 ) is the observable resolution from a timing measurement with start and stop energies corresponding to E 1 and E 2. As σ (E) is plotted on Figure 4.18 (lower line), a visual solution to Equation 4.11 is to simply sum, in quadrature, the jitter values at two energies. Because Equation 4.11 depends on two independent variables, a surface plot has been made, in Figure 4.19, that shows the uncertainty for each combination of energies between 100 kev and 900 kev.

112 112 Chapter Start Energy (kev) Stop Energy (kev) Jitter (ns) Figure 4.19: The calculated timing uncertainty surface of the LaBr 3 detectors in the array for all combinations of start and stop energies between 100 kev and 900 kev LaBr 3 Array Intrinsic Photopeak Efficiency The intrinsic photopeak efficiency describes the probability of photon quanta, incident on a detector, to transfer all of their energy into the detector material [66]. The ways in which this interaction can take place are given in Section 4.1. To calculate the intrinsic photopeak efficiency of the LaBr 3 detectors in the array, a measurement of decay activity of 60 Co was made. A 60 Co source was placed at a known position in front of the LaBr 3, and the positions of each detector in the LaBr 3 array was noted. The activity of the source and the time in which data were recorded was known. The diameter of the forward facing surface for each detector was known, allowing the area to be calculated. An effective area of each detector was calculated by multiplying the true area by the cosine of the angle between the detector surface normal and the line connecting the positions of the

113 Chapter source and detector. This effective area was then used in approximating the solid angle subtended for each detector in the array as a fraction of the total solid angle of a sphere. The solid angle for each detector was calculated as the ratio of the effective area to the area of a sphere with radius equal to the distance between the source and detector. The total solid angle for the LaBr 3 array was found by summing the solid angle contribution from all eight detectors. The total number of counts recorded for the 1333-keV γ-ray transition in the decay of 60 Co were calculated by fitting the data to a Gaussian function, under the assumption of no background, and summed over all eight detectors to provide a total intensity. The activity at half way through the time in which data were recorded was calculated by applying the radiation-decay law using the known activity as measured at a known time, and the known half-life of the decay of 5.3 years. [54]. As data were only recorded for a matter of hours, it was approximated that the activity was constant for the duration and the total number of decays was obtained by multiplying this activity by the time in which the data were recorded over. Using the total number of decays emitted by the source, N, the number of detected decay events at 1333 kev, n, and the fractional solid angle subtended by the array, Ω f, the intrinsic photopeak efficiency, ɛ ip, was calculated according to Equation 4.12, ɛ ip = n NΩ f. (4.12) The intrinsic photopeak efficiency of the LaBr 3 detectors in the array was calculated to be 13(2) % at 1333 kev. This value is shown in Table 4.4, along with the intrinsic photopeak efficiencies for the individual detectors.

114 114 Chapter 4 Detector # ɛ ip peak/total (3) (3) (5) (4) (5) (3) (3) (5) (3) (3) (5) (6) (5) (5) (3) (3) sum 0.13(2) (2) Table 4.4: LaBr 3 -array detector efficiencies and peak-to-total measurements. ɛ ip shows the intrinsic peak efficiency, taking into account the source activity, number of counts in peak, and solid angle coverage. The peak to total was calculated using 60 Co, no background was assumed for the higher energy peak and a linear background was used for the lower energy peak. The peak-to-total ratio, shown in Table 4.4 was also calculated and provides a measure of how much of the total radiation quanta, incident on the detector volume, deposited their full energy compared to those quanta that deposited only part of the full energy. Using the same recorded data as described above, the number of counts recorded for both the 1173-keV and 1333-keV transitions in 60 Ni were calculated. For the 1173-keV transition, the number of counts was obtained by fitting the data to a Gaussian function with a fixed linear function superimposed to describe the background contribution from partially absorbed photons originating from the 1333 kev transition. The intensities of both transitions were summed together and over all eight detectors to give a total full-energy peak count. The data recorded with an energy greater than the electronic noise threshold were summed by simple integration for each detector to give the total number of recorded counts. The peak-to-total ratio was calculated to be 15.88(2) %, the ratio of these two numbers. The efficiency of detecting incident radiation varies depending on the energy of that radiation. To calculate the efficiency as a function of energy, data were recorded using a 152 Eu source due to its large number of energy levels across

115 Chapter the energy range 0 to 2 MeV [54]. The data from each detector in the array were summed after calibration to provide a single spectrum. The intensities of several of the peaks in the spectrum were calculated using the RADWARE software suite [98]. The lowest energy value, at 40 kev, was computed as the intensity-weighted average of 152 Eu decay X-rays [54], as they were not individually resolvable using the LaBr 3 detectors. By dividing each of the recorded intensities by the known probability of observing a photon of the respective energy given a decay of 152 Eu, the efficiency of the LaBr 3 array could be calculated at various energies. A function used to describe the relationship between efficiency and energy [99] is given by Equation { [(A η (E g ) = exp ) + Bx + Cx 2 G ( ) ] } + D + Ey + F y 2 G 1 G, ( ) ( ) Eg Eg x = ln, y = ln, 100 kev 1 MeV (4.13) where η is the efficiency, and A, B, C, D, E, F and G are parameters to be fit while E g is the γ-ray energy. The absolute scale of the efficiencies is arbitrary and depends on the integral number of counts recorded and the solid angle subtended by the array. It was therefore decided to scale all efficiencies by a constant value such that η (1333) = 13 %, thus providing the absolute intrinsic photopeak efficiency as a function of energy. Figure 4.20 shows these intrinsic photopeak efficiencies.

116 116 Chapter 4 Intrinsic Photo-peak Efficiency (%) Energy (kev) Figure 4.20: The intrinsic photo-peak efficiency of the eight LaBr 3 detectors in the array. The efficiency was obtained at various energies using a source of 152 Eu and was scaled to 13 % at 1333 kev, obtained using a 60 Co source The Prompt Response of the LaBr 3 Array The time at which a CFD signal output is generated is dependent on the characteristics of the input signal [69, 100]. Variations in the timing of the output signal occur as a result of phenomena such as charge sensitivity, the charge above the threshold level required to trigger the CFD, input signal amplitude and input signal noise. See Section for further details. The systematic timing uncertainty due to amplitude is referred to as the time-walk of the signal. As outlined by Régis [21] et al., the time-walk of a CFD can be corrected by constructing a prompt response difference function using γ-ray timing data from a 152 Eu source. The prompt response difference function, PRDF, is of the form PRDF (E) = a E + b + ce + d, (4.14) where E is the energy of the photon measured, and a, b, c and d are free parameters. From Equation 4.14, the prompt response difference between two γ-ray energies

117 Chapter can be calculated as PRD (E 1, E 2 ) = PRDF (E 1 ) PRDF (E 2 ), (4.15) where PRD is the prompt response difference, and E 1 and E 2 are the energies of the feeding and decaying transition γ rays of the state to be measured, respectively. The prompt response difference is then used to calculate a lifetime from the centroid difference obtained from time spectra according to the MSCD method [73] described in Section The lifetime can be calculated according to the formula, τ = C PRD (E 1, E 2 ), (4.16) 2 where τ is the lifetime to be measured and C is the centroid difference between a natural time measurement and one with reversed gating conditions. The prompt response difference function was calculated using pairs of γ rays, from a 152 Eu source, with energies (in kev) of: 444, 1086; 964, 444; 245, 867; 1193, 245; 1299, 344; 779, 344; 564, 1086; and 40, Figure 4.21 shows the partial level schemes for 152 Sm and 152 Gd, the two main daughter products of the decay of 152 Eu.

118 * * * Gd * * * * * * * * ( ) 152 Sm ( ) ( ) (0.94) ( ) ( ) Figure 4.21: Partial level schemes of 152 Gd and 152 Sm showing the energies of the states and transition γ rays in both nuclei. Images adapted from References [101] and [102] ( ) ( ) ( ) (1.24) ( ) ( ) Chapter 4

119 Chapter The latter pair of γ-ray energies had a coincidence condition of a 122-keV γ ray required in a focal plane HPGe clover detector, to avoid obtaining a time spectrum with background due to escaped energy from the Compton scattering of higherenergy coincident γ rays. The above energy combinations had little background so no background correction procedure was required [73]. The combination of 344 kev and 779 kev had the largest count rate and so the corresponding γ-ray pair was used to synchronise all of the detector combinations to a common prompt time point. This was achieved by adding the time spectra, for each detector pair in turn, in which the start signal was due to the detection of a 344-keV γ ray and the stop signal due to the detection of a 779-keV γ ray to the time spectra with the start and stop conditions reversed. The mean of this summed distribution then gave the prompt time, which was subtracted from all subsequent measurements to synchronise the time signals to a common prompt value. This procedure is based on the centroid of the natural time spectrum being equal to the prompt response function centroid plus the lifetime of the state. Similarly the centroid of the reverse time spectrum is equal to the prompt response function centroid minus the lifetime of the state. The centroid of the natural and reverse time summed spectrum therefore has the value of the prompt response function centroid assuming an equal intensity in each of the spectra. This is a reasonable assumption as the two time spectra are simultaneously acquired under the same experimental circumstances. It is noted, however, that the two prompt response function centroids are not necessarily equal, which could lead to the introduction of systematic uncertainties in the synchronisation times of the detector pairs [103]. Following the synchronisation procedure, the corresponding time spectra were then averaged by summation over all detector pairs in accordance with the generalised centroid difference (GCD) method [21]. Each of the γ-ray pairs were used to create prompt response differences by taking the centroid difference of the time spectra started by the timing of the first γ ray in the list above and stopped by the second, referred to as the natural distribution, and the time spectra with start and stop conditions reversed, referred to as the reverse distribution. Figure 4.22 displays these prompt response differences (blue

120 120 Chapter 4 squares) as indexed by the first listed energy in the pair. As the prompt response difference depends on two energies, it is necessary to modify the data to be relative to a single energy, so a single function can be used to represent each of the points. The data points were modified by addition of t m = PRDF (E 2 ) PRDF (E ref ), (4.17) where t m is a modification time and E ref is the so-called reference energy, which in this case was 344 kev. It is noted that Reference [103] details a similar function, however, there is no indication of whether the PRD values are from measurement of a source or from the equation defining the energy dependence of the PRD. In the latter case, as assumed by the author from the lack of coincidences between specific reference energies in 152 Eu, the prompt time values are modified by the function in which they are the data used to define the parameters. This scenario therefore requires an iterative procedure to be performed, that modifies the data points by the current function value of t m before calculating the weighted sum of squared residuals in each step of the minimisation routine. Figure 4.22 shows the modified data points as green circles, after the least-squares minimisation routine had completed.

121 Chapter Time (ns) Energy (kev) Figure 4.22: Plot of the prompt response difference function. The square data points (blue) show the unmodified prompt response data. The circular data points (green) have been modified to represent all points as being relative to a start energy of 344 kev from the decay of 152 Eu to 152 Gd. The solid line is the result of a least-squares minimisation procedure to describe the circular data points - see text for details. The uncertainties are smaller than the data points. Parameter value a (ns kev 1 2 ) 8.2(14) b (kev) 31(17) c (ns kev 1 ) (36) d (ns) -0.51(8) Table 4.5: Parameter values describing the prompt response difference function, resulting from the fit to the data as described in the text. All parameters were made free during the fitting procedure. These parameters were used in calculating the prompt response differences.

122 122 Chapter 4 a b c d a 7.23E E E E-03 b 7.57E E E E-02 c 1.74E E E E-08 d -3.96E E E E-04 Table 4.6: Covariance matrix for the prompt response difference parameters listed in Table 4.5. These values are used in propagating the parameter uncertainties when calculating the mean PRD between any two energies. Table 4.5 gives the parameter values of the prompt response difference function, derived from the fitting procedure. Using these parameters, the prompt response difference can be calculated using Equation 4.15 for any pair of energies used to make a timing measurement with the LaBr 3 detectors. Care must be taken when propagating errors due to the correlation of parameters when calculating the PRD according to Equation Table 4.6 gives the covariance matrix generated by the fitting program. This covariance matrix was scaled by the minimised reduced χ 2 of 28.9 to account for any underestimation in data uncertainty that resulted in a minimised reduced χ 2 greater than one [104]. The uncertainties on the prompt response differences given in the present work were propagated using an uncertainties package [105]. 4.7 Chapter Summary Background knowledge on current detector systems has been presented in preparation for the work undertaken in this thesis. A fast-timing array consisting of eight LaBr 3 detectors has been designed, installed and calibrated for use with the experimental work contained within Chapter 5. Using this detector system, the measurement of fast-decaying states ( ps) that are populated via isomeric states and transported though RITU can proceed, allowing the effect of spatial distribution on the measurement of excited nuclear-state lifetimes to be tested.

123 Chapter 5 Investigation into the Effect of Spatial Distribution on Fast-Timing Measurements 5.1 Introduction to Timing of a Spatially Distributed Source Fast-Timing measurements are typically undertaken with detectors arranged at fixed distances to a localised source of radioactivity. The work presented in this thesis is aimed at studying the influence of non-localised, or distributed, radioactivity on the lifetime measurements of nuclear states. An example of a typical timing experiment can be a simple case in which a radioactive source is positioned at the centre of two detectors as shown in Figure 5.1 (top). 123

124 124 Chapter 5 γ rays Radioactive Detector 1 Point-Source Detector 2 Radioactive Source Distribution Possible Decay Points Figure 5.1: A simple drawing of a two-detector experimental configuration used to measure the time between consecutively emitted γ rays from a point-like radioactive source (top). As the distance each γ ray (pictured) has to travel to reach the detector is equal, there is a zero net relative time difference between measurements. The bottom diagram shows how uncertainty in source position can introduce a relative difference in the time of flight of each γ ray in a pair. In focal plane spectrometers, the spatial extension of recoil-isomers that are implanted into a DSSSD can become large compared to the distance between the DSSSD and the surrounding detectors. The subsequent γ-ray emissions from the recoil-isomers can no longer be considered to emanate from a point-like source, illustrated in Figure 5.1 (bottom). If the positions of the recoil-implants can be inferred, it is possible to adjust the recorded times of detected γ-rays to correct for the spatial distribution. The method used to apply this correction is referred to as

125 Chapter the Relative-Photon-Time-of-Flight (RaPToF) technique. Section 5.2 introduces this technique and Section 5.3 tests the technique using experimental data. 5.2 Details of the RaPToF Technique The RaPToF correction accounts for the different flight paths traversed by coincidently emitted γ rays from a point of emission to each detector-pair used to make a timing measurement. The absolute time-of-flight for a given γ ray between its point of emission and a detector is unimportant as timing measurements require two events (two γ rays in this case). It is therefore the time-of-flights of both γ rays in a coincidentally emitted pair that must be considered relative to each other. The method is discussed below and tested using an experimental configuration introduced in Section 4.5 The time-of-flight difference for photons leaving each DSSSD pixel and arriving at each LaBr 3 detector crystal was calculated using Equation 5.1, t pos = p i d p j d. (5.1) c Here, t pos, d and c correspond to the time difference due to the relative-photontime-of-flight between the source and the detector-pair, the position vector for a given DSSSD pixel and the speed of light, respectively, whilst p i and p j correspond to the position vectors of the LaBr 3 detectors providing the start and stop signals, respectively. Figure 5.2 demonstrates this calculation with Detector 2 used to provide the start signal to a TAC and Detector 4 used to provide the stop signal. Figure 5.2 shows how Equation 5.1 is derived geometrically from a scale drawing. Taking the centre of the DSSSD as the origin of the coordinate system, vector arithmetic can be used to calculate the source-to-detector distances for a given DSSSD pixel and two of the LaBr 3 detectors. From Equation 5.1, t pos is equal to the difference in magnitude of the dotted line (red) vectors shown in Figure 5.2.

126 126 Chapter 5 Vacuum Box d O DSSSD p 2 d p 2 p 4 1 p 4 d Figure 5.2: A scale drawing showing the different flight times for photons emitted from a given DSSSD pixel position, d, towards a particular detector pair as illustrated by the difference in magnitude of the dotted line (red) vectors. The figure is shown from a top-down orientation. The solid (blue) vector shows a pixel position relative to the centre of the DSSSD, whilst the dashed (green) vectors show the positions of two LaBr 3 detectors. The origin, O, is defined to be at the centre of the DSSSD. A resultant time offset is calculated as described in the text. 5.3 An Experimental Trial of the RaPToF Correction To investigate the necessity of the RaPToF correction detailed in Section 5.2, an experiment was undertaken to remeasure the known lifetime of the first excited 2 + state in 138 Gd to observe the effect on timing measurements Implementation of the RaPToF Technique using Source Data Due to difficulties in positioning a radioactive source at the implantation position, the source was placed on the outside of the vacuum chamber, shown in Figure 5.2.

127 Chapter Data were collected using a 152 Eu source to test the correction technique. The relative-photon-time-of-flight correction was applied on an event-by-event basis by subtracting t pos, see Equation 5.1, from the recorded TAC value for each pair of detectors. Figure 5.3 shows the effect of the correction being applied to the data as the source was placed into different positions. The timing signal corresponding to the 867-keV (3 + -state decay) and 244-keV (4 + -state decay) γ-ray transitions in 152 Sm were used to serve as the start and stop signals for the TAC connected to Detectors 1 and 8, respectively. The half-life of this 4 + state is known to be 57.7(6) ps [106], which is small compared to the intrinsic time resolution of LaBr 3 detectors and therefore the corresponding timing distribution is approximately Gaussian. The time spectrum represented by a thick line (blue) in Figure 5.3 (a) was made from data collected with the source in the bottom-left of the GREAT vacuum chamber exterior when viewed looking upstream of the recoil axis. The time spectrum represented by a thin line (red) used data collected with the source in the bottom-right of the vacuum chamber. The centroids of these two distributions differ by 979(13) ps. Figure 5.3 (b) shows the same data as in Figure 5.3 (a), but with t pos subtracted from each time measurement in each event. The difference in the centroids was reduced to 50(30) ps. The increase in error on this value is due to the timing correction uncertainty, propagated from uncertainties in the detector and source positions. It is clear that these distributions were more aligned with the correction applied and that the centroid difference is consistent with 0 ps to within two standard deviations. Whilst it is possible that other effects exist, such as the photon penetration dependence into the detector as a function of the angle of incidence to the detector surface, the uncertainties in detector positions in the present experiment were deemed too large to observe them.

128 128 Chapter 5 Figure 5.3: The time difference between the 867-keV and 244-keV internaltransition γ rays in 152 Sm between LaBr 3 Detectors 1 and 8, collected with a 152 Eu source positioned at the bottom-left of the vacuum chamber (thick line, blue) and at the bottom-right of the vacuum chamber (thin line, red) when viewed upstream of the recoil direction axis. The data presented in the inset panels (right) are the sum of the data in the main panels. (a) shows data collected with no correction applied and (b) shows data after being corrected for the relative positions of the source and detectors. The inset to the left of (b) illustrates the path of the γ rays from the outside of the vacuum chamber to the LaBr 3 detectors that correspond to the plot shown in the same style. The DSSSD, shown as a thick horizontal line, is for visualisation purposes only. As the vacuum box exterior was located much closer to the LaBr 3 detectors than the DSSSD, Figure 5.3 shows a greater time-shift than what would be expected from two different implantation positions. This becomes apparent when comparing the difference in magnitude of the dotted line (red) vector-distances shown in Figure 5.2 to the difference in magnitude of the arrows originating from either side of the vacuum box shown on the left inset of Figure 5.3 (b). The maximum magnitude of the time-of-flight correction across the DSSSD was 330 ps. This corresponded to the DSSSD pixel closest to Detector 1 (top-left of DSSSD as

129 Chapter viewed when looking upstream of the recoil axis) acting as a source, with Detector 1 providing the start signal and Detector 8 providing the stop signal. With the source pixel in the opposite corner, closest to Detector 8, then the result was a correction with equal and opposite magnitude. Therefore, in the most extreme case, two timing measurements can differ by 660 ps. It can be seen, from Figure 5.3, that in an experiment measuring decays of recoil implants into the DSSSD, many positions in between the extremes mentioned exist, forming a continuous, wider peak. The effect, if left uncorrected, is a worsened time resolution as the width of this distribution represents additional uncertainty to that of the timing resolution alone Experimental Details An experiment was conducted at the Accelerator Laboratory in University of Jyväskylä using the detector equipment detailed in Section 4.5. A 190-MeV 36 Ar 8 + beam impinged on a 1.1-mg/cm 2 thick 106 Cd target to produce 138 Gd and 136 Sm via the 2p2n and 4p2n channels of a fusion-evaporation reaction, respectively. The beam was produced using the K130 cyclotron at a current of 50 ena over the course of the five-day experiment. The experimental configuration is shown in Figure 5.4.

130 130 Chapter 5 vacuum chamber clover beam target JurogamII recoils MWPC RITU primary beam DSSSD clover LaBr array Figure 5.4: A schematic illustration of the experimental configuration viewed from above. The beam is indicated by the blue arrow while the red arrow demonstrates how the recoils are directed toward the focal plane using RITU. Orange arrows indicate a potential path for a pair of emitted photons having different time-of-flights. The red rectangles on the right show the top half of the LaBr 3 detector array set up in this work. A 44-µg / cm 2 12 C foil backing was used as a charge resetting foil, to suppress the beam components reaching the RITU focal plane by assisting in achieving an average charge state of the recoil products. The beam current was limited by the requirement that the time between consecutive recoil implants at the DSSSD was low compared to the half-life of the isomeric state. Consequently, the measurement rate of the DSSSD was kept below 6 khz throughout the experiment. This corresponded to 170 µs on average between events, which is much larger than the half-life of the isomeric states in either 138 Gd, of 6 µs, or 136 Sm, of 15 µs. The low rate reduced the probability of another recoil arriving and being detected by the DSSSD whilst waiting for a previously implanted recoil to decay, which would adversely affect the ability to determine the position of the decay. The position, in 1 mm 1 mm pixels, of every recoil implantation into the DSSSD was used to identify the position at which the γ-decay of each recoil occurs. The pixel position was determined by taking the maximum energy deposition in each of the horizontally and vertically-orientated DSSSD strips as the point of recoil implantation. An array of eight LaBr 3 detectors, discussed in Section 4.6, was arranged about the DSSSD to detect the timing and energy of fast-transition

131 Chapter ( ps) γ rays of implanted recoil products that are populated by an initial isomeric state. The total data readout (TDR) method was used in which signals from the detectors and TACs were fed into Lyrtech VHS-ADC cards where they were digitised [92]. Digitised signals were processed using moving window deconvolution and time-stamped using a 100-MHz metronome before being further processed by an event builder and subsequently written to tape [107, 108]. The data were sorted using the GRAIN software package [109]. Sorted data were subsequently exported from GRAIN and analysed using the RADWARE software suite [99], GNUPLOT 5.0 [110] and kmpfit [111]. Both GNUPLOT and kmpfit use least-squares minimisation procedures based on the Levenberg-Marquardt algorithm [112, 113] Experimental Results Two triple-coincidence 3D histograms, E i, E j, T AC ij, for each detector pair denoted by i and j, were constructed. TAC values were both corrected using the RaPToF correction from Section 5.2, and not corrected so that comparisons could be made between the two cases. One 3D histogram contained events that were recorded within 18 µs, approximately three times the half-life of the 6-µs 8 isomeric state in 138 Gd [16, 24], subsequent to a recoil implantation into the DSSSD. A second 3D histogram contained events 15 µs prior to the DSSSD implantation and was subtracted from the first histogram to reduce the number of background events. Longer time ranges, beneficial for the observation of the longer-lived 136 Sm isomeric state, appeared to be extremely noisy. The reasons for this were not understood so the same time conditions were implemented for both nuclei. Energy gates set in these 3D histograms were also background subtracted using adjacent energy channels. The widths of the background gates were such that they contained a number of counts approximately equal to the background under the region of interest. Figure 5.5 shows the total projection of the energy coincidence matrix observed when two γ rays were detected within 30 ns of each other (a) along with the resultant projection when selecting γ-ray energies of

132 132 Chapter kev (b) and 384 kev (c), corresponding to the two lowest-lying states in the ground-state band of 138 Gd [76] (a ) Counts / 2 kev (b ) (c) Energy (kev) Figure 5.5: LaBr 3 -energy spectra showing the total projection of the γ-ray coincidence matrix observed in a 30-ns coincidence window (a), those also coincident with a 221-keV γ ray, corresponding to a 138 Gd 2 + state decay (b), and those also in coincidence with 384-keV γ ray, corresponding to a 138 Gd 4 + state decay (c). The shaded areas represent the energy range selection for both events (green, inclining hatch) and background (red, declining hatch). Peak energies listed are in units of kev. The 255 kev and 431 kev peaks correspond to decays in 136 Sm [76]. Only a small number of γ rays are observed in Figure 5.5 as a result of those transitions being below a K π = 8 isomer with a half-life of 6(1) µs [16]. Previous experimental work has estimated the fraction of the total reaction yield that populates this K π = 8 isomeric state to be 1.4 % [28]. A subsequent decay from one of these isomers follows a rotational cascade in which the separation of γ-ray transition energies is large enough to be resolved by LaBr 3 detectors. Because of this isomeric state, and due to recoil selection through RITU in addition to recoil-isomer tagging, it was not necessary to employ an additional clover HPGe

133 Chapter coincidence condition to reduce background events. This would have had the effect of lowering background at a cost of good statistics being reduced by a factor of the HPGe efficiency. The position distribution of implanted recoils, calculated from the maximum energy deposition in the x- and y-strips of the DSSSD, is shown in Figure 5.6. DSSSD y-strip DSSSD x-strip Counts / pixel Figure 5.6: The distribution of recoils implanted over the two side-by-side DSSSDs. Brighter regions indicate a higher number of counts per unit area of 1 mm 1 mm pixels. Dark lines can be seen where data from some DSSSD strips were not recorded, or thresholds increased, due to the presence of electronic noise. To produce the recoil distribution, the timing conditions, above, have been imposed. Whilst not shown due to visual similarity, recoil maps were obtained with an addition condition requiring that an observed coincidence-γ ray had an energy of either 221 kev or 255 kev, corresponding to the decay of the 2 + states in 138 Gd and 136 Sm, respectively. Figure 5.7 shows LaBr 3 time spectra, after selecting on both time and energy as discussed above, with the RaPToF correction applied (right) and not applied (left) for both 138 Gd (top) and 136 Sm (bottom). Shown are both natural time spectra (thick line, blue), where the γ ray populating the given 2 + state was used to start a TAC and the given depopulating γ ray was used to stop the TAC, and reverse time spectra (thin line, red), where the start and stop conditions were reversed.

134 134 Chapter Gd (a ) (b ) Counts / 0.05 ns S m (c) (d ) Time (ns) Figure 5.7: Time spectra of the 2 + states in 138 Gd (top) and 136 Sm (bottom), both with (right) and without (left) the RaPToF correction being applied. The thick blue lines represent the natural time spectra whilst the thin red lines represent the reverse time spectra. Inset into (b) and (d) are the partial level schemes showing the energies of the populating and depopulating γ rays corresponding to the 2 + states in 138 Gd and 136 Sm, respectively. Measured State 138 Gd Sm 2 + RaPToF-corrected no yes no yes C (ns) 0.36(4) 0.36(4) 0.35(19) 0.36(19) PRD (ns) (7) (7) (6) (6) τ (ns) 0.213(20) 0.217(20) 0.20(10) 0.21(9) Table 5.1: Lifetime results obtained from the data shown in Figure 5.7 using the MSCD method. The second row details whether the RaPToF correction was applied. C is the centroid difference between the natural and reverse time spectra. PRD is the mean prompt response difference. The lifetime of the listed state is given as τ. It can be seen that the natural and reverse time data have different mean values, as expected from the MSCD method [73]. The difference in centroids of the natural

135 Chapter and reverse time spectra in each case were computed and listed in Table 5.1 as C. The table also lists the prompt response difference and extracted lifetimes for the specified levels of interest as calculated according to the method outlined in Section The lifetimes of the 2 + states were 213(20) ps and 200(100) ps in 138 Gd and 136 Sm, respectively, for the case with no RaPToF correction having been applied. With the correction applied, the lifetimes were measured to be 217(20) ps and 210(90) ps, respectively. Clearly, the effect of the correction on the lifetime is lower than the standard deviation with the specific LaBr 3 detector geometry used in the present work. This is discussed in detail in Section Experimental Discussion Previous measurements of the lifetime of the 2 + state in 138 Gd have been performed [9, 24], the most recent of which utilised the DDCM in γ-ray coincidence mode to obtain a mean lifetime of 308(17) ps. The measurement made in the present work, 213(20) ps, is about 3.6 standard deviations lower than that of the previous work and lies just outside the standard acceptance range of three standard deviations. While the lifetime result may be correct, one reason for this discrepancy could be that that the time spectra were subtracted using energy windows optimised to provide a clean energy signal, rather than a clean time signal, and so a small systematic error may be expected as a result. To optimise the time signal, at least one of the transitions should have been free from Compton events of higher energy transitions. This was not achievable in the present work and using additional HPGe selection to remove higher energy events was ruled out due to statistics. It is also noted that the main reason behind the experiment was to investigate the effect of DSSSD spatial distribution on lifetime measurements. The 136 Sm 2 + state mean lifetime has been previously measured [25, 114, 115]; the most recent of these measurements was 128(12) ps. This was achieved using the RDM and used both singles measurements and two-fold-coincidence data, with the latter allowing to avoid the possibility of side-feeding contamination. Contamination was mitigated by fitting only distances above 200 µm to remove

136 136 Chapter 5 contributions from short lived feeding transitions. Albeit with a large error on the result obtained in the present work, the two measurements are in agreement within one standard deviation. Due to the lower number of statistics, and a larger relative error observed for the 136 Sm measurement, the 138 Gd measurement is mainly referred to in the analyses of the effect that spatial distribution has on the lifetime measurements. The change in measured lifetime when the RaPToF correction was applied, from the results presented in Table 5.1, was much lower than the uncertainty on the lifetimes for both cases of 138 Gd and 136 Sm. It was therefore impossible to directly infer from the measurements what the effects on the lifetime were as a result of the RaPToF correction due to being unable to resolve the difference between the two cases. This is interpreted as the detector geometry used in the present work not affecting timing resolutions sufficiently enough to be observed with the current level of statistics and thus the effect of the RaPToF correction is also unobservable. A discussion of the effect of spatial distribution follows in Section 5.4 to explain the results presented in this section and to predict when particular detector geometries may require the RaPToF correction to be performed. 5.4 A Method to Quantitatively Link Spatial Distribution to an Increase in Lifetime Uncertainty Due to the limited statistics of the data presented in Section 5.3.3, and the specific distances between the DSSSD and LaBr 3 array in the present work, the effect of the RaPToF correction on the uncertainty of the measured lifetime was not significant. These factors may change between different experimental configurations. For example, at Super-FRS, the focal plane distribution is expected to be larger than in the present work and may require the RaPToF correction to reduce the

137 Chapter uncertainty on measured lifetimes [116, 117]. To demonstrate this, Monte-Carlo simulations have been carried out and are presented in this section. In order to make this calculation relevant to as many experimental configurations as possible, Section introduces the concept of distribution uncertainty. Section builds upon the concept of distribution uncertainty to produce simulated data and test how accurately lifetimes can be measured under various conditions Parameterisation of the Spatial Diffuseness of a Source and the Surrounding Detector Geometry The effect of detector and source geometry on timing measurements is described by t pos in Equation 5.1. Each combination of DSSSD-pixel position and the possible LaBr 3 detector-pair positions can give different values of t pos due to the different path lengths traversed by the γ rays involved. Under the assumption that the position of each DSSSD pixel and each LaBr 3 detector is exactly known, t pos was calculated for all combinations available in the present work and the results were histogrammed. Figure 5.8 (top) shows the resultant t pos time spectrum.

138 138 Chapter Cou n ts / n s Tim e (n s) Figure 5.8: A time spectrum showing the values of t pos calculated for each combination of DSSSD pixel position and pairs of LaBr 3 detectors that demonstrates the effect on timing measurements from the detector geometry used in the present work (top). In the bottom panel, a spectrum corresponding to the intrinsic timing resolution is shown (thick line, blue) along with the result of its convolution with the t pos spectrum (thin line, red). The effect is small enough to not be noticed in the present work and thus performing the RaPToF correction had little impact. Figure 5.8 (bottom) shows both a Gaussian time spectrum that is representative of the intrinsic timing resolution obtained in the experimental work conducted in Section 5.3 (thick line, blue), and the result of the convolution of the t pos spectrum with the intrinsic timing resolution (thin line, red). The intrinsic timing resolution was assumed to have a standard deviation of 0.27 ns and was generated using a random number generator (rng) compatibility layer provided by the numpy python module version [118] that is based on the Mersenne-Twister [62] algorithm. The convolution was achieved by adding each calculated value of t pos to a randomly generated number that followed a Gaussian distribution with a

139 Chapter standard deviation of σ res = 0.27 ns, val = t pos + σ res rng.gaus(), (5.2) where val is the convoluted time value, and rng.gaus() is an abstracted function that returns a number that follows a Gaussian distribution with a standard deviation of one. The thick line (blue) in Figure 5.8 (bottom) represents the time spectrum for a prompt transition, having performed the RaPToF correction to remove the contribution from the detector geometry. The thin line (red) therefore represents the time spectrum of a prompt transition with the RaPToF correction not having been performed. It can be inferred from Figure 5.8 (bottom) that the additional uncertainty due to geometry is small relative to the timing resolution available. The structure of the t pos time spectrum shown in Figure 5.8 is specific to the geometry of the focal plane and the specific placement of the LaBr 3 detectors used in the present work. Instead of proceeding to test the geometrical effect on lifetime uncertainty using the t pos spectrum directly, its standard deviation is computed and used as a single defining parameter to describe the spread of the distribution. This standard deviation parameter, hereby referred to as the distribution uncertainty, is then stated to convolve with the Gaussian time distribution representative of the timing resolution to give a distribution with a standard deviation representative of that which would be measured when not performing the RaPToF correction. In the case of Gaussian distributions, the standard deviation of the convoluted time spectrum is the quadrature sum of the standard deviation of the two convolving distributions. While the t pos time spectrum is not required to be Gaussian, the above statement is shown below to be true for the time spectra shown in Figure 5.8. The standard deviation of each of the time spectra shown in Figure 5.8 was calculated by performing a single pass through each combination of DSSSD pixel and detector-pair positions and summing the number of points, N, the value of t pos at each iteration, t i, and the square of the t pos values at each iteration, t 2 i. From these sums, the standard

140 140 Chapter 5 deviation, σ, was calculated using, σ = 1 N ( N [t 2 i ] 1 N i 2 N [t i ]). (5.3) i The distribution uncertainty was calculated to be σ dist = 117 ps, and the standard deviations of the intrinsic timing distribution and the convoluted distribution were found to be σ res = 270 ps and σ conv = 293 ps, respectively. The quadrature sum of 117 ps and 270 ps yields the value, 294 ps, which is in close agreement to the standard deviation of the convoluted distribution, having a difference of 1 ps. An additional check of the validity of using the distribution uncertainty is provided by considering the experimental configuration of Super-FRS at the Facility for Antiproton and Ion Research (FAIR), in which a fast timing array (FATIMA) surrounds the Advanced Implantation Detector Array (AIDA) [116, 117]. When operating in the 24-cm 8-cm focal plane configuration, assuming a stack of 10 DSSSDs with a strip pitch of mm, a separation of 1 cm between each DSSSD in the stack, and 36 surrounding LaBr 3 detectors, t pos was calculated for each voxel (volume pixel) position and for each detector-pair combination and the results were histogrammed. Figure 5.9 (top) shows the t pos time spectrum obtained for the FATIMA array and Figure 5.9 (bottom) shows the time spectrum corresponding to the intrinsic time resolution (thick line, blue) and the convolution of the t pos time spectrum with the intrinsic timing resolution (thin line, red).

141 Chapter Cou n ts / n s Tim e (n s) Figure 5.9: A time spectrum showing the values of t pos calculated for each combination of DSSSD pixel position and pairs of LaBr 3 detectors that demonstrates the effect on timing measurements from the detector geometry of FATIMA at Super-FRS (top). In the bottom panel, a spectrum corresponding to the intrinsic timing resolution is shown (thick line, blue) along with the result of its convolution with the t pos spectrum (thin line, red). The effect is much larger than that observed in the present work and suggests the RaPToF correction will be effective. The distribution uncertainty and standard deviations of the intrinsic resolution and convoluted time spectra were found to be σ dist = 222 ps, σ res = 270 ps and σ conv = 349 ps, respectively. The quadrature sum of 222 ps and 270 ps yields the value 349 ps, which is equal to the standard deviation of the convoluted distribution. As the distribution uncertainty can be used to find the standard deviation of the convoluted time spectrum, it is used in the following section to simplify the simulations performed. An additional benefit comes from the fact that the distribution uncertainty can be calculated for any source and detector geometry, allowing the effects on lifetime uncertainty to be considered in terms of

142 142 Chapter 5 a parameter that is applicable to all detector geometries, and not just the specific geometry used in the present work. So far, the t pos time spectra have been considered in a way that suggests each pixel, or voxel, has an equal probability of being implanted by a recoiling nucleus, and that subsequently emitted γ rays have an equal probability to be detected in any of the LaBr 3 detectors. Neither of these conditions are met and so the method of calculating the standard deviation is weighted by both the probability, P k, of a given pixel, indexed by k, being implanted given that a recoil is observed and the product of the solid angles, Ω ik Ω jk, subtended by each detector in an associated pair indexed by i and j, i < j. The probability, P k, is obtained from the experimentally measured recoil distribution shown in Figure 5.6. The solid angle subtended by each detector from each pixel position was calculated numerically according to Ω ik = A cos α da, (5.4) r ik 2 where A is the area of the circular front face of the LaBr 3 detector crystals, r ik is the translational vector between the pixel position and the detector surface element, da, and α is the angle between the normal to the surface element and the vector r ik [66]. The weighted standard deviation is then given by, l σ = [w lt 2 l [w l] w l = P k Ω ik Ω jk, l ] ( l [w ) 2 lt l ] l [w, l] (5.5) where w l is the weighting assigned to the lth iteration through all pixel and detector-pair combinations. Using Equation 5.5, the distribution uncertainty for the source distribution and detector geometries in the present work was found to be σ dist = 92 ps. This is lower than the distribution uncertainty calculated under the assumption of uniform weighting (117 ps) due to the increased probability of a recoil implanting a pixel located near the centre of the DSSSD and the increased likelihood that detectors at small distances will observe the γ rays emitted as the implanted recoil

143 Chapter decays. The latter point is due to the inverse-square scaling of the solid angle as a function of the pixel-to-detector distance. The w l weights were also used to calculate a weighted mean of 4.52(25) ps for the t pos time spectra. This is clearly not consistent with zero and results from the non-symmetrical recoil distribution shown in Figure 5.6. It is noted that a non-zero mean value does not necessarily require the RaPToF correction to be performed under circumstances in which perfect symmetry of the recoil distribution and the LaBr 3 -detector positions is unattainable, as was the case in the present work. This is because, in the MSCD method [73], the difference in centroid positions of the natural and reverse time spectra are taken to calculate C. A systematic shift of both centroid positions in the same direction therefore cancels when this difference is taken. Because of this, the following section is aimed at quantifying the increase in the uncertainty on the lifetime as a function of only the distribution uncertainty, rather than the mean Evolution of the Lifetime Uncertainty as a Function of Distribution Uncertainty An investigation into the relationship between the distribution uncertainty and the uncertainty on lifetime measurements was performed using a Monte-Carlo calculation. The following assumptions were made for the calculation: the promptresponse function was Gaussian with a standard deviation was 0.27 ns, as observed for the LaBr 3 detectors in the present work; the lifetime to be measured was 0.3 ns; and the alignment of the prompt-points of the 28 detector-pair components was assumed to have no error. This latter assumption is based on the determination of the synchronisation of the 28 component prompt time points, which had an uncertainty of the order of 1 ps in the present work. Additional error is also introduced in a real system by the prompt response difference correction. The contribution of this to the lifetime uncertainty in the present work was 3 ps, obtained by halving the PRD uncertainty given in Table 5.1, although the work presented in Chapter 6 shows this uncertainty can be reduced further by a

144 144 Chapter 5 factor of two in the present work. It is noted that a distribution uncertainty of 0.11 ns has been used throughout this section to describe the present experimental configuration. This value was obtained before the solid angle weighting was performed in the previous section and results from only weighting the standard deviation by the probability of a recoil being implanted in a given pixel. The difference between 92 ps and 110 ps is, however, small when considering that it is added in quadrature to the timing resolution of 270 ps when convolving the t pos spectrum with a Gaussian defining the timing resolution, and so this choice has a negligible effect on the final results. The distribution uncertainty and number of counts, N, in each spectrum was varied to see how these affected the lifetime uncertainty. The N counts were distributed uniformly across each of the 28 unique detector-pair components. For each count in each of the 28 components, a value was calculated according to Equation 5.6 and binned into a histogram. value = σ res rng.gaus() + σ dist rng.gaus() + τrng.exp(), (5.6) where σ res is the uncertainty due to timing resolution, σ dist is the distribution uncertainty and τ is the lifetime, 0.3 ns. The random number generator, rng, is the same as that defined in the previous section. The function, rng.gaus(), returned a number following a Gaussian distribution with a standard deviation of one. The function, rng.exp(), returned a number following an exponential distribution with a characteristic mean lifetime of one. The value was therefore a convolution of two Gaussian functions and an exponential, about the prompt-point (0 ns). After N values had been produced, all 28 histograms were summed to produce the natural decay distribution. This was repeated, with a τ of equal magnitude and opposite sign, to form the reverse decay distribution that represents the reversal of the start and stop conditions in a TAC. Figure 5.10 shows both natural and reverse distributions for N = 1000 and σ dist = 0.11 ns. The lifetime was subsequently calculated from the distributions shown in Figure 5.10 using the MSCD method [73].

145 Chapter Counts / 0.05 (ns) τ = 0.309(10) ns Time (ns) Figure 5.10: An example of simulated data showing the natural (thick line, blue) and reverse (thin line, red) time spectra with N = 1000 counts in each spectrum and a distribution uncertainty of 0.11 ns. The value of τ shown is calculated from this data using the MSCD method. The above process was repeated times, and the calculated lifetimes histogrammed to produce a distribution whose standard deviation was representative of the uncertainty of any given lifetime measurement. This distribution of calculated lifetimes is shown in Figure 5.11 for N = 1000 and σ dist = 0.11 ns.

146 146 Chapter Counts / ns Time (ns) Figure 5.11: A histogram of the results of lifetime calculations using the MSCD method with N = 1000 counts in each spectrum and a distribution uncertainty of 0.11 ns. The σ parameter shown gives the standard deviation of the distribution, taken to be the uncertainty of a single experimental measurement. Lastly, various combinations of N and σ dist were used to calculate the variation in lifetime uncertainty in each case. It is noted that all calculations produced an absolute lifetime value consistent with the true value of 0.3 ns so the discussion is focused on improvement to the lifetime uncertainty. Figure 5.12 demonstrates how the lifetime uncertainty evolves as a function of distribution uncertainty for fixed values of N.

147 Chapter Relative lifetime uncertainty Lifetime uncertainty (ps) N = 500 N = 1000 N = 2000 N = 5000 N = Distribution uncertainty (ns) Figure 5.12: The evolution in lifetime uncertainty as the distribution uncertainty changes, and for various levels of statistics, N. The top plot shows the absolute uncertainty while the bottom plot shows the uncertainty as a fraction of the value at zero distribution uncertainty. Figure 5.12 (top) shows that the lifetime uncertainty has a large dependence on the statistics available, within the chosen range, compared to the distribution uncertainty. For example, a shift from 1000 counts to 2000 counts at a distribution uncertainty of 0 ns, i.e. for a point-like source, improves the lifetime uncertainty by nearly 4 ps compared to a shift from 0 ns to 0.1 ns distribution uncertainty at 1000 counts improves the lifetime uncertainty by less than 1 ps. The lifetime uncertainty appears to scale approximately with N. Figure 5.12 (bottom) shows the increase in lifetime uncertainty as a function of distribution uncertainty relative to the lifetime uncertainty at 0-ns distribution uncertainty, referred to as the relative lifetime uncertainty. The N dependence is greatly reduced as shown by the significant overlapping of the plot lines. Defining the relative lifetime

148 148 Chapter 5 uncertainty allows an estimate of the lifetime uncertainty to be obtained for any experimental configuration from an estimate of the uncertainty of a point-like source measurement. In the present work, with a distribution uncertainty of 0.11 ns, a relative lifetime uncertainty of 1.03 was obtained using Figure 5.12 (bottom). An estimate of the uncertainty obtainable for a point-like source in the present work can be obtained using the measured lifetime uncertainty, of 0.02 ns for 138 Gd (as shown in Table 5.1) with no relative-photon-time-of-flight correction. By dividing 0.02 ns by the calculated relative increase in uncertainty, of 1.03, ns is obtained. This corresponds to an improvement of less than 1 ps on the lifetime uncertainty obtained with no correction performed, and is consistent with the results presented in this work, i.e. there is no significant increase in error when the spatial distribution is unaccounted for. 5.5 Future Experimental Considerations Experiments may take place at separator focal planes such as RITU or Mass Analysing Recoil Apparatus (MARA) [119] at The University of Jyväskylä, or Super-FRS with the decay spectroscopy detector (DESPEC) at FAIR [65]. With the large diffuseness of recoils at fragmentation facilities and the potential for different recoil-products to be implanted at different positions on the focal plane, the relative-photon-time-of-flight correction in the present work becomes increasingly important. For example, using Figure 5.12 and a value for the distribution uncertainty of 0.22 ns, as was calculated in Section 5.4.1, errors in lifetime measurements will increase by around 13 % if no relative-photon-time-of-flight correction is performed. Cross-section estimates have been made to help guide future experiments. The following cross sections were calculated using PACE [120]. For 138 Gd there was a production cross section of 70 mb at the target position. Over the five-day experiment, roughly 1000 counts were observed in the final TAC spectra. This

149 Chapter corresponded to a statistical error on the measured lifetime of 20 ps. For 136 Sm, a production cross section of 30 mb lead to a measured area of 230(50) counts and an error on the lifetime of 90 ps. It is worth noting here that the experiment was optimised with 138 Gd in mind. Another experiment, using a similar setup and with similar levels of background, could use these values as a rough guide to the order of lifetime that could be measured. These experiments should aim for 1000 counts in each TAC spectrum to achieve about 10% uncertainty on the lifetime measurement with an eight LaBr 3 detector array at the focal plane. In the work by Procter et al. [24], the same reaction, using the RDDS technique at the target position with focal plane recoil decay tagging, had a comparable uncertainty on the lifetime measurement of 17 ps with approximately 5 days of measurement time. The systematic errors in both types of experiment are completely independent and as such both experiment types complement each other. 5.6 Conclusion The effect on a lifetime measurement due to the spatial distribution of recoils implanted into a focal plane DSSSD detector, and the geometry of the surrounding fast-timing detectors, has been studied using experimental data acquired at the Accelerator Laboratory in the University of Jyväskylä. Implanted recoils distributed across this DSSSD were located using the maximum energy deposition in the x and y strips. The subsequent decay of these implanted recoils released γ rays that were measured using an array of eight LaBr 3 detectors to extract both energy information and timing between pairs of γ rays. The positional information was used to apply an event-by-event time shift to all measured time values to correct for the difference in flight time of the recoil-decay γ rays from the corresponding DSSSD pixel position to the known positions of the LaBr 3 detectors. In the present work, the observed recoil distribution and specific placement of the LaBr 3 detectors resulted in no significant improvement in uncertainty being observed when the RaPToF correction was performed, which is interpreted as the geometrical effect on timing measurements being too small to be detected.

150 150 Chapter 5 Simulations were performed to understand the insignificant results and predict when the RaPToF correction may reduce the lifetime uncertainty significantly. A method has been introduced to reduce an arbitrary experimental configuration with a diffuse region of implanted recoils to a single distribution uncertainty parameter. This parameter, combined with Monte-Carlo calculations, has been used to quantify the contribution of the spatial diffuseness of implanted recoils to the uncertainty on a measured lifetime. Using this quantification, a distribution uncertainty of 92 ns was calculated for the experiment conducted in the present work which corresponds to a predicted increase in lifetime uncertainty of 3 % above that of a point-like source. This explains why no noticeable improvement was observed when applying the relativephoton-time-of-flight correction in the present work whilst also demonstrating that situations may exist, such as at Super-FRS, where this correction may be necessary.

151 Chapter 6 A New Fitting Procedure for Obtaining the Individual Time-Walk Parameters with Reduced Complexity for Use with a Multi-Detector Array A method to account for amplitude dependence in the pick-off times of CFDs, or time-walk, has been developed, by the author of this thesis, that treats each CFD as independently contributing to the total time-walk of a multi-detector system. The parameters used to describe the time-walk are simultaneously derived from all prompt response centroid data calculated using prompt time spectra obtained by measurement of a source using each pair of detectors. By using the prompt response centroids directly, instead of calculating the PRD as is done in the MSCD method, twice the number of independent data points are available for use in fitting the time-walk parameters to data. Several of the benefits gained from using the MSCD method [73] are inherent when fitting all centroid data simultaneously, such as a prompt time point for 151

152 152 Chapter 6 each detector pair that is energy independent and the electronic drift affecting both centroids for natural and reverse time spectra equally. In addition, the proposed fitting method reduces the complexity of calculating the time-walk for each detector pair in an array of N > 2 detectors by correctly handling the correlation that arises from the re-use of individual CFDs across multiple detector-pair combinations. Subsequently, the number of parameters used to describe the observed time walk scales with N, instead of N (N 1), as would 2 be required to perform individual fits to the prompt response centroid data from each detector pair. For example, a system with eight detectors requires eight CFDs to generate logic pulses when γ rays are detected. The signals generated by two CFDs are used to start and stop a TAC to produce a time measurement. Such a system has 28 unique detector pairs, in which each CFD is used seven times, e.g. data is recorded when a TAC that is started by the CFD connected to Detector 1 (CFD 1) is stopped by CFDs 2 through 8, resulting in seven unique detector pairs that share the same CFD, each providing the same contribution to the observed time walk. In general, a timing array of N detectors has N CFDs and N (N 1) unique detector pair combinations with each CFD being used in 2 (N 1) of those pairs. Therefore, a multi-detector timing array consisting of three or more detectors will have timing signals with correlated time walks in the observed time spectra for different detector pairs that should be accounted for using the new time-walk correction technique. It is noted, however, that the new time-walk correction is valid even with N = 2 detectors and, by extension, can also be used on summed time spectra akin to the GCD method [21]. This chapter is the result of an independent investigation into the observed time walk of a multi-detector array for use in timing measurements, however, similar concepts have been explored in Reference [21]. This previous work assumed that such correlation exists but did not suggest a means in which to calculate the time walk parameters for each detector-pair individually while accounting for the suggested correlation. In this chapter, the existing method for prompt response correction is considered (in Section 6.1) before the new method, introduced in this work, is presented in

153 Chapter Section 6.2, where it is tested using the experimental data from Chapter 5 and compared to the existing method. 6.1 Existing Prompt Response Correction Method The existing (GCD) method for correcting the systematic deviation in measurement time as a function of amplitude, or time-walk [21], has been utilised in Section to obtain reasonable results i.e. with uncertainty in the order of ps. In this method, the timing data for short-lived, or prompt, transition γ rays is taken for many detector-pairs. The time offset in which two simultaneously emitted γ rays are detected differs for each detector-pair. These offsets are referred to as the prompt points and are denoted by p ij, where the suffixes i and j refer to the indexes of the detector used to start and stop a TAC, respectively. The prompt points for each detector-pair are found using a given prompt transition and all subsequent timing data from the specific detector-pair combinations are synchronised by subtracting these prompt points, allowing the statistics obtained from the timing of each detector-pair to be summed, according to the GCD method [21]. The above time spectra were acquired, using a source of 152 Eu, for each detector pair with two energy coincidence conditions, corresponding to the energies of γ rays that populate and depopulate specific states in the source. The specific energy pairs used, given in Section 4.6.4, result in the collection of data with negligible contribution from Compton background. Due to the presence of non-negligible lifetimes ( 50 ps), with respect to the standard deviation of the PRF ( 300 ps), of the states within the 152 Eu decay products, the measured centroids could not be considered prompt. The resultant time spectra centroids were corrected by the addition (subtraction) of the known lifetimes of the associated state [54] in order to derive the prompt response centroids.

154 154 Chapter 6 It must be noted that the decision was made in the present work to choose an alternate convention (see below) for some of the points so that they would be better distributed over the range of energies. For example, the PRDs were calculated as PRD i = C T (E p, E d ) C T (E d, E p ), where PRD i is the PRD of the ith data point, C T (E 1, E 2 ) is the centroid of the time spectra acquired with a start signal given by a γ ray with energy E 1 and a stop signal energy of E 2, and E p and E d are the energies of the γ rays populating and depopulating the state of interest. Other data points were calculated using the reverse condition, PRD i = C T (E d, E p ) C T (E p, E d ), which is valid due to the symmetry between the natural and reverse time spectra [100]. In a previous work [100], this mirror symmetry condition has been used to provide a second data point in addition to the original data point. As these values are the difference of two measured time spectra centroids (and the mathematical negative of said difference), there is a clear dependence between the two resulting points. These dependent data were then used in fitting the PRDF function parameters (cf. Equation 4.14) which, if not made known to the fitting procedure, would result in incorrect uncertainties being assigned to the parameters. This situation was avoided in the present work by requiring the measured centroid values to only be used in the calculation of a single PRD i value. It is also noted that the energy combinations involve two independent energies and so to correctly display the prompt response differences on a single axis, the data must be modified to be relative to a single energy. In Figure 4.22, the decision was made that all data be relative to a starting energy of 344 kev, one of the transition energies observed in the decay of 152 Eu. It has been shown [21, 73, 100] that modifications to the PRD values are required for measurements made using different reference energies. It is not clear from these texts, however, what procedure should be used to quantify these modifications. A method was developed by the author to allow the data points to be modified by the same parameters that are used to describe those data points, via means of an iterative procedure detailed in Section The iterative procedure used during the fitting process modified the data points by the difference between the

155 Chapter calculated time at the start energy of the given data point and the calculated time for 344 kev, t m in Equation The modified data points were then used in the calculation of the weighted sum of squared residuals, which was minimised during the fit. 6.2 The New Prompt Response Correction Method Motivation for a New Prompt Response Correction Method A new method for time-walk correction has been developed in this thesis that enables the time walk of each CFD in a multi-detector array to be calculated from the observed time walks of prompt centroid data obtained using all unique detector-pair combinations. Equation 4.14, used to describe the prompt response difference, was originally [69] developed to describe the walk of a single CFD rather than the difference between timing measurements. The new method utilises this in parameterising the timing signal in a way that allows each CFD to be independently described with correlations in different detector-pairs being accounted for. This makes it possible to fully describe the time walk without necessitating the summation of different detector-pairs for a given pair of energies, ensuring individual detector-pair information is not lost. Previous works [21, 103] have suggested that performing the time-walk correction using the MSCD method becomes too complicated for timing arrays consisting of many detectors, due to the number of centroids increasing with N (N 1) for 2 an array of N detectors. However, as demonstrated throughout Section 6.2, the majority of data reduction in both techniques is identical e.g. the centroids off all data points are still calculated. Therefore the only increase in complexity using the new technique comes from the way in which the data are fitted, which

156 156 Chapter 6 is offset by using parameters to describe the time-walk of each CFD instead of each pairwise time spectra. The complexity of the new fitting procedure therefore increases linearly with detector number, which is again offset by not having to re-sort data to align to a common synchronised prompt time point in order to sum them in accordance with the GCD method. The modification to the data to make all centroid values relative to a common reference energy, discussed in Section 4.6.4, is avoided by treating the start and stop energies as independent variables and handling the problem two-dimensionally instead of one-dimensionally Details of the New Prompt Response Correction Method The time response of a CFD, wk C, varies as a function of γ-ray energy according to the time walk given by Equation 6.1, w C k (E) = a k E + bk + c k E + d k, (6.1) where E denotes γ-ray energy and a k, b k, c k and d k are parameters to be fitted [21, 100]. The index, k, is used to indicate that each CFD has its own time-walk which can be different from the others in the array. Two detected γ rays, one to provide a start signal and one to provide a stop signal, are required for a time measurement with a TAC. With this parameterisation, the time measurements made using the TAC have an effective walk characteristic, wij. T This characteristic is subsequently referred to as the Time-Measurement Walk (TMW) and is described by Equation 6.2, w T ij (E start, E stop ) = w C j (E stop ) w C i (E start ), (6.2) where the suffixes start or stop denote the nature of the signal, either starting or stopping the TAC. Here, i is the index of the CFD used for the start signal

157 Chapter and j is the index of the CFD used to stop the TAC for a given detector pair, denoted ij. Time measurements also have offsets on their prompt points, p ij, where two simultaneously detected γ rays are detected at non-zero values within the range of the TAC. Variation between detector combinations in these prompt points is present due to asymmetries in the electronic channels, such as different delays in electronic units and differences in connecting wire lengths. A prompt time measurement, subject to the TMW, wij T in Equation 6.2, can be described by Equation 6.3, T ij (E start, E stop ) = p ij + w T ij (E start, E stop ). (6.3) For a TAC started by detector i and stopped by detector j, T ij is the measured prompt time value (including time walk) and p ij is the prompt point of the TAC, that is independent of energy. The T ij values can be found by making time measurements of a source with (almost) prompt states that have a wide range of energies for the corresponding populating and depopulating γ rays, such as 152 Eu. The centroids of these measured energy-dependent prompt response spectra are described by Equation 6.3, and can therefore be used to obtain the CFD time walk parameters (a k, b k, c k and d k k) with a least-squares minimisation procedure. The technique to find the prompt points mimics that discussed in Section 4.6.4, however, it is included below to be discussed using the notation introduced in this section. To obtain the values for p ij, a particular γ-ray pair of a source should be chosen that has high intensity to reduce statistical error. Let the corresponding pair of energies be denoted, E 1 and E 2. For each pair of detectors, a time measurement started by E 1 and stopped by E 2 would have a mean value of p ij + wij T (E 1, E 2 ) ± τ, where τ is the lifetime of the state between energies E 1 and E 2. Similarly, a time measurement started by E 2 and stopped by E 1 would have a mean value of p ij + wij T (E 2, E 1 ) τ. Under the assumption that both measurements were taken simulataneously, the intensity of each time distribution should be equal. It is also assumed that wij T (E 1, E 2 ) wij T (E 2, E 1 ). In this case p ij can be found as the mean of the sum of both measurements. The latter

158 158 Chapter 6 assumption is not strictly true, and is equivalent to the problem of the prompt response function centroid differing between natural and reverse time spectra, as noted when using the GCD method in Section Time spectra are then produced for each detector pair, by measurement of a source, that have energy-coincidence conditions corresponding to fast states, relative to the timing resolution of the detectors, within the daughter product(s) of the source. With the p ij values calculated from source data, a function is constructed that takes four parameters for each present CFD, an array containing the p ij values themselves, the start and stop indexes of the detectors, the energies that were used as start and stop signals, and the mean value of the time measurements made under these conditions. As performed in Section 4.6.4, the mean value of the time measurements must be corrected by adding or subtracting the known lifetime of the corresponding daughter state of the source as appropriate to give prompt points with no lifetime information. The function must then use the i and j indexes to correctly choose a value of p ij and the corresponding set of parameters to pass along to another function that is of the form of Equation 6.3. The function is used fit all centroid data simultaneously, allowing only the CFD parameters to vary, to extract the time-walk parameters for all CFDs in a single least-squares minimisation procedure. Constructing the function in this way allows any time measurement from any detector combination, using any arbitrary energies as start or stop, to be used in finding the correct values of the time-walk parameters for each CFD. It is worth noting that the CFD time walk parameters obtained using the method described above are non-unique solutions and do not describe the reality of the time walk contributions of each CFD individually. However, when a linear combination of two CFD time walk components is made according to Equation 6.2, the resultant TMW correctly describes the time walk for the corresponding detector pair. This is clearly seen when considering just the d k parameters in Equation 6.1, where the difference between any two d k parameters is independent of the value of those parameters; thus the values are arbitrary when taken alone, but the difference of these parameters represents the data. It is therefore only a calculation of the

159 Chapter TMWs that can correctly describe the time walk between two energies. Once the parameters have been found, Equation 6.2 may be subtracted from any subsequent time measurement to correct for the time-walk in a way that is sensitive to the individual CFD units involved in making that measurement. The mean PRD between two γ-ray energies, E p populating a given state of interest and E d depopulating that same state, may be calculated from the CFD time-walk parameters as PRD (E p, E d ) = = 1 i,j W ij 1 i,j W ij N 1 i N 1 i N j=i+1 N j=i+1 W ij [ w T ij (E p, E d ) w T ij (E d, E p ) ] W ij PRD ij (E p, E d ), (6.4) where PRD ij (E p, E d ) = wij T (E p, E d ) wij T (E d, E p ), is the PRD for each detector pair, calculated from the TMWs, and W ij is is the weighting of each PRD ij value. The W ij weights may be made equal to 1/σij, 2 where σ ij is the error on each PRD ij value, or, to account for different intensities observed in different detector pairs, I ij, W ij may be made equal to I ij /σij. 2 Equation 6.4 can be used as a means of directly comparing the new time-walk correction technique to that of the GCD method, however, as the new technique gives individual time-walk information for each detector pair, it is not advised to average these PRDs if the CFD time walks differ. Note that this procedure can be performed with sorting the source data only once, as there is no need to have corrected the data for the prompt points due to not requiring to sum the data over each pair of detectors for a given energy pair. The key point that distinguishes this new method from the GCD method is that the time walk parameters in Equation 6.1 are used to describe components of walk originating from the CFDs in use, and therefore the times of the signals that start and stop a TAC, rather than the resultant time differences. In the GCD method [21] the time spectra for all detector pairs are summed to reduce the number of centroid data points from scaling with N (N 1) for an array of 2 N detector to that of a simple two detector set up. As has been presented in

160 160 Chapter 6 this section, the majority of data analysis between the GCD method and the new method is identical, such as the timing of source states and the resultant centroid calculations. The new method is more complex, by N (N 1), from the addition 2 (subtraction) operations to correct the non-prompt source states to be prompt by correcting for the lifetimes. This is due to this operation being performed on each measured centroid instead of the summed values, however, this is negligible and the new method gains from not requiring to re-sort the data to perform the summation. In addition, the number of parameters used for the new method scales by N, whereas the GCD fitting procedure does not scale. The reason for scaling by N, and not N (N 1), is due to the parameters describing the time walk 2 of each CFD connected to each detector, rather than that of each detector-pair combination separately, owing to the re-use of CFDs in the different combinations that ultimately makes the individual fitting procedure feasible for large detector arrays Experimental Test of the New Prompt Response Correction Method A test of the new time-walk correction, introduced in Section 6.2.2, has been conducted using the experimental data acquired in Section from a radioactive source of 152 Eu and a comparison to the existing time-walk correction method is detailed in Section Section shows that the concept of using a function to describes the start and stop γ-ray energies as independent variables can be used alongside the GCD method equivalently to the existing method of modifying the centroid data to be relative to a single reference energy. Following this, Section applies the time-walk correction to each recorded time measurement made during the experiment conducted in Chapter 5.

161 Chapter Obtaining the CFD Time-Walk Parameters with a 152 Eu Source The majority of data analysis in this section has already been performed using the same 152 Eu source data as presented in Section Therefore a brief explanation is presented here and further details may be found in Section A radioactive source of 152 Eu was placed in front of the LaBr 3 array (cf. Section 4.6) and the time difference between γ rays populating and depopulating states within the daughter products of the source were measured for each detector pair. The γ-ray energy pairs used were (in kev): 444, 1086; 964, 444; 245, 867; 1193, 245; 1299, 344; 779, 344; 564, 1086; and 40, A 122 kev γ ray coincidence was required in a HPGe clover detector to avoid recording Compton events in the last pair of energies given. This set of γ-ray energies provided several data points between 40 kev and 1500 kev that had little or no background events under the energy peaks and so no background correction has been performed. The energy-independent prompt time points, p ij as referred to in Section 6.2.2, were obtained for each detector pair by summing the natural and reverse time spectra using the most intense γ-ray energy combination, 344 kev and 779 kev, and calculating the centroid of the resultant time distribution. Centroids were calculated for each detector pair using the time spectra obtained in coincidence with the pairs of γ rays listed above. This was done for both natural and reverse cases i.e. using the first pair listed above as an example, the natural time spectrum requires that the start signal was from a 444-keV γ ray and the stop signal from a 1086-keV γ ray; the reverse time spectrum required a 1086-keV γ ray start signal and a 444-keV γ ray stop signal. As the lifetimes of the states corresponding to each pair of γ rays were not exactly zero, each of the calculated centroids were modified by the addition (subtraction) of known lifetimes for each state [54]. The result is a list of prompt response centroids, T ij observed for various γ-ray energies used as start, E start, and stop, E stop, signals for each pair of detectors, where i (j) denotes the index of the detector used as a start (stop) signal.

162 162 Chapter 6 The CFD time-walk parameters were computed for each of the eight CFDs used in the LaBr 3 array by fitting the data in accordance with the method proposed in Section The fitting procedure accounted for uncertainties in all three dimensions (E start, E stop, T ij ) by utilising a custom effective variance outlined in Appendix A. Table 6.1 shows the CFD time-walk coefficients obtained from the fitting procedure. k a k (ns kev 1 2 ) b k (kev) c k ( 10 5 ns kev 1 ) d k (ns) 1-4.8(14) 51(35) -24.3(34) -0.4(13) 2-6.3(17) 110(50) -17.0(29) -0.4(13) 3-14(18) 410(500) -12(11) -0.2(14) 4-2.5(6) -23(6) -5.2(24) -0.6(13) 5-2.7(6) -12(12) -1.8(23) -0.6(13) 6-11(6) 130(130) -33(6) -0.1(13) 7-16(22) 500(600) -24(12) -0.1(12) 8-7.6(19) 86(39) -30(4) -0.2(13) Table 6.1: Parameter values used in describing the time-walk contribution of each CFD (with index, k) to a timing measurement using the LaBr 3 array. The uncertainties listed in Table 6.1 have been scaled by the square root of the reduced minimised χ 2, of 12.7, to account for any underestimation of experimental errors [104]. It is noted that the parameter values listed in Table 6.1 have relatively large uncertainties, typically upto 130 % but with greater uncertainty in the d k parameters. Despite these large uncertainties, the TMWs can be calculated with an accuracy of 5 ps (see below) due to large correlation being present between the parameters. For full disclosure, the covariance matrix for these parameters, extracted from the least squares fit routine, is provided by Table B.1 in Appendix B. The time walk for each CFD is shown in Figure 6.1, calculated using the parameters from Table 6.1 with Equation 6.1.

163 Chapter CFD Time Walk (ns) CFD 1 CFD 2 CFD 3 CFD 4 CFD 5 CFD 6 CFD 7 CFD Energy (kev) Figure 6.1: The eight individual wk C CFD time walk components for the LaBr 3 array used in the present work. The zero offset of the time scale is arbitrary as the observed time measurement walk (TMW) is the difference of two values from a corresponding detector pair. See text for details. The zero offset of the time walk shown in Figure 6.1 is arbitrary as a result of only the linear combination being valid for a timing measurement consisting of two detectors. The zero offset of Figure 6.1 can be altered by addition or subtraction of a constant value to all d k parameters listed in Table 6.1; it is only the difference pairs of these d k values that contribute to an observed TMW (cf. Equation 6.2). In order to represent the measured prompt centroid data, surface plots have been made that show the prompt response centroid values, T ij as a function of start energy, E start, and stop energy, E stop, for each detector pair. Figure B.1 has been made that contains a surface plot for every detector-pair combination and is deferred to Appendix B due to its size. A selection of those surface plots are presented in Figure 6.2 for the benefit of the discussion.

164 164 Chapter 6 T 12 T Stop Energy (kev) T 13 T Stop Energy (kev) T 14 T Stop Energy (kev) T 15 T Stop Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Stop Energy (kev) Stop Energy (kev) Stop Energy (kev) Time (ns) Time (ns) Time (ns) Time (ns) Stop Energy (kev) Figure 6.2: Surface plots of the prompt response centroids, T ij, for each detector pair, indexed i and j, as a function of the γ-ray energies used in starting and stopping a TAC. The heatmaps (left) are generated using the pairspecific CFD time walk parameters. The projected 3D plots (right) show the calculated surfaces with the measured prompt response centroid data overlaid.

165 Chapter T 58 T Stop Energy (kev) T 67 T Stop Energy (kev) T 68 T Stop Energy (kev) T 78 T Stop Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Stop Energy (kev) Stop Energy (kev) Stop Energy (kev) Time (ns) Time (ns) Time (ns) Time (ns) Stop Energy (kev) Figure 6.2: Surface plots of the prompt response centroid, T ij, for each detector pair, indexed i and j, as a function of the γ-ray energies used in starting and stopping a TAC (continued).

166 166 Chapter 6 On the left of Figure 6.2, a heatmap is used to present the surface without the obfuscation that occurs for some parts of the plot on the right, where a projection of a 3D plot is shown. The 3D projections (right) show the prompt response centroid data values overlaid onto the calculated surfaces. It can be seen from Figure 6.2 (right) that the shape of the surfaces can differ significantly between different detector pairs. As the prompt points, p ij, are energy independent, they can be subtracted from the T ij surfaces, shown in Figure 6.2, to give the TMWs, wij, T according to Equation 6.3. This is equivalent to the synchronisation of each detector pair to a common prompt point of zero and has the benefit of allowing all surfaces to be displayed using a single colour axis. The time walk observed for different start and stop energies, and for different detector pairs, can therefore be directly compared using Figure 6.3.

167 Chapter w T 12 w T 13 w T 14 w T 15 w T 16 w T 17 w T 18 w T 23 w T 24 w T 25 w T 26 w T 27 w T 28 w T 34 w T 35 w T 36 w T 37 w T 38 w T 45 w T 46 w T 47 w T 48 w T 56 w T 57 w T 58 w T 67 w T 68 w T Time Measurement Walk (ns) Figure 6.3: Heatmaps showing the time measurement walks (TMWs) for each detector pair as a function of the start- and stop-signal γ-ray energies. Brighter regions indicate a low amount of time walk and darker regions represent larger time walks. Whilst not shown, each square has x and y axes corresponding to start and stop energy, respectively, ranging from 40 kev to 1500 kev. The heatmaps were obtained by subtracting the energy-independent prompt points, p ij, from the surfaces that describe the variation in measured prompt response centroids. See text for more details. The absolute magnitude of the TMWs shown have been limited to 25 ps to better present the majority of the data. In order to form a comparison between the new time walk correction method and the existing method, the reduced minimised χ 2 (χ 2 r) values of each is considered. In the existing method, performed in Section 4.6.4, the fitting procedure used to calculate the parameters given in Table 4.5 resulted in a χ 2 r of The fitting procedure of the new method, performed in this section, resulted in a χ 2 r of Both of these values suggest the uncertainties are underestimates, as they are greater than the ideal case of χ 2 r = 1, and is likely the result of underlying

168 168 Chapter 6 systematic uncertainties being unaccounted for. One such cause for systematic uncertainties, present in both methods, is in the calculation of the prompt time values used to synchronise the time spectra, which has been noted at the point of calculation. It is apparent, however, that the new method has a χ 2 r that is closer to unity, and therefore suggests a better fit to the data. This is important when considering that the new method introduces the additional constraint that the time walk contribution of each CFD to the overall time walk of a given detector pair is the same in all detector pairs that use that CFD. An additional comparison is made by computing the mean PRD corresponding to the 2 + state in 138 Gd using the parameters calculated during application of the new time walk correction method [76]. The mean PRD for the keV populating γ ray and the keV depopulating γ ray has been calculated using the existing method in Section and found to be 69(7) ps. The equivalent mean PRD value is calculated, assuming equal intensities in each detector pair, according to Equation 6.4 using the new method and the result is found to be 70.2(32) ps. It is noted that when the relative intensities observed during the experiment conducted in Section 5.3 are accounted for, the mean PRD changes by less than the quoted uncertainty ( 70.0(32)). The two methods therefore produce consistent values for the mean PRD between the two energies stated above, however, the uncertainty on the mean PRD is halved when the new time walk correction method is used over the existing method. This improved uncertainty is likely a result of finding a better local minimum in the overall χ 2 during the least-squares minimisation procedure. Further evidence for this conclusion is provided in the next section. Achieving a similar level of uncertainty to that of the existing method under the imposed correlation restrictions provides evidence that the modelling of the time walk is correct and suggests that each CFD has its own independent contribution to the observed TMW that affects each detector pair similarly. Furthermore, the individual centroid data has been used for each pair of detectors without making any approximations and has resulted in a time walk for the entire array of N

169 Chapter detectors that is completely described by a number of parameters that scales with N. The time measurement walks are seen to exhibit large differences between different detector pairs in Figures 6.2 and 6.3, resulting from each CFD having different time walk contributions. This is perhaps most pronounced in Figure 6.2 (right) in which the shapes of the calculated prompt response centroids are seen to vary significantly, for example in the T 58 and T 67 plots. Given that these individual prompt response centroids can be calculated for each detector pair, it is suggested that the mean PRD need not be calculated. Instead, the time measurements obtained corresponding to a particular state of interest can be corrected before filling an associated histogram during the data sorting procedure. This correction is achieved by subtracting wij T (E start, E stop ) from each measured time value made in which detector i provided the start signal to a TAC by detecting a γ ray with energy, E start, and detector j provided the stop signal by detecting a γ ray with energy, E stop. This technique is explored further in Section Demonstration of the Equivalence of the Fitting Procedures in Both Time-Walk Correction Methods The new time walk correction method has been designed to exploit the correlation in time walks observed between different detector pairs that is present when using an array of three or more detectors to reduce the complexity in deriving the time walk for individual detector pairs. It is, however, also valid to use the new time walk correction method on data acquired from only two detectors (N = 2). Because of this, it is possible to sum the synchronised prompt response time spectra over all detector pairs for detector arrays of N > 2, as is done in the GCD method. The resulting summed spectra can then be treated, in accordance with the new method, as though there are only two detectors. This is performed below for the purpose of demonstrating that the concept of fitting the measured prompt response centroids to a function that treats the start energy and the stop energy as independent variables is equivalent to modifying all of the data points to be

170 170 Chapter 6 relative to a single reference energy, as is done using the MSCD method. After this section, it should be clear that the reduction in the observed uncertainty on the mean PRD values calculated in Section is not a result of using an alternate fitting technique, but rather most likely a result of finding a different local minimum in the value of χ 2 that is closer to, if not equal to, the global minimum. The data analysis performed here is identical to that discussed in Section up to and including the measurement of the p ij prompt points, therefore the discussion below continues from this point to avoid repetition. With the p ij values calculated, the source data, with projected time spectra denoted S ij (E start, E stop ), were re-sorted with p ij subtracted from each time measurement to synchronise each detector pair to a common prompt point of zero. If k is an index for a given time measurement, t k, then the synchronised time spectra, Sij (E start, E stop ), are produced using the values t k, where t k = t k p ij. (6.5) The resultant S ij (E start, E stop ) time spectra were summed over each detector-pair, indexed by i and j, to leave a set of summed spectra obtained in coincidence with a given start energy and a given stop energy, denoted S (E start, E stop ) where S (E start, E stop ) = i,j S ij (E start, E stop ). (6.6) The centroids of the S (E start, E stop ) spectra were computed and modified by the lifetime of the state corresponding to the start and stop γ-ray energies to give a set of energy-dependent prompt response centroids. These summed prompt response centroids represent the mean time measurement walk of the detector array in accordance with the GCD method [21]. To fit these summed prompt response centroids to a function that treats both start and stop energy as independent variables, a hypothetical N = 2 detector array is considered that requires parameters of two hypothetical CFDs to describe

171 Chapter the time measurement walk. The resulting equation used to describe the time measurement walk is w T mean (E start, E stop ) = w C 2 (E stop ) w C 1 (E start ), (6.7) where w C i is the time walk of a CFD, given by Equation 6.1, and index 1 (2) is used for the parameters describing the variation in the time measurement walk with respect to the start (stop) energy. The summed prompt response centroid data were fitted to Equation 6.7, allowing the CFD time walk parameters to vary throughout the least-squares minimisation procedure and the resulting parameter values are given in Table 6.2. k a k (ns kev 1 2 ) b k (kev) c k ( 10 5 ns kev 1 ) d k (ns) (13) 43.3(5) -14.2(5) (29) 28.3(20) (32) (10) Table 6.2: Parameter values used in describing the variation in time measurement walk with respect to start (k = 1) and stop (k = 2) signals due to the detection of γ rays of various energies. The d 1 parameter has been fixed during the least-squares minimisation procedure and set to zero. The d 1 parameter was fixed during the fitting procedure and subsequently set to zero, as only the difference between d 2 and d 1 is required to describe the TMW. The uncertainties given in Table 6.2 have been scaled by the square root of the reduced minimised χ 2 (47.0) to account for any underestimation in experimental errors [104]. Table 6.3 shows the covariance matrix that is used in the propagation of uncertainties on the parameters listed in Table 6.2.

172 172 Chapter 6 a 1 b 1 c 1 d 1 a 2 b 2 c 2 d 2 a 1 3.8E-6-2.3E-5 1.4E E-5-2.0E-4-2.5E E-7 b 1-2.3E-5 4.3E-3-5.8E E-4 5.7E-4 1.4E-9 2.8E-6 c 1 1.4E E-9 5.0E E-8-9.7E-8-1.1E E-10 d a 2 3.2E-5-1.3E-4 1.6E E-3-1.2E-2 1.0E-9-5.7E-5 b 2-2.0E-4 5.7E-4-9.7E E-2 8.2E-2-4.6E-8 3.8E-4 c 2-2.5E E-9-1.1E E-9-4.6E-8 2.2E E-10 d 2-7.9E-7 2.8E-6-2.0E E-5 3.8E-4-1.3E E-6 Table 6.3: Covariance matrix for the eight parameters defining the average time walk contribution from both the start signal energy (index 1) and stop signal energy (index 2) used in the present work. These values are used to propagate errors when calculating the average time measurement walk. The CFD time walk components are shown in Figure 6.4, for the hypothetical CFDs corresponding to start energy (CFD 1) and stop energy (CFD 2) -dependent variation in the observed time walk CFD Time Walk (ns) CFD 1 CFD Energy (kev) Figure 6.4: The individual time walk components used to describe the average time walk for all eight CFDs used to provide start or stop signals to a TAC. The lines are named according to a hypothetical two detector system in which CFD 1 provides the start signal and CFD 2 provides the stop signal. The offset of the time scale is arbitrary as the observed time measurement walk (TMW) is the difference of two values, one from each line shown. See text for details.

173 Chapter The contributions of the start and stop signals to the mean time walk of the array are seen to be different. This is understood in the fact that the average start signal and the average stop signal come from different real CFDs within the array e.g. CFD 1 is never used to provide a stop signal whereas CFD 8 is never used to provide a start signal. The mean TMW is calculated using the parameters listed in Table 6.2 for each combination of start and stop energy in the range 40 kev to 1500 kev and displayed as a surface plot in Figure 6.5. Stop Energy (kev) Start Energy (kev) Start Energy (kev) Time (ns) Stop Energy (kev) Figure 6.5: Surface plots of the mean TMW for the LaBr 3 detector array as a function of the γ-ray energies used in starting and stopping a TAC. The heatmap (left) is generated using the mean time walk parameters as a function of start and stop energies. The projected 3D plot (right) show the calculated surfaces with the measured prompt response centroid data synchronised to a common prompt point of zero and overlaid. The uncertainties on the data are smaller than the markers used to represent them. On the left of Figure 6.5, a heat map displays the mean TMW as a function of start and stop signal energies while the right of the figure shows a projection of the 3D surface plot of the same data with the measured prompt centroid values overlaid as data points. The errors on the data points are smaller than the markers used to represent them. A comparison is made between the mean TMW surface function defined in this section, the individual TMW surface functions defined in Section and the

174 174 Chapter 6 time walk correction procedure used in Section by computing the mean PRD value corresponding to the 2 + state in 138 Gd, as was done for the other techniques. The mean PRD corresponding to this state has been calculated using the new fitting procedure in Section to be 70.2(32) ps. Using Equation 6.4 with the parameters that describe the mean TMW (from Table 6.2), the mean PRD is calculated to be 75.9(33) ps. Both values are in experimental agreement to within two standard deviations. Of importance, these two values, that were derived by fitting the start and stop energies as independent variables, have almost the same uncertainty. As the latter of these values utilises the GCD method to sum the time spectra over the unique detector pairs, the conclusion is that calculating the mean PRD from individual PRD ij values does not improve the uncertainty on the mean PRD over using the GCD method. This mean PRD was also calculated using the GCD method, while performing a 1D fit to modified data points as shown in Section 5.3, to be 69(7) ps. This is in agreement with the result from the mean surface fit performed in this section ( 75.9(33) ps) to within one standard deviation. This reduction in uncertainty when treating both start and stop energies as independent is considered to be due to finding different local minima of χ 2 during both least-squares minimisation procedures. Only a single set of initial parameters, of many trials, were found that resulted in the χ 2 r of 47 and the error on the mean PRD corresponding to the 2 + state in 138 Gd of 3.3 ps. A large majority of trials using different initial parameters resulted in an error on the mean PRD that ranged between 6 ps and 8 ps, which is what was observed for the 1D fit to modified data, and suggests some difficulty is present in finding the global minimum in χ 2 when summing the spectra. No initial parameters have been found for the 1D fit to reduce this uncertainty any further, however, there is a possibility that such parameters exist and so it can not be said with confidence that one technique is better than the other at computing the mean PRD.

175 Chapter Using the New Time-Walk Correction on In-Beam Data To avoid calculating the mean PRD, the TMWs can be subtracted from each timing measurement during the process of sorting recorded data into histograms. The resulting time spectra can then be analysed using the standard MSCD method with no additional prompt response correction. This results in a simplified lifetime calculation of τ = C 2, (6.8) where τ is the lifetime of the state being measured and C is the difference in mean values of the natural and reverse time spectra after being corrected for the TMWs during the sorting procedure. The data, from Section 5.3.3, were sorted again with each time measurement being corrected by subtracting the TMW given by Equation 6.2. Subtracting the prompt points for each detector combination, p ij, from the measured time values synchronised the detector combinations to the same prompt value of zero. The synchronised, TMW-corrected spectra were summed over all detector combinations to produce a single spectrum for both the natural and reverse case as shown in Section Figure 6.6 displays the time spectra corresponding to the 2 + state in 138 Gd without the TMW correction (a) and after performing the TMW correction on each recorded event (b). Table 6.4 shows the lifetime obtained using the new prompt response correction method in comparison to that obtained using the existing method, copied from Table 5.1.

176 176 Chapter 6 Counts / 0.05 ns 120 (a) Existing 100 method Gd (b) 138 Gd New method Time (ns) Figure 6.6: Spectra of natural (thick line, blue) and reverse (thin line, red) time data corresponding to the 2 + state in 138 Gd. Data are shown both unmodified (a) and modified (b) to correct for the energy dependent prompt time response during the sorting procedure. Measured State 138 Gd 2 + TMW-corrected no yes C (ns) 0.36(4) 0.44(4) PRD (ns) (7) 0 τ (ns) 0.213(20) 0.220(20) Table 6.4: Lifetime results obtained from the uncorrected data, shown in Figure 5.7, and the corrected data, shown in Figure 6.6, using the MSCD method. The second row details whether the data were corrected for time-walk whilst sorting. C is the centroid difference between the natural and reverse time spectra. PRD is the prompt response difference. The lifetime of the listed state is given as τ. The data that were corrected for the TMW during the sorting procedure resulted in a lifetime for the 2 + state in 138 Gd of 220(20) ps, shown in Table 6.4. This result is consistent with the value obtained using the MSCD method, of 213(20) ps, to within a single standard deviation suggesting both techniques of time walk correction are equally valid. Unfortunately, the uncertainties on the C parameters in both cases are large and a comparison of the TMW correction procedure during the sort with the mean PRD is hindered, however, it is noted that the change in C from 360(40) ps to 440(40) ps suggests an effective PRD of -80(60) ps is

177 Chapter applied during the sort. The effective PRD is simply calculated as the difference in the C parameters with no TMW correction during the sorting procedure and with the correction applied. The scale of the uncertainty on the effective PRD results in agreement with all equivalent values obtained in the previous section, though the true uncertainty may be infered to be negligible compared to the error on C as it remains unchanged after applying the correction. More generally speaking, there may be cases in which the mean PRD is not a good representation of the overall time walk, such as when the number of detectors is small and when the time walk of the individual CFDs differ. In such cases the method presented in this section allows the time walk to be corrected for by considering the individual TMWs of each detector pair as a function of both start and stop signal energies. 6.3 Chapter Summary A new fitting procedure has been introduced to reduce the number of parameters required to individually describe the time walk for each detector and energy combination from scaling with N (N 1) to scaling with N. The new procedure 2 uses the correlation between the contributions of individual CFDs to the observed time walk for a detector-pair. Such correlation has been mentioned in previous works [21] but no means of actually extracting the correlated parameters was discussed. The present work provides a method to extract these correlated parameters to describe the time walk of a multi-detector array. The new procedure has also been shown to be compatible with the existing GCD method [21] and benefits by having no requirement to modify prompt response centroid data to be relative to a single reference energy. The complexity of fitting each centroid individually has been mentioned [21], however the present work has found that the time required for data preparation is almost the same when using the MSCD method. Energy selection conditions can

178 178 Chapter 6 be scripted to be applicable to all detector pairs as easily as the sum alone. Once these conditions have been defined they can be applied to all energy-calibrated spectra, and manually checked to remove any coincident time spectra if statistics are too small for use in calibration. The new fitting procedure required an amount of time in the order of minutes for an array of eight detectors, performed in the Python language using a single core of a standard i7 desktop computer. It is therefore predicted that significant gains on this computation time would be achieved by using a language allowing for compilation, such as C++, and would compensate for the increased computation time when more detectors are used. By contrast, sorting the source data a second time to perform the summation for the GCD method took approximately 40 minutes. Time is also saved by having no requirement to modify centroid values to be relative to a single reference energy when using the new technique, as the function treats the start energy and the stop energy as independent variables. The new technique therefore handles the additional complexity of using all centroids in the fitting procedure well enough to be feasible for use in large number detector arrays and offers an alternative to the GCD method in cases where taking a mean time walk may not be a good approximation i.e. where the time walk of the individual CFD units differ.

179 Chapter 7 A New Analysis Technique Providing an Improved Accuracy in Fast-Timing Measurements A new analysis technique has been developed in this thesis that uses a single set of parameters to simultaneously describe both the natural and reverse time spectra obtained for lifetime measurements of excited nuclear states. This technique uses a functional form of the natural time spectrum and its mirror-symmetric reflection about the prompt point to perform a fitting procedure that exploits this symmetry and directly provides the lifetime from a least-squares minimisation procedure. The functional form is presented in contrast to the established MSCD method [73], discussed in Section 3.2.2, where no knowledge of the time distribution is utilised. The new technique is tested using simulated data and compared to the MSCD method, and limitations of both methods are discussed. 179

180 180 Chapter The Symmetrised Convolution Lifetime Measurement (SCLM) Technique When making a measurement of an excited nuclear-state lifetime that is much smaller than the intrinsic timing resolution of the detector system used, the PRF is observed, as detailed in Section 6.1. When the energies of the γ rays that provide the start and stop signals to a TAC are of a similar magnitude, the PRF is a symmetrical function and can be approximated as being Gaussian [103]. Recorded time data follow a distribution that is a convolution of the PRF and an exponential distribution characterised by the lifetime of the nuclear state being observed [121]. Under the assumption of a Gaussian PRF, the convoluted function describing the data is [ ( f n (t) = a ) ] 2 σ 2τ exp τ t µ 2 τ [ ( f r (t) = a ) ] 2 σ 2τ exp τ + t µ 2 τ [ σ erfc τ 2 t µ σ 2 [ σ erfc τ 2 + t µ σ 2 ], ], (7.1) where a is the peak area, µ and σ are the respective mean and standard deviations of the convoluted Gaussian component, and τ is the lifetime of the convoluted exponential component. The suffixes n and r denote whether the function describes natural time spectra or reverse time spectra, as defined in Section The reverse function, f r (t), is derived by reflecting the natural function, f n (t), in time about the prompt point µ i.e. (t µ) (t µ). This exploits the symmetry between the natural and reverse-case timing measurements. A given pair of natural and reverse time spectra can be simultaneously described by Equation 7.1 so long as the measured times have been corrected for the TMWs as discussed in Section Once corrected for TMW, the natural and reverse data can be simultaneously fitted in a process that involves minimisation of the sum of the squared residuals of the individual spectra, weighted appropriately to account for the uncertainty on the data. This is equivalent to the minimisation of the total

181 Chapter χ 2, defined as the sum of the χ 2 from the individual data sets to be, χ 2 = i ( fn (t i ) f i σ i ) 2 + j ( fr (t j ) f j ) 2, (7.2) σ j where t is the measurement time, f is the number of counts at time t and indices i and j describe the data from the natural and reverse time spectra respectively. Symmetry about the common prompt point, µ, is intrinsically included in such a method as the centroid of the data is parameterised as µ + τ and µ τ in the natural and reverse cases respectively. By requiring the same parameters be used in both equations, this symmetrical condition is satisfied. This technique is henceforth referred to as the Symmetrised Convolution Lifetime Measurement (SCLM) method. It is noted that at very short (below 15 ps) lifetimes, convergence issues arise while performing the fitting procedure using Equation 7.1, owed to inaccuracies in floating point arithmetic as the exponential term becomes too large. An alternative form of the equation may be more useful in such circumstances. 7.2 Applying the SCLM Technique to Simulated Data Natural and reverse time spectra were simulated using the same method as that discussed in Section The simulations assumed a timing resolution of 0.27 ns, 1000 counts per time spectrum, N, and a lifetime of 0.3 ns, as these values approximated the experimental conditions achieved in Chapter 5. Figure 7.1 shows an example of simulated data from which a lifetime can be extracted.

182 182 Chapter Counts / 0.05 ns Time (ns) Figure 7.1: Simulated data representing the natural (thick line, blue) and reverse (thin line, red) time spectra observed when measuring a nuclear state with a lifetime of 0.3 ns and a timing resolution parameterised by a standard deviation of 0.27 ns with 1000 counts in each spectrum. The parameters used to draw the lines were calculated using the SCLM method. Both the MSCD and SCLM methods were employed to ascertain the known lifetime. The fitted line shown in Figure 7.1 results from the SCLM method, that directly provides the lifetime. The lifetime extracted using the MSCD method, described in Section 3.2.2, assumes no distribution of the data and is simply a function of the mean values of the spectra, with no prompt response difference present in the simulated data. Simulated spectra were generated times, chosen as a compromise between accuracy and the required computation time. The lifetime and error on the lifetime was calculated for each generated spectra and the results were histogrammed. Figure 7.2 shows the resultant histograms.

183 Chapter (a) Counts / 0.1 ps µ = (12) σ = (8) µ = (6) σ = (4) Counts / 2 ps (b) Uncertainty on mean lifetime (ps) µ = (36) σ = (25) (c) Mean lifetime (ps) µ = (28) σ = (20) Figure 7.2: (a) The lifetime uncertainty provided by the fitting program using the SCLM method (thick line, blue) and propagated from centroid uncertainties using the MSCD method (thin line, red). The bottom histograms show lifetimes calculated from different simulated spectra using both the SCLM (b) and MSCD (c) methods. The mean, µ, and standard deviation. σ are listed for each peak. Figure 7.2 (a) shows the uncertainty for both the SCLM method (left), as provided by the fitting program, and MSCD method (right), propagated from the uncertainty on the means of the natural and reverse data. Figure 7.2 (b) shows the lifetime calculated using the SCLM method and Figure 7.2 (c) shows the lifetimes calculated using the MSCD method. The variance on the unbiased variance estimator, s 2, for a normal distribution is given as Var s 2 = 2σ4 n 1, (7.3) where n and σ are the size and standard deviation of a sample distribution, respectively [122]. The uncertainty on s, s, can then be propagated from the

184 184 Chapter 7 variance in Equation 7.3 to give, s = σ 2n 2. (7.4) The uncertainties on the σ parameters given in Figure 7.2 were calculated according to Equation 7.4 assuming s = σ. The standard deviation of the lifetimes calculated using the MSCD method, 9.012(20) ps, is in agreement with the mean of the uncertainties given on the right of Figure 7.2 (a), (6) ps. By contrast, the two methods of uncertainty calculation disagree using the SCLM technique, as (25) ps is not consistent with (12) ps. The mean uncertainty given by the fitting program using the SCLM method, (12) ps in Figure 7.2, is approximately half of that calculated for the MSCD method, (6) ps, suggesting that the SCLM method gives more precise answers. However, when the uncertainty in the SCLM method is calculated by taking the standard deviation of calculated lifetimes, the result is an uncertainty that is slightly bigger, (25) ps, than that obtained with the MSCD method, 9.012(20) ps. Figure 7.2 (b) also shows a mean value for the lifetime of (36) ps, which differs from the known value used in the simulations, 300 ps. The above observations suggest a systematic bias is present in the calculation of lifetimes using the SCLM technique. To investigate further the presence of any systematic effects when utilising the SCLM method, the mean lifetime values have been calculated using both methods and the known lifetime (300 ps) was subsequently subtracted. These deviations from the known lifetime have been calculated for various levels of statistics, N, and are shown in Figure 7.3.

185 Chapter Deviation from true lifetime (ps) (a) 0.04 (b) N (counts) Figure 7.3: Deviation of measured lifetimes from the known value, of 300 ps, as a function of N for both SCLM (a) and MSCD (b) methods. N is the number of randomly generated events used in each distribution for which a single lifetime measurement was made. See text for more details. Figure 7.3 (a) shows clearly that there is little agreement, within the uncertainties shown on the data points, between the lifetimes calculated using the SCLM method and the known value. Furthermore, as statistics increase, the deviation from the true lifetime becomes smaller in magnitude. This trend does not hold for the value calculated at N = 500. The reason for observing this specific trend is unclear at present. In contrast, lifetimes calculated with the MSCD method, shown in Figure 7.3 (b), appear to be consistent with the known value as each point is within a single standard deviation from zero. Further testing is needed to gain an understanding of the cause of the deviations presented when using the SCLM method to determine lifetimes. If solved, the SCLM technique may offer reduced uncertainties on lifetime measurements made both previously and in future work.

186 186 Chapter Limitations of Both MSCD and SCLM Analysis Techniques Limitations of the MSCD Method A systematic deviation has been observed when applying the MSCD method to simulated time spectra corresponding to a small nuclear state lifetime (10 ps). Time spectra were generated according to the method specified in Section with a given lifetime of 10 ps and a resolution defined by a standard deviation of 300 ps. The MSCD method was applied to natural and reverse time spectra to calculate a lifetime. Lifetimes were calculated for separately generated time spectra pairs for each level of statistics, denoted N. Figure 7.4 shows the difference between the mean calculated lifetime and the known lifetime of 10 ps for various levels of statistics Deviation from true lifetime (ps) N (counts) Figure 7.4: Deviation of the measured mean lifetime from the known value, of 10 ps, as a function of counts, N, utilising the MSCD method. N is the number of randomly generated events used in each distribution for which a single lifetime measurement was made. See text for more details.

187 Chapter It is clear from Figure 7.4 that there is a systematic deviation between the mean calculated lifetimes and the known lifetime. It appears that calculated lifetimes tend to be larger than the true lifetime and that this overestimation becomes greater when analysing a lower level of statistics: becoming significant at around 5000 counts for a lifetime of 10 ps. With 1000 counts, the calculated lifetime was 1.5 ps, or 15 % larger than the known value. At 500 counts, this deviation rises to 3.3 ps, or 33 % above the known lifetime. Further work is needed to ascertain the cause of this systematic overestimation. The MSCD method involves taking half of the difference between two mean values, when prompt response differences are ignored. It is therefore not apparent, from such a simple procedure, how systematic bias may be introduced. Given that no such deviations are observed in Figure 7.3 (b), it appears that both the magnitude of the lifetime being measured, perhaps in relation to the timing resolution, and the number of statistics obtained in the measurement contribute toward the systematic overestimation observed in Figure Limitations of the SCLM Method As described in Section 7.1, the functional form used to describe the data is only strictly valid if the γ rays used to provide start and stop signals are of similar energy. In tests conducted using time spectra obtained from states populated and de-populated by several energy combinations in a 152 Eu source, the difference between the centroids obtained from a Gaussian fit and those obtained numerically was small, having a standard deviation of 18 ps. It is noted that the more extreme differences appeared to be the result of low statistics, typically less than 200 to 1000 counts, and that when statistics were larger, in the best cases exceeding counts, the difference was much less than 1 ps. It is therefore quite likely that statistics dominated the differences observed. It may be possible to describe the PRF with a function that is more generally applicable and can account for asymmetries in the observed distribution. Such a consideration would likely not be trivial and, for example, rely on simulations to account for effects such as

188 188 Chapter 7 the dependence of light collection as the interaction point within the detector crystal changes. Alternatively, it may be possible to measure the PRF and create a discrete distribution following its shape; the resultant distribution could be used either directly as the PRF, and numerically convoluted with the exponential nuclear decay probability distribution, or used to find an approximate analytical function to describe the PRF and convolute it with the exponential equation. Due to computational errors in handling floating-point numbers, there is a limit, of 15 ps, on how small the lifetime can be as the exponential term in Equation 7.1 becomes too large. A test using 128-bit floating-point numbers, as opposed to 64-bit, successfully lowered this limit to an approximate range of between 7 ps and 10 ps, however a better solution would be to find an alternate form of the expression that avoids this issue. 7.4 Chapter Summary A potential method of extracting lifetimes from natural and reverse time spectra has been developed in the present work. This new technique has been created to reduce the uncertainty on the measured lifetime. Conflicting evidence of whether it successfully reduces the lifetime uncertainty, compared to that obtained using the MSCD method, has been presented. The uncertainty reported, on the lifetime parameter, by the fitting program suggests a two-fold improvement on the uncertainty calculated when using the MSCD method, however, the results of a Monte-Carlo calculation suggest a slight increase in the uncertainty instead. The increase in uncertainty has been attributed to an observed systematic shift in the measured lifetimes, when compared to the true lifetime. The cause for this systematic shift is unknown, though is thought to be from certain parameters in Equation 7.1, particularly σ and τ, being somewhat indistinguishable. Further work on the new technique is required before it may be of use, including generalising it to work on asymmetrical PRFs if possible.

189 Chapter A small systematic deviation has been found when using the MSCD method on simulated data corresponding to a small lifetime (10 ps) in the limit of low statistics (< 5000). As the cause for such a systematic deviation is unclear, it is suggested that the test be repeated using a different software library to recreate the simulation in order to eliminate the potential of it being caused by software bugs (e.g. in the random number generation library).

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191 Chapter 8 Summary and Outlook 8.1 Summary The effect that spatial distribution of a radioactive nuclear source has on timing measurements has been investigated. An experiment was conducted, using the GREAT focal plane spectrometer at the University of Jyväskylä, to test the effectiveness of correcting for differences in the time-of-flight of γ rays between a recoil-isomer implanted into the DSSSD, and the surrounding detectors. To achieve this, a new array consisting of eight LaBr 3 detectors was constructed and affixed to the GREAT spectrometer, located downstream of the focal-plane DSSSD. Due to geometrical constraints provided by the extension of the vacuum chamber surrounding the DSSSD, the LaBr 3 detectors were positioned approximately 10 cm from the nearest point on the DSSSD surface. The position of implanted recoilisomers was extracted by using a low beam current, so that any implanted recoil would decay before another implant arrived at the DSSSD, taking the x- and y-direction DSSSD strips that had the maximum energy deposited in the event to form a pixel. From a known position, vector addition was used to correct for any differences in time-of-flight of coincident γ rays detected by any two LaBr 3 detectors. The lifetimes of the 2 + states in both 138 Gd and 136 Sm were measured to be 217(20) ps and and 210(90) ps with the RaPToF correction applied, respectively, and 213(20) ps and 200(100) ps without the RaPToF correction being applied, 191

192 192 Chapter 8 respectively, showing an insignificant difference. To explain this observation, and provide a methodology in which to quantitatively predict the effects of spatial distribution on timing measurements for an arbitrary focal-plane and surrounding detector configuration, the concept of distribution uncertainty was defined. This definition depends not only on the spatial distribution of the recoils, but also on the relative positions of the surrounding detectors. A method to calculate this value computationally was put forward and involved computing the difference in time-of-flight between each pixel position of the DSSSD, and each pair of detectors. A weighting was used for each pixel position and was calculated as the percentage of recorded recoils in each pixel over the course of the experiment. An additional weighting was used that accounted for the solid angle available between each pixel position and the detector pair involved. The weighted standard deviation of these values was then taken to give the distribution uncertainty. A Monte-Carlo calculation was performed that simulated the time spectra observed in the experiment for different values of the distribution uncertainty. From this calculation, Figure 5.12 was produced, showing the uncertainty on lifetime measurements as a function of the distribution uncertainty. It was concluded from Figure 5.12 that this effect is best described as an increase in the uncertainty from that of a point-like source to avoid a dependency on the number of recorded events. Using this quantification for the GREAT DSSSD, and the positions of the LaBr 3 detectors in the present work, the distribution uncertainty was calculated to be 92 ps. This corresponds to a predicted increase in the uncertainty on the measured lifetime of 3 % compared to what is expected from measuring a point-like source. As this is much lower than the relative error on the measured lifetime, of 9 %, confidence in this quantification is drawn from the insignificant difference observed when the time-of-flight difference between two coincident γ rays is accounted and corrected for. A new method of correcting for amplitude dependence on the time pick-off, or time-walk, of a CFD has been proposed. By treating each time measurement as a linear combination of two CFD pick-off time events (stop time minus start time), it is apparent that correlation exists between measurements using different

193 Chapter pairs of detectors that share a common CFD. For instance, a measurement made with Detectors 1 and 2 shares the same CFD (connected to Detector 2) as a measurement made with Detectors 2 and 7. In this example, the time-walk exhibited by CFD 2 will be manifest in both measurements. To account for this correlation, each CFD in use was parameterised as having its own independent time-walk profile and, subsequently, the effects of these time-walks were only observable indirectly as TMWs. To find the parameters used to describe the CFD time-walks, a source of 152 Eu was used to provide γ rays corresponding to population or depopulation of states with known lifetimes over a range of energies. A new fitting procedure was defined to calculate the time-walk parameters from the prompt-time centroids at different energies, for different detector combinations and different CFDs. The fitting procedure uses the observed correlation to reduce the complexity of finding parameters used to describe the time walk of each detector pair independently so that the number of required parameters scales with N and not N (N 1). The newly-proposed method has been shown to be 2 equivalent to the existing method when calculating the mean PRD for the detector array used in the present work, however, has the benefit of being applicable in cases where the mean PRD may not be a good approximation to the overall observed time walk. A new method to measure nuclear state lifetimes has been proposed that builds upon the MSCD method to potentially reduce uncertainty for a given level of statistics. The SCLM method assumes a Gaussian PRF to generate a symmetrical pair of equations that describe the coincidence time distribution of γ rays populating and depopulating a nuclear state. Evidence for the reduction of statistical error on the measured lifetime values has been presented in the form of a reduced uncertainty provided by the fitting program when compared to the MSCD method. Evidence of increased statistical error has been shown from the standard deviation of lifetime values resulting from many lifetime calculation using simulated data, which may be due to the demonstrated systematic deviation also present in the technique. If the problem of the systematic deviation is solved, the SCLM technique could potentially as much as halve the uncertainty of measured

194 194 Chapter 8 lifetimes. A systematic overestimation has been found when using the established MSCD method in the specific case of measuring small lifetimes with a small number of statistics. This overestimation has been shown to be in the order of picoseconds using simulated data with a lifetime of 10 ps when the number of counts populating the individual time spectra are in the low thousands. The possibility of this being caused by errors in computer code libraries has not yet been ruled out. 8.2 Outlook The work conducted in Chapter 5 is currently being written up for publication in the journal, Nuclear Instruments and Methods. This work is useful for future experimental configurations that aim to measure the time between coincident γ-rays. The distribution uncertainty can be calculated for an arbitrarily-shaped focal-plane recoil distribution and for an arbitrary number of arbitrarily-positioned γ-ray detectors. The broadening of the time distribution width, when compared to that of a point-like source, can be estimated using Figure 5.12, allowing a decision to be made on the necessity of performing a correction. For example, it may be more desirable to run an experiment at the focal plane of a separator using a high reaction rate, increasing the number of counts in a given measurement at the cost of reliably locating the position. As the centroids of the timing spectra are used to extract the lifetimes using the MSCD method, and the centroid can be more accurately defined with an increase in the observed number of counts, there is a compromise in timing resolution when running at lower beam currents that must be considered for each experimental configuration separately. In the limit of a perfect counting experiment, i.e. with no background events, the uncertainty on a lifetime measurement is proportional to σ/ N, where σ is the timing resolution and N is the number of recorded counts. Any increase in the effective timing resolution, σ eff due to geometrical effects is then cancelled by an equal increase in N or, more intuitively, an increase of N that is the square of

195 Chapter the factor, σ eff /σ. In the case of the LaBr 3 detector array used in the present work at the GREAT spectrometer, the effective resolution due to detector geometry was 3 % more than that for a point-like source, so (σ eff /σ) = = , meaning an increase in N by 6 % would cancel the additional uncertainty on the lifetime due to detector geometry. The decision should be made such that the beam current is not limited by the requirement to identify individual pixel positions if the resultant reduction in the total number of statistics observed during a similar experiment is reduced by 6 % or more. One of the experiments in which correcting for the relative difference in coincident γ-ray time-of-flight becomes increasingly important is that of DESPEC at FAIR [65]. As discussed in Chapter 5, a distribution uncertainty of 0.22 ns was calculated for an assumed focal-plane recoil distribution of 24 cm 8 cm 10 cm, corresponding to a 13 % increase in the error on a given lifetime measurement compared to that of a point-like source. While this is an upper limit, as the recoil distribution will be smaller than the total size of the stack of DSSSDs, an increase in uncertainty of this magnitude should be either corrected for or compensated by increasing the reaction rate to achieve a 28 % ( ) increase in number of recorded counts, assuming a σ/ N relationship in the error on a lifetime measurement as above. It is possible for future time-walk corrections to be performed in accordance with the surface fitting method described in Chapter 6, in which the energies of both the start and stop signals of a TAC are considered to be independent variables. The surface fit has no requirement that the prompt centroid data, acquired from timing measurements of a source, be corrected to be relative to a specific reference energy and thus avoids the potential to introduce additional uncertainty into the mean PRD. In addition, by not requiring a reference energy, it is easier to use prompt response function centroid data acquired from multiple sources, e.g. using 60 Co in addition to 152 Eu. Further work on parameterising the PRF distributions will be useful to increase the usability of the SCLM method. Most importantly for this technique, further

196 196 Chapter 8 work is required to eliminate the systematic inaccuracies present in the current implementation. Lastly, it is acknowledged that, while the signal timing was performed using analogue electronics in present work (cf. Section ), it is possible to directly digitise the voltage pulses at the output of the PMTs [123]. Doing this requires a high sampling frequency as the LaBr 3 signal pulses decay quickly (cf. Section 4.3). A sampling rate of at least 2 GHz is required for optimal timing resolution [123] which is notably an order of magnitude times more information to be stored in comparison with the present work. In addition, signal traces must be stored so that digital processing algorithms can be used to perform the functions of, for example, the CFDs to extract the timing of the signals. Both the high sampling frequency and storage of traces mean a large amount of data must be stored, which may cause issues when using triggerless acquisition systems. One of the benefits of digitising the PMT outputs is that all signal processing can be done offline. For example, one can find the optimal fraction of the maximum pulse height to set as the triggering threshold for the digital CFD algorithm [123]. In the analogue system, this optimisation is performed on an actual CFD unit prior to recording experimental data and cannot be revisited. Additionally, the CFD and TAC (and pre-amplifier) units are not required when using digital electronics, saving on the required cost to implement a detector system and run an experiment. It is therefore likely that future fast-timing experiments may be done by digitising the PMT outputs directly

197 Appendix A An Effective Variance to account for Uncertainties in Two Independent and One Dependent Variable(s) in a Least-Squares Minimisation Routine applied to a Surface Function utilising Existing Algorithms It has been shown [124] that, for an arbitrary one-dimensional function mapping a dependent variable to an independent variable, an effective variance can be defined that allows uncertainties in both variables to be accounted for when using minimisation routines. The effective variance for the one dimensional case is V eff = ( ) 2 f V x + V y, (A.1) x 197

198 198 Appendix A where V eff is the effective variance, f is the function mapping the independent variable, x, to the dependent variable, y, and V x and V y are the variances of x and y respectively. Equation A.1 is derived from considering the most probable parameter values required to map the function, f(x), to a given set of data using the method of maximum likelihood [124, 125] to first order approximation. It can equivalently be derived by minimisation of the squared distance, in units of standard deviation, between a line, y = ax + b, (A.2) and a data point, where y is a polynomial of degree one and represents the first order approximation of any one-dimensional function and a and b are the parameters that map x onto y [126]. The squared distance is then given by, ( ) 2 ( ) 2 D 2 xi x yi y = +, (A.3) σ x where D is the distance between the data point, (x i, y i ), and σ x and σ y are the standard deviations estimated as the uncertainties on the data point. It can be shown that the effective variance given by Equation A.1 can be derived by minimising the squared distance given by Equation A.3. This latter proof (not shown) is extended to the case in which there are two independent variables, given by x and y, and one dependent variable, given by z. As x and y are independent variables, there are no terms in the function mapping them to z, f (x, y) = z, that are functions of one another. The functional form for f(x, y) need only be equivalent to the first order expansion of a general surface to be the analogue of Equation A.2. With this in mind, Equation A.4 shows the simplest case in which a change in x or y can affect z, σ y z x = a; z y = b, (A.4) where a and b are scalar magnitudes of the change in z due to a change in x and y respectively. It follows from Equation A.4 that a change in both x and y would

199 Appendix A 199 affect z by summing the total change from each independent variable according to Equation A.5, dz = a x + b y. (A.5) Integrating Equation A.5, keeping in mind that x and y are independent, gives the function, f (x, y) = z = ax + by + c, (A.6) where c is a constant resultant from the integration. It may be noted that Equation A.6 is a standard equation of a plane. This makes sense when considering the situation from a geometrical standpoint, the plane equation is as simple an approximation of a surface function as the line equation is to a one dimensional function. Using Equation A.6, an expression, similar to Equation A.3, can be constructed to give the squared distance between a data point, (x i, y i, z i ), and the planar equation to be ( ) 2 ( ) 2 ( ) 2 D 2 xi x yi y zi z = + +. (A.7) σ x σ y σ z To minimise Equation A.7, it is necessary to differentiate it with respect to both independent variables and make equal to zero. Equation A.6 is substituted into Equation A.7 to give ( ) 2 ( ) 2 ( ) 2 D 2 xi x yi y zi ax by c = + +. (A.8) σ x Equation A.8 is differentiated with respect to x to give where V j = σ 2 j σ y dd 2 dx = 0 = x i x V x + a (z i ax by c) V z, (A.9) for j in x, y, z and dy dx Solving Equation A.9 for x gives σ z = 0 due to the independence of x and y. x min = V zx i + av x (z i by c) V z + a 2 V x, (A.10)

200 200 Appendix A where x min is the minimum value of x. Similarly, the derivative of Equation A.8 is taken with respect to y to give dd 2 dy = 0 = y i y V y + b (z i ax by c) V z. (A.11) Solving Equation A.11 for y gives y min = V zy i + bv y (z i ax c) V z + b 2 V y, (A.12) where y min is the minimum value of y. Substituting equations A.10 and A.12 into Equation A.8 gives the minimum value for the squared distance between a point and the plane described by Equation A.6, D 2 min, as D 2 min = (z i f (x i, y i )) 2 By defining the effective variance, V eff, as Equation A.13 can be simplified to (a2 V x + V z ) (b 2 V y + V z ) ( a 2 b 2 V xv y V z ) + V z ( ) a 2 b 2 V xv y V z V eff = (a2 V x + V z ) (b 2 V y + V z ) 1. (A.13) + V z, (A.14) D 2 min = (z i f (x i, y i )) 2 V eff. (A.15) Finally, Equation A.4 is used to substitute a and b in Equation A.14 to give ( ( f ) ) ( 2 ( f ) ) ( 2 ( f ) 2 ( ) 2 f V x V y V eff = V x + V z V y + V z x y x y V z + V z ) 1. (A.16) Equation A.15 represents the squared-difference between the z-value of a data point, z i, and the function value computed from the x and y-values, x i and y i, weighted by the inverse of V eff. For a given set of many data points, the value for D 2 min is calculated for each data point in the set and the sum of D 2 min over all

201 Appendix A 201 data points is minimised using existing least-squares minimisation algorithms.

202 202 The remainder of this page has been intentionally left blank.

203 Appendix B More Information on the Time-Measurement Walk (TMW) Parameters This appendix contains detailed information about the parameters, derived in Section 6.2.3, that are used to describe the TMWs (cf. Section 6.2.2). The covariance matrix corresponding to the parameters listed in Table 6.1 is given in Table B.1. In addition, the full compliment of surface plots used to describe the prompt response centroid as a function of start and stop energy are shown in Figure B

204 a 1 b 1 c 1 d 1 a 2 b 2 c 2 d 2 a 3 b 3 c 3 d 3 a 4 b 4 c 4 d 4 a 1 1.6E-1-3.7E+0 3.7E-6-1.2E-2-1.0E-2 2.7E-1-1.8E-7-3.4E-3-3.9E-2 1.0E+0-2.7E-7-2.6E-3-1.2E-3 1.4E-2-4.4E-8-3.7E-3 b 1-3.7E+0 9.9E+1-7.7E-5 3.1E-1 2.5E-1-6.0E+0 4.6E-6 1.2E-1 1.8E+0-4.7E+1 1.2E-5 7.6E-2 3.4E-2-3.5E-1 1.4E-6 1.3E-1 c 1 3.7E-6-7.7E-5 9.1E E-7-2.0E-7 5.6E-6-3.6E E-9-1.7E-7 3.4E-6-1.6E E-8-2.3E-8 3.1E-7-8.3E E-8 d 1-1.2E-2 3.1E-1-2.1E-7 1.3E-1 1.7E-3 4.4E-2 8.5E-8 1.3E-1-3.4E-2 1.1E+0-1.5E-7 1.3E-1 1.4E-3-8.9E-3 7.2E-8 1.3E-1 a 2-1.0E-2 2.5E-1-2.0E-7 1.7E-3 2.2E-1-6.0E+0 3.6E-6-8.7E-3-9.5E-2 2.4E+0-6.8E-7 4.1E-3-1.4E-3 1.8E-2-5.1E-8 1.3E-3 b 2 2.7E-1-6.0E+0 5.6E-6 4.4E-2-6.0E+0 1.8E+2-8.9E-5 3.1E-1 4.1E+0-1.1E+2 2.8E-5-6.5E-2 5.1E-2-5.4E-1 2.0E-6 5.4E-2 c 2-1.8E-7 4.6E-6-3.6E E-8 3.6E-6-8.9E-5 6.4E E-8-5.9E-7 1.2E-5-5.2E E-8-1.8E-8 2.8E-7-6.9E E-8 d 2-3.4E-3 1.2E-1-8.7E-9 1.3E-1-8.7E-3 3.1E-1-8.8E-8 1.3E-1-3.2E-2 1.1E+0-1.3E-7 1.3E-1 1.4E-3-9.0E-3 7.2E-8 1.3E-1 a 3-3.9E-2 1.8E+0-1.7E-7-3.4E-2-9.5E-2 4.1E+0-5.9E-7-3.2E-2 2.5E+1-7.0E+2 1.5E-4-7.5E-1-1.1E-3 1.5E-1 4.4E-7-3.5E-2 b 3 1.0E+0-4.7E+1 3.4E-6 1.1E+0 2.4E+0-1.1E+2 1.2E-5 1.1E+0-7.0E+2 2.0E+4-4.2E-3 2.1E+1-1.4E-2-3.7E+0-1.4E-5 1.2E+0 c 3-2.7E-7 1.2E-5-1.6E E-7-6.8E-7 2.8E-5-5.2E E-7 1.5E-4-4.2E-3 9.2E E-6-2.4E-8 1.1E-6 2.0E E-7 d 3-2.6E-3 7.6E-2-1.2E-8 1.3E-1 4.1E-3-6.5E-2 9.6E-8 1.3E-1-7.5E-1 2.1E+1-4.5E-6 1.5E-1 1.4E-3-1.3E-2 5.8E-8 1.3E-1 a 4-1.2E-3 3.4E-2-2.3E-8 1.4E-3-1.4E-3 5.1E-2-1.8E-8 1.4E-3-1.1E-3-1.4E-2-2.4E-8 1.4E-3 2.8E-2-2.7E-1 1.1E-6-6.0E-4 b 4 1.4E-2-3.5E-1 3.1E-7-8.9E-3 1.8E-2-5.4E-1 2.8E-7-9.0E-3 1.5E-1-3.7E+0 1.1E-6-1.3E-2-2.7E-1 2.9E+0-1.1E-5 1.1E-2 c 4-4.4E-8 1.4E-6-8.3E E-8-5.1E-8 2.0E-6-6.9E E-8 4.4E-7-1.4E-5 2.0E E-8 1.1E-6-1.1E-5 4.7E E-9 d 4-3.7E-3 1.3E-1-1.6E-8 1.3E-1 1.3E-3 5.4E-2 7.8E-8 1.3E-1-3.5E-2 1.2E+0-1.6E-7 1.3E-1-6.0E-4 1.1E-2-8.5E-9 1.3E-1 a 5-1.6E-4 1.2E-2 2.7E-9 1.7E-3-2.1E-3 6.3E-2-3.0E-8 1.8E-3-6.0E-3 1.4E-1-4.4E-8 1.9E-3-4.4E-5 1.6E-3 1.8E-9 1.7E-3 b 5-7.9E-4-5.9E-2-9.1E-8-1.2E-2 4.3E-2-1.1E+0 7.3E-7-1.4E-2 4.0E-1-1.1E+1 2.7E-6-2.4E-2 4.7E-4-1.3E-2-2.5E-8-1.2E-2 c 5-5.3E-9 4.9E-7 1.6E E-8-5.8E-8 2.0E-6-8.1E E-8 1.7E-7-5.5E-6 8.0E E-8 3.5E E-8 1.3E E-8 d 5-3.8E-3 1.3E-1-1.8E-8 1.3E-1 1.3E-3 5.4E-2 7.8E-8 1.3E-1-3.5E-2 1.1E+0-1.5E-7 1.3E-1 1.3E-3-8.3E-3 7.0E-8 1.3E-1 a 6-1.5E-2 5.6E-1-1.6E-7-2.8E-2-3.2E-2 1.2E+0-2.7E-7-2.8E-2 1.4E-1-4.1E+0 9.2E-7-3.3E-2-2.2E-3 5.0E-2 1.2E-8-2.9E-2 b 6 3.1E-1-1.2E+1 3.0E-6 7.0E-1 6.2E-1-2.5E+1 4.4E-6 6.9E-1-3.0E+0 8.4E+1-2.0E-5 8.0E-1 3.9E-2-9.9E-1-7.0E-7 7.2E-1 c 6-1.9E-7 6.6E-6-2.3E E-7-4.1E-7 1.5E-5-4.1E E-7 1.9E-6-5.5E-5 1.2E E-7-2.8E-8 5.9E-7-1.3E E-7 d 6-3.2E-3 1.1E-1-1.1E-8 1.3E-1 2.4E-3 9.8E-3 8.8E-8 1.3E-1-4.1E-2 1.3E+0-1.9E-7 1.3E-1 1.4E-3-1.0E-2 7.0E-8 1.3E-1 a 7-8.3E-2 2.9E+0-1.1E-6 8.0E-1-1.3E-1 5.6E+0-9.3E-7 8.0E-1 8.2E-1-2.2E+1 5.6E-6 7.8E-1-9.3E-3 3.2E-1 1.2E-7 8.0E-1 b 7 2.0E+0-7.0E+1 2.3E-5-2.2E+1 3.0E+0-1.3E+2 1.9E-5-2.2E+1-1.9E+1 5.1E+2-1.4E-4-2.1E+1 2.0E-1-7.5E+0-4.2E-6-2.2E+1 c 7-5.1E-7 1.8E-5-7.0E E-6-8.3E-7 3.5E-5-7.0E E-6 5.5E-6-1.5E-4 3.6E E-6-6.7E-8 2.0E-6 1.2E E-6 d 7-1.4E-3 4.5E-2 1.3E-8 1.1E-1 4.9E-3-1.0E-1 1.1E-7 1.1E-1-5.9E-2 1.8E+0-3.2E-7 1.1E-1 1.6E-3-1.8E-2 6.7E-8 1.1E-1 a 8-5.3E-3 1.3E-1-1.1E-7 4.9E-4-1.1E-2 3.2E-1-1.6E-7 7.0E-4-9.7E-2 2.5E+0-6.6E-7 3.1E-3-9.7E-4 1.2E-2-3.9E-8 2.9E-4 b 8 1.0E-1-2.3E+0 2.3E-6 8.0E-3 2.1E-1-5.5E+0 3.4E-6 4.0E-3 2.9E+0-7.6E+1 1.9E-5-7.3E-2 2.4E-2-2.5E-1 9.9E-7 1.2E-2 c 8-9.5E-8 2.6E-6-1.8E E-8-2.2E-7 6.6E-6-3.0E E-8-8.0E-7 2.0E-5-6.1E E-8-1.2E-8 2.1E-7-5.4E E-8 d 8-3.6E-3 1.2E-1-1.3E-8 1.3E-1 1.7E-3 4.3E-2 8.4E-8 1.3E-1-3.1E-2 1.1E+0-1.3E-7 1.3E-1 1.4E-3-8.8E-3 7.1E-8 1.3E-1 Table B.1: Covariance matrix for the 32 parameters defining the time walk contributions from the eight CFDs used in the present work. The values given in this matrix are used to propagate errors when calculating the time measurement walk for a given detector pair. 204 Appendix B

205 a 5 b 5 c 5 d 5 a 6 b 6 c 6 d 6 a 7 b 7 c 7 d 7 a 8 b 8 c 8 d 8 a 1-1.6E-4-7.9E-4-5.3E-9-3.8E-3-1.5E-2 3.1E-1-1.9E-7-3.2E-3-8.3E-2 2.0E+0-5.1E-7-1.4E-3-5.3E-3 1.0E-1-9.5E-8-3.6E-3 b 1 1.2E-2-5.9E-2 4.9E-7 1.3E-1 5.6E-1-1.2E+1 6.6E-6 1.1E-1 2.9E+0-7.0E+1 1.8E-5 4.5E-2 1.3E-1-2.3E+0 2.6E-6 1.2E-1 c 1 2.7E-9-9.1E-8 1.6E E-8-1.6E-7 3.0E-6-2.3E E-8-1.1E-6 2.3E-5-7.0E E-8-1.1E-7 2.3E-6-1.8E E-8 d 1 1.7E-3-1.2E-2 8.8E-8 1.3E-1-2.8E-2 7.0E-1-2.3E-7 1.3E-1 8.0E-1-2.2E+1 4.2E-6 1.1E-1 4.9E-4 8.0E-3 1.9E-8 1.3E-1 a 2-2.1E-3 4.3E-2-5.8E-8 1.3E-3-3.2E-2 6.2E-1-4.1E-7 2.4E-3-1.3E-1 3.0E+0-8.3E-7 4.9E-3-1.1E-2 2.1E-1-2.2E-7 1.7E-3 b 2 6.3E-2-1.1E+0 2.0E-6 5.4E-2 1.2E+0-2.5E+1 1.5E-5 9.8E-3 5.6E+0-1.3E+2 3.5E-5-1.0E-1 3.2E-1-5.5E+0 6.6E-6 4.3E-2 c 2-3.0E-8 7.3E-7-8.1E E-8-2.7E-7 4.4E-6-4.1E E-8-9.3E-7 1.9E-5-7.0E E-7-1.6E-7 3.4E-6-3.0E E-8 d 2 1.8E-3-1.4E-2 9.0E-8 1.3E-1-2.8E-2 6.9E-1-2.2E-7 1.3E-1 8.0E-1-2.2E+1 4.2E-6 1.1E-1 7.0E-4 4.0E-3 2.3E-8 1.3E-1 a 3-6.0E-3 4.0E-1 1.7E-7-3.5E-2 1.4E-1-3.0E+0 1.9E-6-4.1E-2 8.2E-1-1.9E+1 5.5E-6-5.9E-2-9.7E-2 2.9E+0-8.0E-7-3.1E-2 b 3 1.4E-1-1.1E+1-5.5E-6 1.1E+0-4.1E+0 8.4E+1-5.5E-5 1.3E+0-2.2E+1 5.1E+2-1.5E-4 1.8E+0 2.5E+0-7.6E+1 2.0E-5 1.1E+0 c 3-4.4E-8 2.7E-6 8.0E E-7 9.2E-7-2.0E-5 1.2E E-7 5.6E-6-1.4E-4 3.6E E-7-6.6E-7 1.9E-5-6.1E E-7 d 3 1.9E-3-2.4E-2 8.3E-8 1.3E-1-3.3E-2 8.0E-1-2.9E-7 1.3E-1 7.8E-1-2.1E+1 4.0E-6 1.1E-1 3.1E-3-7.3E-2 3.9E-8 1.3E-1 a 4-4.4E-5 4.7E-4 3.5E E-3-2.2E-3 3.9E-2-2.8E-8 1.4E-3-9.3E-3 2.0E-1-6.7E-8 1.6E-3-9.7E-4 2.4E-2-1.2E-8 1.4E-3 b 4 1.6E-3-1.3E-2 5.3E-8-8.3E-3 5.0E-2-9.9E-1 5.9E-7-1.0E-2 3.2E-1-7.5E+0 2.0E-6-1.8E-2 1.2E-2-2.5E-1 2.1E-7-8.8E-3 c 4 1.8E-9-2.5E-8 1.3E E-8 1.2E-8-7.0E-7-1.3E E-8 1.2E-7-4.2E-6 1.2E E-8-3.9E-8 9.9E-7-5.4E E-8 d 4 1.7E-3-1.2E-2 8.7E-8 1.3E-1-2.9E-2 7.2E-1-2.4E-7 1.3E-1 8.0E-1-2.2E+1 4.2E-6 1.1E-1 2.9E-4 1.2E-2 1.5E-8 1.3E-1 a 5 3.3E-2-5.0E-1 1.1E-6-3.6E-4-5.5E-3 1.1E-1-7.0E-8 2.0E-3-5.6E-3 9.8E-2-4.7E-8 1.9E-3-1.6E-3 3.7E-2-2.2E-8 1.8E-3 b 5-5.0E-1 1.1E+1-1.6E-5 1.9E-2 1.7E-1-3.5E+0 2.1E-6-1.9E-2 5.1E-1-1.2E+1 3.2E-6-2.7E-2 1.7E-2-3.8E-1 2.4E-7-1.3E-2 c 5 1.1E-6-1.6E-5 4.2E E-8-1.1E-8-2.1E-7-6.5E E-8 3.9E-7-1.2E-5 1.7E E-8-5.1E-8 1.3E-6-6.6E E-8 d 5-3.6E-4 1.9E-2 1.3E-8 1.3E-1-2.9E-2 7.1E-1-2.4E-7 1.3E-1 8.0E-1-2.2E+1 4.2E-6 1.1E-1 3.2E-4 1.1E-2 1.6E-8 1.3E-1 a 6-5.5E-3 1.7E-1-1.1E-8-2.9E-2 2.4E+0-5.5E+1 2.5E-5-1.2E-1 2.6E-1-6.3E+0 1.7E-6-3.6E-2-3.2E-2 8.3E-1-3.8E-7-2.8E-2 b 6 1.1E-1-3.5E+0-2.1E-7 7.1E-1-5.5E+1 1.3E+3-5.6E-4 2.7E+0-5.6E+0 1.3E+2-3.7E-5 8.8E-1 7.0E-1-1.8E+1 8.3E-6 6.9E-1 c 6-7.0E-8 2.1E-6-6.5E E-7 2.5E-5-5.6E-4 2.8E E-6 3.2E-6-8.0E-5 1.9E E-7-3.5E-7 9.2E-6-4.3E E-7 d 6 2.0E-3-1.9E-2 8.8E-8 1.3E-1-1.2E-1 2.7E+0-1.2E-6 1.4E-1 7.9E-1-2.1E+1 4.1E-6 1.1E-1 1.4E-3-1.8E-2 2.9E-8 1.3E-1 a 7-5.6E-3 5.1E-1 3.9E-7 8.0E-1 2.6E-1-5.6E+0 3.2E-6 7.9E-1 3.8E+1-9.6E+2 2.1E-4-2.6E-1-1.1E-1 3.3E+0-7.1E-7 8.0E-1 b 7 9.8E-2-1.2E+1-1.2E-5-2.2E+1-6.3E+0 1.3E+2-8.0E-5-2.1E+1-9.6E+2 2.5E+4-5.4E-3 5.4E+0 2.5E+0-7.6E+1 1.3E-5-2.2E+1 c 7-4.7E-8 3.2E-6 1.7E E-6 1.7E-6-3.7E-5 1.9E E-6 2.1E-4-5.4E-3 1.2E-9-1.8E-6-7.6E-7 2.1E-5-6.3E E-6 d 7 1.9E-3-2.7E-2 7.7E-8 1.1E-1-3.6E-2 8.8E-1-3.3E-7 1.1E-1-2.6E-1 5.4E+0-1.8E-6 1.2E-1 3.5E-3-8.2E-2 3.7E-8 1.1E-1 a 8-1.6E-3 1.7E-2-5.1E-8 3.2E-4-3.2E-2 7.0E-1-3.5E-7 1.4E-3-1.1E-1 2.5E+0-7.6E-7 3.5E-3 2.8E-1-5.4E+0 5.1E-6-1.3E-2 b 8 3.7E-2-3.8E-1 1.3E-6 1.1E-2 8.3E-1-1.8E+1 9.2E-6-1.8E-2 3.3E+0-7.6E+1 2.1E-5-8.2E-2-5.4E+0 1.2E+2-9.0E-5 2.5E-1 c 8-2.2E-8 2.4E-7-6.6E E-8-3.8E-7 8.3E-6-4.3E E-8-7.1E-7 1.3E-5-6.3E E-8 5.1E-6-9.0E-5 1.0E E-7 d 8 1.8E-3-1.3E-2 8.9E-8 1.3E-1-2.8E-2 6.9E-1-2.2E-7 1.3E-1 8.0E-1-2.2E+1 4.2E-6 1.1E-1-1.3E-2 2.5E-1-2.3E-7 1.3E-1 Table B.1: Covariance matrix for the 32 parameters defining the time walk contributions from the eight CFDs used in the present work (continued). Appendix B 205

206 206 Appendix B T 12 T Stop Energy (kev) T 13 T Stop Energy (kev) T 14 T Stop Energy (kev) T 15 T Stop Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Stop Energy (kev) Stop Energy (kev) Stop Energy (kev) Time (ns) Time (ns) Time (ns) Time (ns) Stop Energy (kev) Figure B.1: Surface plots of the prompt response centroids, T ij, for each detector pair, indexed i and j, as a function of the γ-ray energies used in starting and stopping a TAC. The heatmaps (left) are generated using the pair-specific CFD time walk parameters. The projected 3D plots (right) show the calculated surfaces with the measured prompt response centroid data overlaid.

207 Appendix B 207 T 16 T Stop Energy (kev) T 17 T Stop Energy (kev) T 18 T Stop Energy (kev) T 23 T Stop Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Stop Energy (kev) Stop Energy (kev) Stop Energy (kev) Time (ns) Time (ns) Time (ns) Time (ns) Stop Energy (kev) Figure B.1: Surface plots of the prompt response centroid, T ij, for each detector pair, indexed i and j, as a function of the γ-ray energies used in starting and stopping a TAC (continued).

208 208 Appendix B T 24 T Stop Energy (kev) T 25 T Stop Energy (kev) T 26 T Stop Energy (kev) T T Stop Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Stop Energy (kev) Stop Energy (kev) Stop Energy (kev) Time (ns) Time (ns) Time (ns) Time (ns) Stop Energy (kev) Figure B.1: Surface plots of the prompt response centroid, T ij, for each detector pair, indexed i and j, as a function of the γ-ray energies used in starting and stopping a TAC (continued).

209 Appendix B 209 T 28 T Stop Energy (kev) T 34 T Stop Energy (kev) T 35 T Stop Energy (kev) T 36 T Stop Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Stop Energy (kev) Stop Energy (kev) Stop Energy (kev) Time (ns) Time (ns) Time (ns) Time (ns) Stop Energy (kev) Figure B.1: Surface plots of the prompt response centroid, T ij, for each detector pair, indexed i and j, as a function of the γ-ray energies used in starting and stopping a TAC (continued).

210 210 Appendix B T 37 T Stop Energy (kev) T 38 T Stop Energy (kev) T 45 T Stop Energy (kev) T 46 T Stop Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Stop Energy (kev) Stop Energy (kev) Stop Energy (kev) Time (ns) Time (ns) Time (ns) Time (ns) Stop Energy (kev) Figure B.1: Surface plots of the prompt response centroid, T ij, for each detector pair, indexed i and j, as a function of the γ-ray energies used in starting and stopping a TAC (continued).

211 Appendix B 211 T T Stop Energy (kev) T 48 T Stop Energy (kev) T 56 T Stop Energy (kev) T 57 T Stop Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Stop Energy (kev) Stop Energy (kev) Stop Energy (kev) Time (ns) Time (ns) Time (ns) Time (ns) Stop Energy (kev) Figure B.1: Surface plots of the prompt response centroid, T ij, for each detector pair, indexed i and j, as a function of the γ-ray energies used in starting and stopping a TAC (continued).

212 212 Appendix B T 58 T Stop Energy (kev) T 67 T Stop Energy (kev) T 68 T Stop Energy (kev) T 78 T Stop Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Start Energy (kev) Stop Energy (kev) Stop Energy (kev) Stop Energy (kev) Time (ns) Time (ns) Time (ns) Time (ns) Stop Energy (kev) Figure B.1: Surface plots of the prompt response centroid, T ij, for each detector pair, indexed i and j, as a function of the γ-ray energies used in starting and stopping a TAC (continued).