.::., ,,;.Iw..f:. ...,.....,.....:.1. ”(.3 .. 31:... 2:5; .51: .21....1 A... ...:..;..2 MICHIGAN STATE U llllHH 3 1293 00910 \|\ This is to certify that the dissertation entitled CHARGED PARTICLE CORRELATIONS FROM INTERMEDIATE MGY NUCLEAR REACTIONS presented by Daniel Alan Cebra has been accepted towards fulfillment of the requirements for Eh,D . degree in Ehxsics and Astronomy Major professor Date June 2], 1990 MSU Ls an Affirmalive Action/Equal Opportunity Institution filfillllllllflfil UBRARY Michigan State 3 University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE MSU Is An Affirmative Action/Equal Opportunity Institution c:\clrc\detedue.pm3—p.1 CHARGED-PARTICLE CORRELATION S FROM INTERMEDIATE ENERGY NUCLEAR REACTIONS By Daniel Alan Cebra A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1990 3“}5Q: fl45~ ./ ABSTRACT CHARGED PARTICLE CORRELATIONS FROM INTERMEDIATE ENERGY NUCLEAR COLLISIONS by Daniel Alan Cebra Charged particle correlations have been used to analyze experimental data with the purpose of determining the answers to questions concerning the bulk properties of nuclear matter. A two-particle analysis has been performed on a set of 500 MeV p+Ag and p+Be data. In this analysis questions were studied concerning the density and the temperature of the interaction region reached during nuclear collisions. The radii that are extracted vary between 2.0 and 12.0 Fm depending on which particle pair is considered. The temperatures that are extracted vary between 3.0 and 5.0 MeV. These results are not significantly different from those of similar studies using heavy-ion projectiles to probe nuclear systems. A multi-particle analysis has been performed on a set of data from the system Ar+V at a range of energies from 35 to 85 MeV/ nucleon. These data were analyzed in an effort to determine at what energy the multi-fragmentation reaction channel is opened. This channel is expected to become the dominant reaction mechanism at energies which are sufficient to allow the system to expand into a region of mechanical instability. The event shape is used to determine whether a sequential decay or a simultaneous multi-fragmentation is the dominant disassembly mechanism at a given incident energy. A clear signature of the multi—fragmentation mechanism is observed at and above bombarding energies of 45 MeV/ nucleon. ii ACKNOWLEDGEMENTS My advisor, Dr. Gary Westfall, deserves a great deal of recognition for his constant support through my long and sometimes reckless graduate career. He has served in a variety of roles for me. He has been an excellent mentor who has constantly provided me with accurate advice and direction, he has been a tireless colleague who was always willing to work to the point of exhaustion to get the experiments going when things were rough; he has been an upbeat teammate who would not give up even when the softball team was down 21 to 5, and he has been a genuine friend. I have received substantial support from the Nuclear Science Division at the Cy- clotron laboratory. I am deeply indebted to those who served as associate director of nuclear science during my years at the lab (Drs Austin, Crawley, and~Gelbke). The research assistantships, which were continued even while I was not in the best of standing, enabled me to concentrate on experimental physics. In addition, the support that I received for outside experiments and for conferences broadened my experiences and made me feel like a member of the international physics community. Drs Westfall, Crawley, Benenson, and Kashy demonstrated a tremendous patience with me during my period of misadventures with the comprehensive exams. Without their assistance and encouragement I would have surely left Michigan State. A great deal of credit should go to the laboratory staff without whom most of the experiments described in this work would not have been possible. The N SCL has, without doubt, the best working atmosphere of any laboratory within which I have worked. I have enjoyed working in the 47r research group. The team that has been as- sembled for the construction and utilization of this experimental device is extremely iii cohesive and talented. The weekly group meetings were essential for co—ordinating the efforts of the 10 to 20 persons who were actively working on the project at a given time. It has taken a great deal of effort to build and operate this device, but I feel that there has been a tremendous reward. I find the results that are currently being generated to be extremely interesting. Of course, I had chosen to come to MSU be- cause I had wanted to do this type of physics, so it is natural for me to be fascinated with these results. I made a decision in my fourth year not to finish my dissertation on the two particle studies but to stay and see the 47r through to operation. It has certainly been worth staying. I would like to thank Lawrence and Lori Heilbronn, whose friendship throughout the years have made my sojourn in the midwest much more pleasurable. And finally, I would like to deeply thank my wife, Karen. Her steady impatience with my lack of progress through my graduate program served as a constant reminder of the ultimate goals. iv Contents LIST OF TABLES x LIST OF FIGURES xi I Introduction 1 A The Equation of State of Nuclear Matter ................ 1 B Phases and Phase Transitions ...................... 2 C Reaction Trajectories .......... q ................. 4 D Statistical Properties of Nuclei: The Question of Temperature . . . . 6 E The Density of Nuclear Matter ..................... 7 F The Transition from Sequential Decay to Multi-fragmentation . . . . 7 G Organization of the Thesis ........................ 8 II Two Particle Correlations from 500 p+Ag and p+Be 10 A Introduction and Experimental Justification .............. 10 I Background ............................ 10 2 The Fireball Model ........................ 11 3 Gamma Emitting States ..................... 13 v III The A B 4 Two—particle Correlations Review ................ 5 Experimental Goals ........................ Description of the Experiment ...................... Extraction of Source Sizes ........................ 1 Basic Method ........................... 2 Source Sizes from p+Ag ..................... 3 Source Sizes from p+Be ..................... 4 The Effect of Energy Cuts on the Extracted Source Radii . . . 5 Summary of Extracted Source Sizes ............... Extraction of Nuclear Temperatures ................... 1 Basic Method ........................... 2 Backgrounds for Independent Emission ............. 3 Efficiency Calculations ...................... 4 Populations of Particle Unstable States ............. 5 The Nuclear Temperature .................... Conclusions ................................ Search for Multi-Fragmentation Introduction ................................ Theoretical Studies ............................ 1 Statistical Multi-fragmentation ................. 2 Transition-State Multi—fragmentation .............. vi 18 21 23 24 24 26 33 42 44 45 45 46 48 49 53 57 60 60 64 64 65 3 Geometrical Multi-fragmentation ................ 73 C Experimental Studies ........................... 74 D Conclusions ................................ 87 IV Multi-Particle Correlations 90 A Introduction to Multi-particle Observables ............... 90 B Experimental Details ........................... 92 1 Exp 87008A ............................ 93 2 Exp 87008B ............................ 93 3 Exp 88012 ............................. 94 C Event Characterization .......................... 94 D Event Shape Analysis ........................... 96 E Events Shape Distributions as a Function of Incident Energy ..... 104 F Comparison to Published Models .................... 108 1 First Order Observables ..................... 108 2 GEMINI .............................. 111 3 Lopez ............................... 111 C Other Effects That May Induce an Observed Elongation ....... 113 1 Finite Multiplicity Effect ..................... 115 2 Rotational Distortions ...................... 117 3 Collective Elongations ...................... 118 H The Sequential Simulation ........................ 120 vii I The Simultaneous Simulation ...................... 1 Randomized Directions, Energy and Momentum Conserved . . 2 Randomized Directions, Multiplicity Constrained ....... 3 Coulomb Trajectories ....................... J Determination of the Reaction Mechanism ............... K Conclusions ................................ V Conclusions APPENDICES A Technical Descriptions of the Detectors Systems A The 4—by-4 Close Packing Array ..................... B The Multi-Wire Proportional Chamber ................. C The Multiplicity Array Detectors .................... D The MSU 47r Array ................. - ........... B The Light Response of Plastic Scintillator A Scintillation Theory ............................ B The Calibration Experiments ...................... 1 The Detectors ........................... 2 Elastic Scattering ......................... 3 Degraded Calibration Beams ................... 4 Fragmentation Beams ...................... viii 125 132 133 134 135 147 149 152 152 152 153 157 157 161 161 164 164 165 168 172 C The Light Responses of the Scintillators ................ 1 Response Function of the Slow Scintillator ........... 2 Response Function of the Fast Scintillator ........... D Conclusions ................................ C Phoswich Analysis Routines LIST OF REFERENCES 173 178 184 194 List of Tables II.1 A summary of the source radii extracted from the 500 MeV p+Ag and p+Be systems. The results are compared to an earlier study of 35 MeV / nucleon N+Ag [Fox88]. ...................... 45 II.2 A list of all of the particle unstable states that could be studied with this detector system. ........................... 47 II.3 The temperatures extracted from this study. .............. 57 IV.1 The mid—rapidity charge gates for the various impact parameter bins for the various systems that were studied. ............... 98 IV.2 The average values for the sphericity and coplanarity parameters from the analysis of central collisions from the system “Ar +51 V at six different beam energies ........................... 104 IV.3 The input parameters for the simulation for the six beam energies that were studied. The simulation attempted to reproduce central events . from the reaction 40Ar +51 V. ...................... 132 A.1 Characteristics of the fast and slow scintillation material ........ 153 B.1 A listing of the specifications of each of the four types of phoswich detector that were employed in the studies of the light response. . . . 165 B2 The thicknesses of the aluminum degraders that were used in the cal- ibration of the phoswich detectors and the energy deposited in each scintillator ................................. 168 8.3 The energies of the observed fragment species during the fragmentation runs. .................................... 174 L 11 II II. List of Figures 1.1 1.2 1.3 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 A diagram of the predicted phases of nuclear matter in Pressure- Density space [Bond85a] .......................... A schematic diagram of the stages of a nuclear reaction and the sub- sequent deacy of the highly excited system ................ EOS of nuclear matter. The solid line iS'an isentrope. The shaded region on the left is the spinodal region. The shaded region on the right corresponds to initial conditions that will lead to breakup after expansion .................................. A schematic diagram of the the reaction 14N +98 Ag assuming the fireball ball. The effective temperature extracted from a study of the populations of gamma emitting states [Morr84]. The expected ratios between states as a function of temperature. . . . A schematic diagram of the reaction p +98 Ag. The hashed area repre- sents the thermalized region ........................ OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOO The diagram of the experimental setup for the study of two- -particle correlations from p+Ag and p+Be. OOOOOOOOOOOOOOOOOOO The correlation function for proton- proton pairs from the 500 MeV p+Ag system ................................ The correlation function for proton-deuteron pairs from the 500 MeV p+Ag system ................................ The correlation function for proton-alpha pairs from the 500 MeV p+Ag system ................................ The correlation function for deuteron-deuteron pairs from the 500 MeV p+Ag system ................................ 11.10 The correlation function for deuteron-alpha pairs from the 500 MeV p+Ag system ................................ 11.11 The correlation function for triton-triton pairs from the 500 MeV p+Ag system .................................... 11.12 The correlation function for proton-proton pairs from the 500 MeV p+Be system. ............................. 11.13 The correlation function for proton-deuteron pairs from the 500 MeV p+Be system. ooooooooooooooooooooooooooooooo 12 15 17 22 25 27 30 31 32 34 35 36 11.14 The correlation function for proton-alpha pairs from the 500 MeV p+Be system. ............................... 11.15 The correlation function for deuteron—deuteron pairs from the 500 MeV p+Be system ................................ 11.16 The correlation function for deuteron-alpha pairs from the 500 MeV p+Be system ................................ 11.17 The correlation function for triton-triton pairs from the 500 MeV p+Be system .................................... 11.18 The correlation functions for a. variety of particle pairs from the 500 MeV p+Ag system. The diamonds correspond to events selected on high total energy for the two particles of the pair, while the crosses are data selected on low total energy. .................... 11.19 The detection efficiency of the MWPC ................. 11.20 The relative detection efficiency for four different particle pairs. The peak efficiency has been normalized to unity for each pair. ...... 11.21 The relative momentum spectrum for proton-triton pairs (squares) and an estimate of the independent background (dashed line). The lower portion displays the counts above background (squares) and fits to the individual peaks (dotted and solid lines) ................. 11.22 The relative momentum spectrum for deuteron-triton pairs (squares) and an estimate of the independent background (dashed line). The lower portion displays the counts above background (squares) and fits to the individual peaks (dotted and solid lines). ............ 11.23 The relative momentum spectrum for deuteron-a pairs (squares) and an estimate of the independent background (dashed line). The lower portion displays the counts above background (squares) and fits to the individual peaks (dotted and solid lines) ................. 11.24 The relative momentum spectrum for triton-a pairs (squares) and an estimate of the independent background (dashed line). The lower por- tion displays the counts above background (squares) and fits to the individual peaks (solid lines). ...................... 111.1 The measured mass yields from the reaction p+Xe. The energies of the proton beam varied from 80 to 350 GeV. The exponent of the solid curve is fit to the data. This analysis was inspired by the Fisher droplet model which predicts da/dA or A‘T [Finn82] ............... 111.2 Predictions of a statistical multi-fragmentation model. The mean mul- tiplicity and its dispersion as a function of the excitation energy are shown [Bond85b] .............................. 111.3 The average temperature T as a function of the excitation energy E*/Ao. The onset of fragmentation occurs when the energy fluctua- tions are large enough to break internal bonds (crack formation). The dashed line illustrates the temperature of a free nucleon gas [Bond85b]. 111.4 Predictions from a micro-canonical statistical multi-fragmentation model. The relative probabilities of evaporation (E, solid), fission (F, dashed), and cracking (C, dot-dash) like events for 131Xe are shown [Gr0587]. . 38 39 40 41 43 49 50 52 54 55 56 63 66 67 68 . III III III III. 111.5 The excitation energy as a function of the internal temperature calcu- lated from a statistical multi-fragmentation [Gr0587]. ......... 69 111.6 The partial widths I‘N for the breakup of ”0571 into N fragments with mass numbers A>10 as a function of the excitation energy of the source. The curves are labeled by the value of N [L6pe89b]. . . . 71 111.7 Fragment velocity correlation functions for (a) all fragments (b) charged fragments only and (c) heavy fragments (A>4) only. A=150, Z262, and E'=5 MeV/nucleon [L6pe89a]. ................... 72 111.8 Superposition of 200 events in the sphericity-coplanarity plane for A=150, Z262, and E’:5 MeV/nucleon [L6pe89a] ............ 73 111.9 The observed IMF (Z>2) multiplicity distributions from 200 MeV/nucleon Au+Au. The five curves represent five participant charge multiplicity bins [Doss87, Harr87]. .......................... 75 111.101nclusive production cross—sections as a function of the fragment charge from the reaction Nb+Be at 11.4, 14.7, and 18.0 MeV/nucleon [Wozn88, Char88b]. ................................. 77 111.11A plot of the charge observed in the first detector against the charge observed in the second detector. The upper right corner shows a spec- trum of Z1 + Z2 [Bowm87] ......................... 78 111.12A plot of Z1 against Z2, where Z1 and Z; are the charges of the two heaviest fragments produced by a microcanonical multi-fragmentation simulation of the decay of 146Nd‘ [Gr0588] ................ 79 111.13Relative velocity distributions between IMFs. The solid dots are the experimental data with statistical errors [Klot89], the dashed line is the prediction from a sequential decay code, and the solid curve is the calculation from a simultaneous multi—fragmentation [Gros89]. . . . . 80 111.14Calculations for two systems (130+197Au, and a+197Au). The left side shows the effective mass (Adi) of the compound system as a function of the original excitation energy; the right side displays the centroid of the relative velocity distribution [Poch89b] ................ 81 111.15A plot of the average IMF multiplicity as a function of excitation energy for various beams on silver (top) and gold (bottom) targets [Trau89]. 82 111.16The relative velocity correlation functions from 84 MeV/nucleon 130 on 1“Au (top) and "“‘Ag (bottom). The three curves are calculations from a sequential decay code with an adjustable decay time scale [Trau89]. 84 111.17The relative angles (left) and relative velocities (right) between pairs of IMFs from the reaction Ne + Au at 60 MeV/nucleon. The particles are ranked from heaviest to lightest [Boug89b] .............. 85 111.185hape analysis for data from the reaction 43 MeV/nucleon Kr + Au, Ag, and Th. The ND value represents the number of IMFs detected [Boug89a] .................................. 86 111.19Shape analysis plots for the breakup of 160 into four as (c). For com- parison, the predictions from a multi~fragmentation (a) and a sequen— tial decay (b) are shown [Pou189]. .................... 88 IV.1 The mid-rapidity charge as a function of the known impact parame- ter. Both the total charge and the quantity that passed the filtering requirements are displayed [Ogil89a]. .................. 97 IV.2 The relative contributions that different impact parameters make to each of the bins [Ogi189a]. ........................ 97 1V.3 Illustration of a typical event. a) The lengths of the vectors corresponds to the velocity of the particles in the laboratory reference frame. b) The same event transformed to the center—of—mass reference frame. . . 99 1V.4 A display of which areas of the S~C space correspond to which spheroidal shapes .................................... 102 1V.5 The average sphericity parameter extracted from a set of 70 MeV/nucleon La+La data as a function of the velocity with which the data were transformed ................................. 103 1V.6 A contour plot of the event shape distribution gated on central events from the system 35 MeV/nucleon 40Ar —i—51 V. ............. 105 IV.7 A contour plot of the event shape distribution gated on central events from the system 45 MeV/nucleon 40Ar +51 V. ............. 105 1V.8 A contour plot of the event shape distribution gated on central events from the system 55 MeV/nucleon 40Ar +51 V. ............. 106 IV.9 A contour plot of the event shape distribution gated on central events from the system 65 MeV/nucleon 40Ar +51 V. ............. 106 IV.10A contour plot of the event shape distribution gated on central events from the system 75 MeV/nucleon 40A? +51 V. ............. 107 IV.11A contour plot of the event shape distribution gated on central events from the system 85 MeV/nucleon 40Ar +51 V. ............. 107 IV.12The first order observables for the system 40Ar+51V at 35 meV/nucleon. a) The identified multiplicity, b) the total multiplicity, c) the mass distribution, d) the charge distribution, e) the proton kinetic energy spectrum, and f) the helium kinetic energy spectrum .......... 110 IV.13The results of the GEMINI simulation (circles) compared to the ex- perimental data (crosses). ........................ 112 IV.14The results of Lopez’s simulation (circles) compared to the experimen— tal data (crosses) .............................. 114 IV.15The trajectory of the S and C centroids as a function of multiplicity. . 116 IV.16The trajectories of the S and C centroids as a function of multiplicity for emission from a rotating source. ................... 119 IV.17An expanded view of the trajectories from the rotational simulations in the region of the multiplicities that are measured for the experimental data ..................................... 119 IV.18An overview of the relative magnitudes of the various effects that in‘ duce elongationS. ............................. 121 IV.19The results of the sequential simulation (circles) compared to the first order observables from the experimental data for the system Ar+V at 35 MeV/nucleon (crosses) ......................... 126 xiv IV IV 1V.20T he results of the sequential simulation (circles) compared to the first order observables from the experimental data for the system Ar+V at 45 MeV/nucleon (crosses) ......................... IV.21T he results of the sequential simulation (circles) compared to the first order observables from the experimental data for the system Ar+V at 55 MeV/nucleon (crosses) ......................... IV.22T he results of the sequential simulation (circles) compared to the first order observables from the experimental data for the system Ar+V at 65 MeV/ nucleon (crosses) ......................... IV.23T he results of the sequential simulation (circles) compared to the first order observables from the experimental data for the system Ar+V at 75 MeV/ nucleon (crosses) ......................... IV.24The results of the sequential simulation (circles) compared to the first order observables from the experimental data for the system Ar+V at 85 MeV/ nucleon (crosses) ......................... IV.25A scatter plot showing the sphericity and coplanarity values for 100 events generated from the sequential simulation (crosses) and for 100 events from the simultaneous simulation (circles). ........... IV.26Distributions of the relative velocity and angle between any two parti- cles from central events from the reaction 100 MeV/ nucleon Ar+V. IV.27A comparison of the predicted event shape distributions from the a) sequential and c) simultaneous simulations to that measured for the b) experimental data for the reaction Ar+V at 35 MeV/ nucleon. IV.28A comparison of the predicted event shape distributions from the a) sequential and c) simultaneous simulations to that measured for the b) experimental data for the reaction Ar+V at 45 MeV/ nucleon. IV.29A comparison of the predicted event shape distributions from the a) sequential and c) simultaneous simulations to that measured for the b) experimental data for the reaction Ar+V at 55 MeV/ nucleon. 1V.30A comparison of the predicted event shape distributions from the a) sequential and c) simultaneous simulations to that measured for the b) experimental data for the reaction Ar+V at 65 MeV / nucleon. 1V.31A comparison of the predicted event shape distributions from the a) sequential and c) simultaneous simulations to that measured for the b) experimental data for the reaction Ar+V at 75 MeV/ nucleon. 1V.32A comparison of the predicted event shape distributions from the a) sequential and c) simultaneous simulations to that measured for the b) experimental data for the reaction Ar+V at 85 MeV / nucleon. IV.33’I‘ he trajectories of the sphericity and coplanarity centroids as the beam energy is increased from 35 to 85 MeV/ nucleon. ............ 1V.3-’-1:5A,,.,,.age and C Average for the two simulations and for the experimental data at each of the six beam energies ................... 1V.35A plot of S Average as a function of the incident energy for the two simulations and for the experimental data. ............... A.1 A side view of four of the detectors .................... 127 128 129 130 131 136 137 138 139 140 141 142 143 145 146 146 154 ' '_' ‘4‘ A2:— A.3 A.4 A.5 B.1 B.2 8.3 8.4 R5 B.6 B.7 B8 B9 A.2 Perspective view of the 4-by-4 array stacked for installation behind the MWPC. .................................. Scatter plots of the X and Y positions determined by the MWPC when in coincidence with phoswichs 1-16 ................... A schematic diagram of the MSU 47r Array. .............. A schematic diagram of a single module of the 47r Array. ....... The experimental configuration for a calibration of the response to protons and deuterons. .......................... A scatter plot of the magnitude of the signal from the slow scintillator against the calculated energy of the recoil proton from the reaction p[p,p]p. .................................. A scatter plot of the magnitude of the signal from the slow scintillator against the recoil proton energy ...................... The centroids of the light distribution were determined for each recoil energy. a) displays the proton response function, and b) displays the deuteron response function. The solid curves correspond to fits to the experimental calibration data. ...................... The light response as a function of energy for the slow scintillator for deuterons and a particles. ........................ A scatter plot displaying the raw data from a fragmentation calibration run. These data are from a single detector and a single rigidity setting. Several different isotopic species are simultaneously produced for each element. .................................. This figure displays the light response of the slow scintillator as a func- tion of energy for several different particle types. These data are from the fragmentation run in the 60 inch chamber using type I detectors. This figure displays the light response of the slow scintillator as a func- tion of energy for several different particle types. These data are from the fragmentation run using the S320 spectrometer and 6 type III de- tectors .................................... This figure displays the light response of the slow scintillator as a func- tion of energy for several different particle types. These data are from a fragmentation run at GANIL [Gont90] ................. B.10 An illustration of the ionization density as a function of depth in scin- tillation material for a 100 MeV deuteron (solid line) and a 25 MeV deuteron (dotted line). .......................... B.11 A compilation of the data from the various calibration runs. This figure displays the light response of the fast scintillator as a function of energy for several different particle types. .............. 154 156 158 159 166 167 169 170 171 175 176 176 177 179 181 B.12 A calibration of the AE scintillator for energy near the punch-in region. 182 CI The electronics diagram for the analysis of signals from the phoswich detectors used in the MSU 47r Array. .................. 185 C.2 0.3 0.4 C.5 C.6 An idealized phoswich signal for a charged particle incident on a de- tector with a fast plastic front scintillator and a slow plastic stopping scintillator. ................................ A scatter plot of the signal in the fast gate versus the signal in the slow gate. The data are from a 41r phoswich at 23°, the reaction is 35 MeV/nucleon Ar+V. ........................... AL—L plots for 16 phoswichs. The gains are set for as. The data are from the reaction 500 MeV p+Ag ..................... A summation of AL—L spectra from the same 16 phoswichs displayed in the previous figure ............................ The predicted particle bands (dotted)and the particle identification gates (solid) ................................. xvii 187 188 190 192 193 bot dis sys Wh lat iso SUI as ph. cle ert Chapter I Introduction A The Equation of State of Nuclear Matter In 1881, van der Waals demonstrated that a system containing particles that exhibit both a weak attraction and a strong short range repulsion will contain at least two discrete phases. This was demonstrated in the study of the equation of state of systems of molecules. The equation of state used by van der Waals was: _ — (1.1) where P is the pressure, R is the gas constant, T is the temperature, I) is the mo- lar volume, and a. and b are constants. For temperatures below a critical value, isotherms on a Pressure-Volume (P-V) diagram are multivalued with respect to pres- sure. A Maxwell-Gibbs construct can be used to explain this area on the diagram as a metastable region in which there is a coexistence of both the liquid and the gas phases. The force between nucleons is similar in nature to the intermolecular forces. N u- cleons exhibit an attractive force and a short-range hard-core repulsion. Estimates of the nuclear equation of state suggest that the matter should exhibit the same prop- erties and phase transitions as a van der Waals system [Lamb78, Dani79, Barr80, Fre oft wht tha def C01] 011 V8.1 Usi dis; var a 11' reg ph: be Fre181, Schu82, Curt83, Bert83, Jaqa83, Bond85a]. Using a Skryme parameterization of the nuclear force, one can determine an equation of state for nuclear matter (EOS): 0+2T 0+21 2 l P=2 — o+1v 0' v2 <7(ace [Bond85a]. A phase transition is expected to occur if a homogeneous system enters a region instability. The requirements for stability are a positive heat capacity (thermo- rnarnic stability), a positive compressibility (mechanical stability), and a positive essure [Barr80]. A negative pressure can not be excluded for any physical reasons and59]. For the EOS suggested for nuclear matter, the heat capacity is positive for temperatures. The compressibility, however, is negative at low temperatures and nsities. In this mechanical instability region (spinodal region) a homogenous sys- is unstable against fluctuation growth [Bert83, Peth87, Heis88, Sura89, Boal89]. these density fluctuations are able to grow, the system will separate into distinct uid and vapor components. The magnitude of the fluctuation growth will be deter- ned by the duration of time spent in the spinodal region and the growth rate of the ctuations [Heis88]. The growth rate of the fluctuations is related to the velocity of SOL‘ gen mu [Ila rea pat the The unt COII will indi Whe The sound in nuclear matter and hence to the EOS [Sura89]. C Reaction Trajectories A nuclear reaction should proceed through several stages as diagrammed in Figure 1.2. These stages can be characterized by their thermodynamic properties. The first stage of the reaction occurs from the point of initial contact between the projectile and the target. During this phase, the interaction region is compressed and heated. Entropy is generated during the course of this compression as the kinetic energy of the projectile nucleus is converted into internal excitation energy of the system. The second stage is marked by the expansion of the thermalized region. This expansion stage is expected to be isentropic [Curt83]. If the system has sufficient energy, it will fragment upon reaching a freeze-out density, which is defined as the density for which the mean free path of a nucleon is equal to the size of the thermalized region. During this stage of the interaction, the system will disassemble into an array of excited nuclear fragments. These newly formed fragments will continue to de-excite in the post-interaction phase until they have dissipated all of their internal energy. Figure 1.3 displays an alternate EOS diagram for nuclear matter [Bert83]. On :his figure one can follow the trajectory of a reaction (dashed line). The solid line orresponds to the S = 0 unperturbed state. The initial compression of the system ill generate entropy which will place the excited system above the S 2: 0 line as 1dicated in the figure (dashed line). The relationship between the energy put into he system and the density should be [Bert83] E : a.(n - no)2 (1'4) here a is the binding energy, n is the density, and no is the normal nuclear density. he system will follow the trajectory suggested by the dashed line on Figure 1.3. Pm C0 Th Ex Fr; Sn of I FigUI deac) Pre-Reaction G Compression / - Thermalization Expansion Fragmentation Secondary Decay of Excited Fragments igure 1.2: A schematic diagram of the stages of a nuclear reaction and the subsequent eacy of the highly excited system. Folio The i was s spino Figur on th initia the t1 llowing the initial compression of the system, it will then expand along an isentrope. e isentropes follow the same form of the S :: 0 line. If the initial excitation energy 5 sufficient to place the system in the overstressed region, it will expand into the inodal zone. 25 IO ~ 6: 32‘ Lué 5 >— Z .J 2 L3: //\\ Fancyggfimrm \\\\\\\ \\\\\/ E Z O //\ /X n# {.‘x \\\\\\\\:\&\\ — c: 5 E // ;/3\ \w . . . \ \ ‘ ‘ .J z 5 //// // \\ ;\\:\\\\ \\\\ q 2 - h ///l \ \\ 5‘; K/ s «\ \ L‘.‘ //‘ \\ \ \ \\ .2. 40- ‘ \\\\ OVERSTRESSED ,»\ \\ ZONE :5 ~ , ’ l6” 4 . . o 5 i :75 2.0 DENStTY n/no gure 1.3: EOS of nuclear matter. The solid line is an isentrope. The shaded region the left is the spinodal region. The shaded region on the right corresponds to tial conditions that will lead to breakup after expansion. The dashed line suggests e trajectory of the compression stage of a reaction [Bert83]. ' Statistical Properties of Nuclei: The Question of Temperature the previous discussion of phases of nuclear matter it has been tacitly assumed that tistical concepts can be applied to nuclear systems. However, nuclear interactions duce systems that contain fewer than 500 particles (the experiments described in s particular work consider systems of approximately 100 nucleons). The relaxation e of these nuclear systems is of the same order of magnitude as the duration of the eraction. Therefore, the simplifying assumptions of infinite matter and thermal ilibriurn can not necessarily be applied. One method of probing the statistical nai du: Du shc em ex; SCV be [rag esti 8011 mm con safe is i: win hea ature of thermalized nuclear systems is to study the apparent temperatures formed curing the course of an interaction. If a true equilibrium has been achieved, all ethods of measuring the temperature should yield similar values. The Density of Nuclear Matter During the course of a nuclear reaction, the temperature and density of the system hould vary. One expects that the hottest and most dense period will be at the initial tages of the reaction. As the system equilibrates, it will expand and cool through mission of energetic particles. At some point during this expansion, a freeze-out is xpected. At this point the expansion will cease, and the system will contract into everal droplets or fragments. The kinetic energy dissipated in the reaction shall >e converted into potential energy which manifests itself as a set of less well bound ragments. This freeze-out process is expected to occur at a Specific density. An :stimate of this density can be made by determining the dimensions and mass of the ource of fragments. F The Transition from Sequential Decay to Multi- fragmentation t low excitation energies, a heated nuclear system is able to de-excite through the mission of gamma rays. The time-scale for electro-magnetic interaction is large mpared to the nuclear time-scale, and for these emissions the nuclear system can fely be assumed to have reached a thermal equilibrium. As the excitation energy increased, channels open for the emission of neutrons and protons. These decays ill be favored over the gamma emitting channels due to their shorter lifetimes. For eavy nuclei fission channels may also be open. For these energies the assumption of ha de pa sp. als of de- thermal equilibrium still holds. This form of de-excitation is analogous to evaporation. For higher excitation energies this slow equilibrium process of nucleon emission is no longer sufficiently rapid to dissipate the excitation energy of the expanding system. The system reaches a region of mechanical instability. The internal energy is expended breaking nuclear bonds within the nucleus itself. This form of rapid de-excitation is known as simultaneous multi-fmgmentation. It is a disassembly mechanism that is qualitatively different than the thermal emission of particles. For even higher energies, the internal temperature of the system should rise above a critical value. For such systems the distinction between the differing phases is lost. The system will completely vaporize. G Organization of the Thesis This thesis addresses the main issues raised in the introduction. Specifically, do nu- clear systems attain a thermal equilibrium during the course of a reaction, at what density does the nuclear freeze-out occur, and can nuclear matter be understood in terms of phases? The first question is addressed in chapter II through a study of the temperatures attained during the course of nuclear interactions. Temperatures were measured through a study of the distribution of excited states of fragments that had been emitted from the decaying systems. The excited state distribution was etermined through the measurement of the correlation functions of light-charged articles. The correlation functions contain peaks at the relative energies that corre- pond to the particle unstable excited states of heavier nuclei. The second question is ISO addressed in chapter II. The same set of data is analyzed to determine the sizes f the sources that emit light fragments. From estimates of the source sizes one can etermine the freeze-out density. local add: whih a glo com: the C a hot evide Thes Senti; first detai anajy raw ( The question of whether a multi-fragment reaction mechanism exists and the acation of the transition point where it becomes the dominant form of decay is .ddressed in chapters III and IV. Chapter III reviews efforts by other research groups vhile chapter IV details the results of a study performed at MSU. This study employs I. global event shape analysis to search for evidence of decays that occur early in the :ourse of the reaction. The existence of such decays suggests a large time scale for ;he disassembly process which indicates that the decaying nuclear system behaves as it hot compressible liquid drop as modeled by Baym et al. [Baym71]. A failure to find evidence of early decays that dissipate large amounts of excitation energy is taken to ingest that the reaction occurs simultaneously, which indicates that the transition :0 a rapid decay mechanism has taken place. § The final chapter of the thesis reiterates the major conclusions reached within :hapters II and IV. The technical details of the experimental detection systems and ".he basic procedures employed to analyze the raw data are detailed in the appendices. l‘hese sections address points which are crucial to this analysis, but would not be es- sential to another experimenter doing similar studies with a different apparatus. The irst appendix describes the physical features of the detectors; the second appendix letails how the energy calibrations were determined; and the third outlines the basic .nalysis procedures that were employed to extract the particle identification from the aw data. Chapter II Two Particle Correlations from 500 p+Ag and p+Be A Introduction and Experimental Justification 1 Background Intermediate energy heavy—ion collisions create short lived nuclear systems which contain a finite number of particles. These systems are sufficiently ephemeral that one can reasonably question whether it is possible to consider equilibrium quantities such as temperature, density, entropy, or pressure. Simple models of nuclear reactions at these energies predict that the incident energy of the projectile will be converted into center of mass motion and excitation energy of the fused system. The system will thermalize, eXpand, and decay. The disassembly of the system is considered to be a process which can be described as freeze-out. Whether this freeze-out occurs Lt a characteristic density or temperature is currently an open question. Attempts 0 measure the nuclear temperature and density at freeze-out can lead to a better Inderstanding of nuclear matter. 10 11 2 The Fireball Model The fireball model is a simple model for nuclear collisions [West76], which assumes the establishment of a thermal equilibrium. It has been applied successfully to describe the inclusive cross-sections of fragments emitted from nuclear collisions in the energy range from 20 to 200 MeV / nucleon [J aca83]. The primary consideration of this model is the geometry of the system for a given nuclear reaction. Figure 11.1 provides a schematic view of the evolution of a reaction as considered in this model. The portion of the projectile that overlaps with the target is considered to form a fireball. The velocity of this fireball can be determined from the incident energy of the beam and the relative weights of the projectile and target contributions. The portions of the target and the projectile that did not overlap during the reaction are considered to continue in their original trajectories relatively unchanged by the collision. Their velocities are slightly modified, and some heating occurs. This model has been applied to experimental data through the use of moving source fits. This has become a standard technique in the analysis of inclusive data. It is used for the extraction of source velocities, temperatures, and production cross- sections [Jaca83, J aca87, Wada89]. Typically, one considers the detected fragment spectra to be formed by the superposition of three Maxwellians that correspond to emission from three distinct thermal sources. These sources each have a characteristic velocity and temperature. The emission is assumed to be isotropic in the rest frame of a given source. The energy spectrum is a Maxwellian; the peak corresponds to the Coulomb repulsion between the source and the emitted fragment, and the slope parameter is supposed to correspond to the temperature of the source. The three sources correspond to the projectile remnant (velocity approximately Vbeam, low tem- perature), the interaction region (velocity approximately wam/Z, high temperature), Fig- 12 ‘ Heavy-Ion Induced Reactions Before: 'ROJECTILE TARGET = IMPACT PARAM. -_— _ _ ——— - ////\ PR ‘HOT‘ ‘ INTERACTION [w Vpaos "COLD REG [ON/ i:\\\i > $3315; \///////v I/ZVPR _ % gure 11.1: A schematic diagram of the the reaction 14N +98 Ag assuming the fireball ll. be dis 13 and the target remnant (almost at rest in the laboratory frame, low temperature). The slope parameters of the intermediate velocity source which are extracted from experimental data using these moving source fits are considered to be the tempera— tures of the thermalized regions. The slope parameters scale with incident energy. Therefore this observable seems to have the basic features that one would expect of a nuclear temperature. 3 Gamma Emitting States Basic Theory: An alternate method of measuring the nuclear temperature was proposed by Morrissey [Morr84]. This method assumes that an equilibration has been established by the time of fragment production. Therefore, the excited state distributions of fragments produced from a thermal source should be representative of the temperature of the source. In order to extract an estimate of the nuclear temperature, one may measure the population distribution of a set of excited states for a given fragment type. The simplest cases are those for which there are only a small number of low lying states. fone assumes that all of the fragments that are emitted from the thermal source re either in the ground state or in one of these low lying gamma emitting states, hen a measure of the total number of fragments of a given type will yield the total population of all states, while a measure of the gamma rays detected in coincidence With these fragments will indicate the populations of various excited states of the nucleus. One corrects the raw measures of the gamma populations for the absolute efficiency of the detection system. For a nucleus with a single low—lying gamma state Li7 or Be7 for example), one can use the Boltzmann two level formula to extract the emperature. The ratio between the population of the excited state and the ground b. 60 IL. state is given by: nez. : 2.25:. + 1 e—AE/Tgf, (11.1) 7119.5. 2.79.3. + 1 where n“, is the population of the excited state, 11.9... is the population of the ground state, j”. and jg,,_ are the spins of the two states, AE is the relative energy between the states, and T,” is the effective temperature. Temperature Measurements: A series of experiments was performed at the N.S.C.L. at MSU by Morrissey et a1. [Morr84, Morr85, Morr86, Bloc86]. These ex- periments employed silicon telescopes to detect isotopes of lithium and beryllium. Gamma rays were detected with an array of NaI and bismuth-germanate scintilla- tors. Gamma events were only recorded when in coincidence with a detected charged fragment. The temperatures that were extracted from these studies are displayed in Figure II.2. All of these temperatures are less than 1 MeV. For comparison, the slope parameters of the kinetic energy distributions for the lithium and beryllium frag- ments yielded temperature estimates of 15 MeV. There is clearly a large discrepancy between these two methods of estimating the nuclear temperature. This method of determining temperatures was checked at low bombarding ener- gies [Morr86, Lee90a, LeeQOb] where nuclear reactions are known to proceed through compound nucleus formation. The compound system is assumed to be long—lived and to reach a state of equilibrium prior to de-excitation through fragment evaporation. Morrissey et a1. studied the system 14N +12 C for a range of incident energies from 87.5 to 350 MeV [Morr86]. The temperatures extracted for bombarding energies below 112 MeV using the populations of gamma emitting states agreed with the ex- pected compound nucleus temperature. Above 112 MeV, the temperatures remained constant at 0.5 MeV. Lee et a1. studied “Ar -l—12 C at 8, 10, and 12 MeV/nucleon [Lee90a, Lee90b]. Their results showed that the populations of excited states reflected Fi 15 2.0 to 0 Ave. all angles 2‘35 A New measurement 0 Old measurement Ag + MN E/A=35 MeV 7Be 6Li 8Li 0.43 3.65 0.98 |.O- 7Li _ Teff 0.48 A (MeV) . A + + 4 4 Ave. + ‘i A. 50° 70° 90“ m V0. 70° 90°. 70° 90° 70° 90° 0 o a. 50° 70° 00 50 70 90 Figure II.2: The effective temperature extracted from a study of the populations of gamma emitting states [Morr84]. 16 the temperature expected from the compound nucleus and were consistent with the temperature determined from the slope parameter of the kinetic energy distributions. Corrections for Sequential Feeding: At intermediate energies, the tempera- tures that were extracted using the populations of the gamma—emitting states were completely different from those extracted from the slope parameters of the kinetic energy spectra [Morr84, Morr85, Bloc86]. This discrepancy was very troubling and several explanations were proposed [Boa184, Hahn87, Frie88]. The most widely ac— cepted of these explanations was the suggestion that population distributions that were measured corresponded to final state distributions and not to the distributions that had been established at the time of nuclear freeze—out [Morr84, PochSSa]. The primordial distribution that had been established at the time of freeze-out contained highly excited fragments. For high temperature systems, many of these fragments would be in particle unstable states which would decay prior to detection. It is these decays of the highly excited resonances that distort the estimates of the nuclear ltemperature. When these resonances break apart, they preferentially feed the ground states of the lighter nuclei [Hahn87]. This causes the experimenter to overestimate the ground state population with respect to that of the gamma emitting excited states, and thus yields a low estimate for the temperature. Theoretical calculations for the effect that this sequential feeding would have upon the temperature measurements were performed using a quantum statistical model [St6c83, Hahn87]. Figure II.3 displays the magnitude of this effect on the excited state ratios that had been used for the extraction of temperature. It is evident that for temperatures above 2 MeV the ratios of these gamma emitting states will not yield accurate measurements of the temperature. Hahn and Stocker’s reanalysis of the earlier data yielded temperature estimates from 4 to 8 MeV. This temperature range corresponds to the range of val- 17 1.75 N , °r.1(2.2 MeV) 1.50 ~ 7 -- °n(g.s.) _ Li(4.63 MeV) 1.25 - 7Li(g-s-) _— - 1.00 ' oy’om'm'» ”it ””9633 in» - m Ignaz, mom»! . 0.50 ~ _, 5,- , """" . “ ....... —Boltzrnum 025 llflmmi- """" P ‘ “P- _ ;, -------------------- -‘--p=0-5p. ooo . "'VP'MP' 7 . - a . .- mgoyn MeV) nagosa MeV) Li(g.s.) Li(g.s.) 0.0 . . 0 10 0 10 20 TEMPERATURE (MeV) igure II. 3. The expected ratios between states as a function of temperature. In this Odel, the initial population of a state is given by its chemical potential, statistical eight, and the temperature. The calculation includes 40 stable and 500 unstable nu~ lear levels (up to mass 20). The excited states that had been populated in the initial hase then decay following the known branching ratios. The solid line corresponds ratios predicted using the simple two level formula. The dotted, dashed, and dot- ash lines include sequential decay effects for three possible freeze- out densities. The ashed region indicates the experimentally observed ratios [Hahn87]. 18 res for which the prediction assuming a 0.1 pg freeze-out density (dotted line) agrees vith the experimental measurement (hashed region). A study by Xu et al. increased the total number of gamma states used for the :stimate of nuclear temperature [Xu86, Xu89]. The results were corrected for the feeding using the quantum statistical model of Stocker. The effective temperature estimated from this study was 3—4 MeV, which was still well below the 15 MeV that was measured for the slope parameter of the kinetic energy distributions. 4 Two-particle Correlations Review A partial solution to the problem of feeding from higher states could be found by measuring the populations of the particle unstable states. These states are less sus- ceptible to distortions because they are less likely to be populated by decays from heavier fragments. As a system decays, excitation energy is converted into kinetic energy and into mass (lighter fragments are less well bound). Thus a decay cascade iwill preferentially feed the lowest states of the daughter nuclei. The unstable state J{populations were studied through the detection of both decay particles at small rela— ive momentum. This also had the added advantage that it allowed the experimenter 0 study the size of the emitting source. he Nuclear Temperature easurements of the temperature were carried out using the populations of particle nstable states by several experimenters [Poch85a, Chit86, Chen87d, Bloc87, Ga1087, 0x88, Sain88, Naya89]. These experiments studied asymmetric heavy—ion induced teactions and covered a range of beam energies from 35 to 94 MeV/nucleon. The tarliest studies focused on the states accessible through detection of light charged )articles (p, d, t, a) and found temperatures of around 5 MeV for the systems 35 19 MeV/nucleon N—l—Ag [Chit86, Chen87a, Chen87c, Chen87d, Fox88], 60 MeV/nucleon Ar+Au [Poch85a, Poch85c, Sain88], and 94 MeV/nucleon O+Au [Chen87d]. This work was complemented by a series of experiments using a technique for studying neutron emitting excited states [Deak87]. These experiments measured temperatures of 3 MeV for the 35 MeV/nucleon N+Ag system [Bloc87, Bloc88], and 1 MeV for the N+Ho system [Kiss87, Galo87]. Nayak et al. extended these to charged—particle studies to heavier fragments. However, the extracted temperatures remained in the 4 to 5 MeV range [Naya89]. Attempts to understand the extracted temperatures had little success. Chen et al. found no dependence of the temperature on the impact parameter as measured by the fission fragment folding angle [Chen87a]. Fox et al. found no dependence on the emission angle [Fox88]. Saint-Laurent et al. found no dependence on either the charged-particle multiplicity or the energy of the detected fragments [Sain88]. However, Deak et al. did find that the population distributions did depend on the energy of the detected particles [Deak89]. In their study, the ratio of fragments emitted in the excited state dropped as a function of the total energy of the measured particles, which indicates a lower temperature. With the exception of the last result, the measured temperatures seemed to be a constant of nature. The Sizes of Nuclear Systems Hanbury-Brown and Twiss introduced a method for measuring the radii of distant stars through the use of photon-photon correlations [Hanb54]. This correlation tech- nique was adapted to the physics of particles by Goldhaber [Gold60]. However, the method of Goldhaber had to be extended to the case of multiple particle emission, which is common in heavy-ion reactions. Kopylov and Podgoretskii proposed a def— inition for the correlation function that resolved the problems of multiple particle 20 emission [Kopy74]: C(p1,p2) : ———'—/__* (H'Z) where p1 and p2 were the momenta of the two detected particles, (7 was the total production cross-section for that type of particle, 360/6p18p2 is the double differen— tial cross-section for the production of two particles with momenta p1 and p2, and Baa/apl and 830/8p2 are the inclusive differential cross-sections of the production of a particle with momentum p1 or p2 respectively. This correlation function is then compared to predictions which assume emission from a source of a given size. Several studies have been performed to measure the source radius and to determine on what it depends [Chit85, Poch86a, Kyan86, Poch87, Chen87a, Fox88, Sain88, DeYo89]. Many of these studies have also been used to address the question of nuclear temperature. Therefore, the systematic studies concerning source size focus on the same variables as the studies of temperature. The extracted radii have been explored as a function of energy of the detected particle pair [Poch86a, Chen87, Poch87], as a lfunction of detection angle [Fox88], and as a function of associated charged particle multiplicity [Kya1186]. The source radii were found to decrease with total energy of the detected particle pair. This was presented as evidence that the collision initially formed a small hot spot. The first particles that were emitted from the collision would have come from this small region of space. As the system equilibrated and the remaining portion of the system was thermalized, the temperature of the source region dropped. Therefore, one expected the low energy region of the distribution to be dominated by the final stage of the reaction (a large, equilibrated source), while the high energy tails should corne primarily from the initial stage (a small, hot spot) [Poch86a, Chen87, Poch87]. The detection angle was expected to influence whether the detected particle spectra were dominated by decays of the projectile, of he target, or of the interaction zone. Within the uncertainties of the measurements, 21 evidence was found to suggest that the extracted radii depended on the detection ;le [Fox88]. The associated charged particle multiplicity is considered to be a sure of the impact parameter of an event. It was expected that the most central its should display the largest source radii, because these events created the largest raction regions. This expectation was confirmed by Kyanowski et al. [Kyan86]. Experimental Goals possible that the dynamics of the heavy~ion induced reaction had clouded the ure. The spectator regions of the target and the projectile should contribute to observed fragments (refer to Figure 11.1). The remnants should be cold, possibly ing a depressed estimate of the temperature of the interaction region. In ad- jbn, the impact parameter could not be measured, therefore the relative sizes of target remnant, interaction region, and projectile remnant were not well defined. refore an experiment was proposed to study proton induced reactions, because reaction dynamics are less complicated. Figure 11.4 displays a schematic diagram proton induced reaction. The systems chosen for study were 500 MeV p+Ag Be. The p+Ag study complemented the 35 MeV/nucleon N+Ag studies that been performed at MSU. The total energy of the nitrogen projectile (490 MeV) almost equal to that of the proton (500 MeV), and the target nucleus was the . Therefore, the target can potentially receive the same input excitation energy, ver, the manner in which this energy is delivered to the system is different. The ion dynamics should be less complicated in the proton induced case, because the ctile remnant (proton) would not be an additional source of energetic fragments. ntermediate velocity source would essentially be contiguous with the target rem- The proton is expected to undergo several nucleon-nucleon interactions as it through the target nucleus. This will create a thermalized zone within the 22 Proton Induced Reactions Before: —_——-—- During: After: 3 11.4: A schematic diagram of the reaction p+98 Ag. The hashed area represents ermalized region. 23 'get as opposed to an independent thermalized source which has been proposed for : heavy-ion induced reactions. It was felt that the p+Ag study would provide a s complicated system, which would help resolve the questions that had been raised the N+Ag studies. The p+Be system would be even less complicated than the p+Ag system. As .11 the p+Ag system, the projectile would not complicate the emission spectra. For 5 system, however, the target remnant would be even simpler. One would not Ject to have a large bulk of spectator matter in a system that contained only 10 :leons. The small size of the system also reduces the effect that feeding from higher tes will have upon the final fragment distributions. Additionally, the assumption :quilibrium seems absurd for a system with so few nucleons. One does not expect ltiple interactions as the incident proton passes through the bulk of the target terial, instead only a single collision is likely. Description of the Experiment 3 experiment was performed in the Proton Hall of the Tri-University Meson Fa- .y (TRIUMF), which is located in Vancouver, British Columbia. This laboratory tains a large room temperature cyclotron that accelerates H“ beams up to 500 V. The beam can be stripped within the cyclotron to create a proton beam, which ctracted and focused down the beamlines in the Proton Hall. Alternatively, the n can be stopped in a production target in order to create secondary meson beams , which are focused to experimental areas in the Meson Hall. This latter use of beam is the primary function of the facility; the experimental proton program parasitically off of the meson program. The detection apparatus was set up in the Simon Frasier University (SFU) 60” 24 tering chamber. This chamber is located close to the cyclotron, and the vault walls een the accelerator and the experimental vault are thin by TRIUMF standards. thin vault walls resulted in a high radiation background which did have an effect he performance of the detection apparatus in this experiment. Beams of 504, 300 200 MeV protons were extracted. These beams were incident on targets of Ag, CH2, and CD2. The 300 and 200 MeV beams and the CH2 and CD2 targets were exclusively for detector calibration. he detector apparatus consisted of a close-packed array of phoswich detectors h was placed immediately behind a multi-wire proportional chamber (MWPC). phoswichs provided particle identification and energy determination for light ged fragments while the MWPC provided information on the emission angles. Lils concerning the dimensions, performance, and calibration of these detectors vswichs and MWPC) are given in the appendices. The central angle of the ar— was chosen to be 60". A diagram of the experimental configuration is given in re II.5 Extraction of Source Sizes Basic Method :very pair of light particles (p, d, t, 4He), a correlation function was generated. correlation function was constructed by dividing the relative momentum spec- for a given pair of light particles by a spectrum generated from random particle . These random events were created by analyzing particles from different real .5. This method of generating the divisor for the correlation function calculation )erior to using the measured singles distributions (equation 11.2) because it prop- ncludes the biases implicit in the requirement that two particles are detected in 25 Beam Axis Array of I6 fast/slow telescopes MWPC 5: The diagram of the experimental setup for the study of two-particle us from p+Ag and p+Be. 26 :idence in the detector array. For this analysis, 40 random pairs were generated ach real event. This minimized the contribution to the statistical error related :e random spectrum. 'he extraction of source size was done by comparing the experimental correlation ions to predictions which were generated by a model that considered indepen- emission from a source region that is parameterized as a Gaussian in space .ime. The source acts as an ensemble of independent emitters rather than as a 'ing excited state [Boa186]. This model calculated the partial waves using both )mb and nuclear potentials. The nuclear phase shift data for a given particle pair .lyzed using a Woods-Saxon potential. The charge is assumed to be distributed 7me within a sphere of a given radius. In this manner, expected correlation ons could be calculated for an arbitrary source radius. These predictions were ired to the experimental measurements of the correlation function. The esti- l source radius was taken as the calculated correlation function that reproduced :perimental data. lource Sizes from p+Ag n-Proton: The pair that was detected most frequently was proton-proton, >re it is appropriate that this be the first correlation function discussed. Fig- 6 displays the correlation function (R(AP) + 1) for the p—p pair from the 500 +Ag system. In this analysis, a value of the correlation function equal to unity vonds to uncorrelated emission. Values below unity represent anti—correlations, alues above unity are enhancements. One expects that the correlation function drop to zero at small relative momentum, because two charged particles emit- ;e together in phase space will exhibit a strong repulsion due to the Coulomb :ion. The data presented in this work unfortunately demonstrate a sharp peak 27 500 MeV p + Ag 5 ‘li I r 1 l I I I I l I I I T l f I I I 4 .I ., ) ...... =I _I I I .: ‘1 (I " [I — I I I ,I _ ‘I I, -I l ,’ . . —I I’ - J . ‘~. 1 I I ‘ I 4L l 1 I I l l I I I ml 1 m 1 I ‘1 O 2 O 6 8 O 40 Ap (MeV/c) ‘ The correlation function for proton-proton pairs from the 500 MeV p+Ag 28 relative momentum. This is due to a slight misalignment of the detector array. llowed single particles to scatter between neighboring phoswich detectors, thus 'ng a genuine coincidence. An effort has been made to estimate this contamina- lhe dotdash line on Figure 11.6). A Gaussian was fit to the contamination peak. bntroid was constrained to be at zero relative momentum and the width was ined by the precision of the detection apparatus. At large relative momentum, relation function is fixed at unity. A normalization constant is determined h pair to ensure that the correlation function approaches unity. There should :orrelated or anti—correlated emission at large relative momentum because the .ual fragments should be too widely separated in phase space to interact. At 20 MeV/c, there is a broad peak in the correlation function. This peak can ight of as an attractive s—wave nuclear interaction, or alternatively as the de- 2He. The dashed curve represents the Coulomb contribution to the expected tion function if one assumes emission from a source of radius 9.0 Fm [Boal86]. lid curve includes the nuclear interactions and corresponds to emission from a source, which was the radius for which the model best fits the experimental I-Deuteron: The proton-deuteron correlation does not contain any nuclear ces. There is only the Coulomb repulsion that results in an anti-correlation at :lative momentum. Figure 11.7 displays this correlation function. The dotdash [1 estimate of the contamination, and the solid curve is a prediction for emission source of radius 11.8 Fm. It has been suggested that the p-d correlation is best for extracting a reliable estimate of the source size [Poch86b]. This 136 the proton and the deuteron have different charge—to-mass ratios. This 11 each particle receiving a different acceleration in the Coulomb field of the 29 arget residue. The result of this distortion is an overestimate of the source I-Alpha: The ground state of 5Li is not stable with reSpect to proton emis- .erefore there is a strong nuclear resonance in the p-a correlation function. (11.8 displays this correlation function. The nuclear resonance appears as a tnt peak at Ap of 55 MeV/c. The contamination is estimated by the dot- ve, the Coulomb background is displayed for emission from a 9.0 Fm source line), and the predicted correlation based on a calculation that includes that interaction is shown by the solid line for emission from a 1.8 Fm source. No- the predicted curves fail to account for the experimental data in the region to 25 MeV/c. There is an excess in the experimental correlation function. :ak’ has been demonstrated to be associated with the partial detection of the >dy decay 9B ——+ p + a + a [Poch85b]. on-Deuteron: The deuteron—deuteron correlation function contains no ev— f nuclear resonances because excited states of 4He are stable with respect :hannel. Therefore there is only the contribution related to the Coulomb . of independently emitted deuterons. Figure 11.9 displays the measured In function, the estimate of the contamination (dot—dash line), and the pre- ssuming emission from a 9.4 Fm source. Chitwood et al. noted that the 11 function could not be associated with the decay of unstable resonances the extracted radius was substantially larger than that estimated for the Chit85]. I—Alpha: The deuteron-alpha correlation contains several peaks that are with particle unstable states of 6Li. Figure 11.10 displays this correlation 30 500 MeV p + Ag l T Ifi fi—l I I IiI T I ‘1 I I l I I I I I mi __ Ail _' ..Hfi_.I.II'|'1|_ ...... n... Elwilfl' The correlation function for proton—deuteron pairs from the 500 MeV 31 500 MeV p + Ag 1 1T1 I Fl I l I I l | I ITTfi l I p—Ot 5O '75 Ap (MeV/c) The correlation function for proton-alpha pairs from the 500 MeV p+Ag 32 500 MeV p + Ag ‘17 I f I I I W *1— I ‘l’ I I 1' T j d — d _‘ ///-E/nfi' ’""—'—"_'—_m—E / a a r _ m E! - III E11 ‘3 ‘iil_l l l I I l I l I l A I l_lm I l I 100 150 200 Ap (MeV/c) The correlation function for deuteron—deuteron pairs from the 500 MeV n. 33 Also displayed are an estimate of the contamination (dot-dash curve), a assuming only Coulomb interactions (dashed curve, 9.0 Fm source radius), :ediction including nuclear interactions (solid line, 2.0 Fm source size). The k at 42 MeV/c corresponds to the 2.186 MeV state of 8Li, while the broad nd 90 MeV/c corresponds to two unresolved states, one at 4.31 MeV and at 5.65 MeV. riton: The triton-triton pair has no measurable nuclear resonances. As Ind d-d, the correlation function (Figure 11.11) displays only the valley at re momentum which is due to the Coulomb interaction. The prediction is a 11 Fm source radius (dashed line). rce Sizes from p+Be : reaction should create a system whose dimensions are smaller than that the p+Ag reaction because there should be fewer nucleons involved. There- ource radii extracted from the p+Be data should be different from those Ibove. Figures 11.12 to 11.17 display the correlation functions that were ex— m the p+Be data for the same six particle pairs that have been discussed Ag system. The basic features are similar to the p+Ag case for all of the rs considered. For all six plots, the solid line corresponds to the calculation II from a source of a given radius. The radii for which the calculation best are: 3.3 Fm for the p-p case, 12.2 Fm for the p~d case, 1.6 Fm for the p-a for the d-d case, 1.9 Fm for the d-a case, and 10 Fm for the t-t case. —V-- .1,— 34 500 MeV p + Ag 171W flinfi 1% 171 l f Ifi' I d—OI lOO Ap (MeV/C) : The correlation function for deuteron-alpha pairs from the 500 MeV n. 35 500 MeV p + Ag IlllTIfill—l’llll’lllllrllilli t—t l ....... I l _ trait i II 1 . I: I L I _I_.A_ I J 1 I I l I I l I l I I l ) 25 5 J_lml | l I | O 75 100 125 150 Ap (MeV/c) The correlation function for triton-triton pairs from the 500 MeV p+Ag 36 500 MeV p + Be 0 I I I I l I I I I j I , I P‘P 40 Ap (MeV/c) 2: The correlation function for proton-proton pairs from the 500 MeV :rn. 37 500 MeV p + Be l l l l l I l l l I I l l l l l I l l l I I I 40 60 Ap (MeV/c) O (\3 O : The correlation function for proton-deuteron pairs from the 500 MeV 11. 38 500 MeV p + Be 5 l I | I | I I I l I I l l l l I I l I l I l l l I I p—O( 3“ I i l l l I | l I l l l l l l l l l I l I | I l l I l I l O 25 5O 75 100 125 150 Ap (MeV/c) : The correlation function for proton—alpha pairs from the 500 MeV p+Be 39 O 500 MeV p + Be I I | I I | l I I I I l I ' I 1 I I I | I l I 1 I —d 'liii 1 l I l l l l l l I I I l l I l l l l I l l l l l l I I O 25 5O '75 100 125 150 Ap (MeV/c) : The correlation function for deuteron-deuteron pairs from the 500 MeV n. 40 500 MeV p + Be O'Hfl‘fl'l'H'l'H'I'H'" 5 d—a A )_ _3 3:“ i .3 E- —i ..,.I,.L,lflllnnnL@..t.l,...1 25 100 125 150 .8 . 50 75 Ap (MeV/C) 3- The correlation function for deuteron-alpha pairs from the 500 MeV .<\D ~/f\l Fi SY 41 500 MeV p + Be I T r ‘T I I I I I I I I I l I I I I I | l I t—t t _ 1 LI I I I I I I l l I I I I I I I I 14L 1; I I I | I 25 5O '75 100 125 150 Ap (MeV/c) The correlation function for triton-triton pairs from the 500 MeV p+Be L— 42 he Effect of Energy Cuts on the Extracted Source Radii een proposed that by placing cuts on the total energy of the tw0 particles in ne can select upon the class of events [Poch86a, Chit86]. It is argued that the energy pairs should come predominantly from the hottest reactions. Evidence n presented that suggests that the source radius extracted from higher energy smaller than that extracted from low energy pairs [Poch86a, Chit86, Chen87]. 5 been put forth to suggest that thermal equilibrium has not been established reactions. correlation functions for the p+Ag system have been analyzed in this manner ffort to confirm these results. High and low total energy gates were set for Lil' type. Figure II.18 displays several different particle pairs each with this total energy gate. The data from the high energy gate are represented by monds on each plot, while the data for the low energy portion of the pair are represented by the crosses. There is no significant difference between the d low energy data for any pair except p-p and p-d. For the p-p pair, the ergy data (diamonds) show a stronger correlation than the low energy data ). Since p-p is a pair which contains a nuclear resonance, a large value of the ion function corresponds to a small source radius. For the p-d pair, the high ata show a stronger anti-correlation near zero relative momentum. Since p-d no nuclear resonances, the size of the source is related to the magnitude of -correlation. A larger valley corresponds to a smaller source. Thus both the the p—d pair suggest a smaller source size for the high energy selection. For e other pairs, the data for the high and low energies are indistinguishable. Ly suggest that protons and deuterons are emitted early during the course eaction. They may be emitted from the hot dense interaction region that 43 o 20 40 6O 0 20 40 so so 0 25 50 75100:.251591 ' 2.0 IIIIIIjIIIIIII—Ifi'III_ IrTI—IIIIIIrI—I—IIIIIIIIIEIIIIIIIIIIIIIIIIIIIIIIIII—IIII 1.5 $9 p—p {If de IL— 3 :: XX°¢ X E; 1.0 1%?” fig» ._ 2 0.5 : : 1 lIIIIIIIIIIIII IIIIIEIIIIIIIII IIIIIIIII :I 0,0EIIII IIIIIIIIIII IIIII IIIIIIIIIIE;IIIIIIrITI—WIIIIII 3L WED—(LR d: 2 _ 2‘ as 3% with 5 X5 @fi :3 %%W ° 'E 13¢ ‘:_ I:IIIIIIIIIIIIIIIIIIIIIIIIIIIIIZEIIIIIIIIIIIIJ_1I_LLIII OEIIIIIIIIIAIIIIIIIIIIIIIIIIIT;III IIIIIIIIIIII 20;— ‘ 15:— 10:— E4... " 0 so 100 150 200 O I 0 2040 60801000 50 100150 0 18: The correlation functions for a variety of particle pairs from the 500 MeV tern. The diamonds correspond to events selected on high total energy for articles of the pair, while the crosses are data selected on low total energy. 44 created at the start of the event. This may be considered as emission prior to ermalization of the entire system. The more complex particles, on the other hand, ow no influence of total energy on source size. This may indicate that they are all litted at the final stage of the reaction, after complete equilibration of the source. Summary of Extracted Source Sizes re source sizes that were extracted are summarized in Table 11.1. In addition, results im an experiment performed earlier with the same detection system are presented ' the 35 MeV/nucleon N+Ag system [Fox88]. The most striking feature of this data ; is the wide range of values extracted. The pairs that contain resonant states (p-p, 1, and d-a) all exhibit extremely small source radii for both the p+Ag and the -Be system. The pairs which contain no resonances (p-d, d-d, t-t) all yield large rimates for the source radius. These radii range from 9 to 12 Fm. This suggests that 3 pairs that contain nuclear resonances do not accurately measure the size of the :rmalized system. Instead these pairs are dominated by the decay of the emitted gments. The radius measured for these pairs is the radius of the parent fragment :1 not the radius of the system that emitted that fragment. A simple estimate of a clear radius in given by: R 21.12.41/3 (II.3) ere A is the mass of the nucleus. Using this equation, one notes that the nuclear .ius for SM is 1.9 Fm which compares remarkably well with the radius values racted from the p-a pair. The nuclear radius for GM is 2.0 Fm, which agrees with values extracted from the d-a pair. The p-p pair does not compare well with an mate for a mass 2 nucleus, but one does not expect 2H6 to be well—bound. The freeze-out density can be estimated from the measured source radius from the rs without resonances. These pairs are not influenced by the decays of particles Table 11.1: A summary of the source radii extracted from the 500 MeV p+Ag and p+Be systems. The results are compared to an earlier study of 35 MeV/nucleon N+Ag [Fox88]. 45 Pair 500 MeV p+Ag 500 meV p+Be 35 MeV/nucleon N+Ag p-p 4.0 d: 0.5 3.3 :I: 1.1 4.3-4.8 p-d 11.8 j: 1.6 12.2 3: 2.6 p-a 1.8 :I: 1.2 1.6 $1.8 d—d 9.4 :t 2.1 9 :l: 5 6.7-8.8 d—oz 2.0 :I: 1.1 1.9 d: 1.2 4.9—5.2 t-t 11 :I: 8 10 :i: 6 emitted in highly excited states and therefore give a truer estimate of the size of the nuclear source. For the case of 500 MeV p+Ag, the source radius, as estimated from the p-d, d-d, and t—t pairs, ranges from 9 to 12 Fm. The radius of a silver nucleus at rest is 5.2 Fm. If one assumes that the thermalized system contains the same number of nucleons as the target, one estimates the freeze-out density to range from 0.19 to 0.08 p0, where p0 is normal nuclear density. Of these three estimates of the source size, the values from the d-d correlation function is the most reliable. The p-d pair will lead to an overestimate of the source radius due to the unequal acceleration lof the two particles in the Coulomb field of the emitting source [Poch86b], and the uncertainty on the t-t measurement is extremely large due to the low counting rate. Therefore, the best estimate of the freeze out density can be attained using the radius value extracted from the d-d pair, which yields pfreeuout = 0.17:8:55p0. Extraction of Nuclear Temperatures 1 Basic Method he populations of particle unstable excited states are studied through the detection fthe two daughter fragments that are associated with the breakup of the resonance. 46 These nuclear resonance states can be observed as peaks in the relative energy spec- trum for a given pair of light particles. By measuring the counts above background in these peaks and correcting for detection efficiency one can determine the excited state populations and then extract the effective temperature. The various states that can be studied with this detector apparatus are listed in Table 11.2. For this analysis however, we will consider only pairs that contain multiple breakup channels for a given parent fragment. This allows one to study the extracted temperature between two states for which the detection biases are the same. To make comparisons between the populations of particle unstable states (detected with the array) and the ground state populations (measured using silicon telescopes) requires many corrections to ensure that no experimental biases affect the final result. Additionally, comparisons between particle-unstable states and stable states are complicated by differential feeding. Considering only the highly excited particle-unstable states facilitates the analysis and reduces potential systematic errors I because the same detection system with the same experimental biases has been used to measure all states. A measurement of the ground state would require only a single particle to be detected by the array. The requirement of only a single hit may sample a different class of events then the multiple hit requirement necessary to study unstable resonances. 2 Backgrounds for Independent Emission The relative momentum spectra contain events that are associated with the decay of unstable resonances as well as events that are random coincidences betWeen two particles which were independently emitted from the same thermalized system. The random background must be subtracted from the spectra in order to get estimates of the contribution associated with the decay of particle unstable excited states. An 47 Table 11.2: A list of all of the particle unstable states that could be studied with this detector system. Pair Parent Energy J1r A p Reference MeV MeV/c p—p 2H6 22 p—d 3H6 p—t 4He 20.1 0‘L 20.47 [Fiar73] p—t 4H6 21.1 0' 42.62 [Fiat73] p-t 4He 22.1 2- 56.69 [Fiar73] p—3He 413i 24.5 2- 81.04 [Fiar73] p—‘He 5Li g.s. 3/2‘ 54.18 [Ajze88] p—‘He 5L1 16.66 3 /2+ 166.6 [Ajzess] p—4He 5L1 18 (1 /2+) 172.6 [Ajze88] p—4He 5Li 20.0 (3/2, 5/2)+ 181 [Ajze88] d—d 4He 26.4 2“ 69.6 [Fiar73] d—t 5H6 16.76 3/2+ 11.58 ‘ [Ajze88] d—t 5He 19.8 (3/2, 5/2)+ 83.25 [Ajze88] d—3He 5Li 16.66 3/2+ 24 [Ajze88] d—4He 6Li 2.186 3+ 42.05 [Ajze88] d—‘He 6Li 4.31 2+ 83.99 [Ajze88] d—4He 6Li 5.65 1+ 98.43 [Ajze88] t—t 6H6 13.6 60.3 [Ajze88] t—3He BLi 21.0 2‘ 120.6 [Ajze88] t—3He 6L7 21.5 0— 126.2 [Ajze88] t—3He 6Li 25 4- 160 [Ajze88] t-3He “Li 26.6 3- 175 [Ajze88] t—4He 7Li 4.630 7/2- 83.06 [Ajze88] t—4He 7Li 6.68 5/2- 116.0 [Ajze88] t—‘He 7Li 7.456 5/2‘ 126.2 [Ajze88] t—4 He 7Li 9.67 7/2‘ 151.6 [Ajze88] 3He—3He 686 23 4- 179 [Ajze88] 3H6—3H6 6Be 26 2' 201 [AjzeB8] 3He—SHe 6Be 27 3‘ 208 [Ajze88] 3He—4H6 736 4.57 7/2‘ 97.6 [Ajze88] 3He—4He 7Be 6.73 5/2- 128.2 [Ajze88] 3He—4He 7Be 9.27 7/2- 157 [Ajze88] 3Hts—“He 736 9.9 3/2' 163 [Ajze88] 3He—4He 71% 11.01 3/2- 173 [Ajze88] 4Hes—“He 8Be g.s. 0+ 18.5 [Ajze88] 4He—“He 8Be 2.94 2Jr 106.3 [Ajze88] 4He*4He 8Be 11.4 4+ 207 [Ajzess] 48 estimate of this independent background was made using the Koonin model [Koon77] from which the Coulomb only correlation function predictions were generated. The source radius used for these estimates of the independent background was 9.0 Fm. This value came from the d-d correlation. There is some reason to believe that the p—d correlation may yield a slight overestimate of the source size [Poch86b], and the uncertainty on the t-t measurement was extremely large. The value extracted from the d-d correlation was therefore deemed to be the most reliable. 3 Efficiency Calculations The efficiency of the detector array varies as a function of the breakup channel and the relative momentum between the daughters. This efficiency function must be calculated before the yields of the particle unstable states can be determined. In general, the efficiency function is zero at small relative momentum, because in this case the two decay products will both be focused into the same phoswich element. The efficiency is also small for high relative momentum, because in this case the opening angle between the two fragments will be larger than the acceptance of the detector array. The efficiency of the detection system used in these experiments was optimized for relative momenta between 20 MeV/c and 100 MeV/c which corresponds to the locations of the most important resonances. In addition to the geometric efficiency, the detection efficiency must also be con- sidered. For the phoswich detectors this was not a problem. The efficiency of the MWPC, however, drops as a function of energy for p, d, and t. Figure 11.19 displays the detection efficiency of the MWPC as a function of particle type and energy. The overall efficiency functions were determined using a Monte Carlo program. The inputs to the program included the detector geometry, the detection efficiency of the MWPC, and the kinetic energy spectrum of the parent particle. The program 49 35 MeV/nucleon N+Ag 10-‘ ' ' ‘ ' '..'..'. fifrrfr'irWWllflTlH _ . “"6me ‘r_ _. He __ 0.8- —~ C t _ 50.6— a — fl — _. .9. _ _. .9. _ _ 530.4? 13 _ O.2—- — r : 0.0—1 1 l l l 1 1 m1 I 1_1 1 1_I 1L1 1— O 50 100 150 200 Energy (MeV) Figure 11.19: The detection efficiency of the MWPC assumed that the excited state distribution was independent of the kinetic energy of the emitted fragment. A simulation was run in which parent particles were directed at random locations on the array. The fragments decayed in flight with a given separation energy. An evaluation was then made to determine whether both of the daughter particles would have been detected and properly identified. Events were considered failures if: either of the two particles failed to hit an element of the array, both daughters hit the same phoswich element, the MWPC efficiency excluded either fragments, or a fragment hit a edge between two detectors. The efficiency was the ratio ofsuccessful detections to total events. Figure 11.20 displays the overall efficiency function for the p-t, d-t, d-a, and the t—oz pairs. 4 Populations of Particle Unstable States An unstable nuclear resonance will decay through particle emission. These will cause emission lines with relatively discrete values in the relative momentum spectra. The width of these peaks are broadened by the experimental detection resolution. The resolution is affected by uncertainties in both energy and emission angle which causes 50 10 III I IIIIIIIIII IIIIIIIII E —t 1 d—t i 0.8 l" p ‘27 —_ t i: : — —— a 06 r“ “:3‘ ‘3 >‘ Z I I 0 0.4 r —__— - 5 E E E g 0.2 _ —:— t :1 I I l I I::I I l I I I I II I I I ILLLII m 00 El | I I I l I | I I I l I I I I I IEEI I I I I I II II I I I I II E E 0.8 9 dfl—f‘r VOL? 2;; : 2 "Q"; 0.6 E— 4i— —: D: 7 I: I 0.4 E :— __ : I: 1 0.2 _— ‘1.— d O O _ 1 I 1 l 1 1 1 1 l 1L] 1 I141_1 : 1 1 1 l m1 11_I_11_1_1 I 111 1 0 50 100 150 0 50 100 150 200 Ap (MeV/c) Figure 11.20: The relative detection efficiency for four different particle pairs. The peak efficiency has been normalized to unity for each pair. 51 the peak width to increase. An estimate of the background independent emission is subtracted from the measured relative momentum spectrum. Peaks are fitted to the remaining resonant decay contribution, and a correction is made for detector efficiency as a function of relative momentum. One then has an estimate of the relative population for various particle unstable excited states from which an effective temperature can be extracted using equation 11.1. Proton-Triton: The method for determining the relative populations of states is illustrated in Figure II.21. The top portion of the figure displays the relative momen— tum spectrum for p-t pairs. The estimated independent contribution is also displayed. The portion above this background can be associated with the contribution from the decay of unstable states of 4H6. The lower portion of the figure displays only the counts above background. Three gaussians are fit to the distribution. These three gaussians represent the first three excited states of 4He: the 20.1 MeV state, the 21.1 MeV state, and the 22.1 MeV state. The centroids of the gaussians were constrained by the energy of the states. The widths were determined by the detector resolution and in a few cases by the known width of the state (only for a few very broad states). The fitting routine varied only the relative heights of the various peaks. The dotted curves in the lower part of the figure display the contributions from the individual gaussians, while the solid curve is the sum of all three. For this pair, the extracted temperature is 5.6 i 1 MeV. The uncertainty is estimated from the fitting routine and does not consider possible systematic errors. Deuteron-Triton: The d-t pair contains peaks associated with the decays of 5H6 from the 16.76 and 19.8 MeV states. The upper portion of Figure 11.22 displays the relative momentum spectrum and an estimate of the independent background. 52 5000 4000 3000 2000 1000 IIIIIIIIIIIIIIIWI‘I‘I‘FITI— 1 \ \ \ \ \ 1500 Counts 1000 500 IIIIIITIIITIIIIIII .75 100 f§5 Ap (MeV/c) O (\3 ._ U‘. 01 C Figure 11.21: The relative momentum spectrum for proton-triton pairs (squares) and an estimate of the independent background (dashed line). The lower portion displays the counts above background (squares) and fits to the individual peaks (dotted and solid lines). 53 The lower portion of the figure displays the counts above background and fits to the excited states of 5H6. The temperature extracted from the ratio of these two peaks is 4.8 i 1 MeV. Deuteron-Alpha: The d-a spectrum contains peaks associated with the 2.186, 4.31, and 5.65 MeV states of 6L1. The lowest peak appears well above the background in the upper portion of Figure 11.23. The two higher energy states are broad and are not resolved as individual peaks with this detection system. The temperature extracted from the ratio of these states is 3.5 d: 1 MeV. Triton-Alpha: There are four states which contribute to the t-a relative momen- tum spectrum. The 4.63, 6.68 (which is quite broad), 7.46, and 9.67 MeV states. The analysis of this state is further complicated because there are relatively poor statistics for this particle pair as can be seen in Figure 11.24.- The relative heights of the highest three peaks are not well defined. The extracted temperature is 3 :l: 2 MeV. 5 The Nuclear Temperature A summary of the temperatures extracted in this study is contained in Table 11.3. The errors quoted correspond to the uncertainty in fitting the populations of the indi- vidual states above the estimated background. Therefore, it is essentially a statistical error and should not be considered an estimate of the potential systematic error. This analysis is susceptible to substantial systematic errors. The main effect that would introduce a systematic error is the manner is which the independent background is estimated. The background affects the relative heights of the peaks and therefore affects the resulting temperatures. A smaller source size would have resulted in a dif- ferent background and, therefore, a lower temperature. Additionally, had the nuclear 600 500 400 300 200 100 150 Counts 100 50 54 flIIII IIIIIIIIIIIIfiIfIIIIE :- q E1: d—t “i :— f CMWEDDE —: : / ‘ CI 2 ;' E1,” \~ ‘7 §_ [ch/ DEFUQJZEFQJ _E a Cljj 55%] E707/1l1111l1111l111al1WNW: :TIIIII I I I I : : i 7 7. L- .. a Z G a C1 : EFF 1lflF1iq1Qn flan-1D? 100 125 75 Ap (MeV/c) Figure 11.22: The relative momentum spectrum for deuteron-triton pairs (squares) and an estimate of the independent background (dashed line). The lower portion (115- plays the counts above background (squares) and fits to the individual peaks (dotted and solid lines). 55 1250 i— I I I I I I I quj I ff fl I Til fl I j—I r I I T 3 1000 I'— D a C1-O( 7 750 :1;— 13 DD —;l 500 E— a in a (\1 01 o g: iii IIILLIIIIIIIIIIIIIIIBIII "LL11 Figure 11.23: The relative momentum spectrum for deuteron-oz pairs (squares) and an estimate of the independent background (dashed line). The lower portion displays the counts above background (squares) and fits to the individual peaks (dotted and solid lines). 56 1 _1 m ': 1" 1 - E - l l l _ 60 r- _ :3 ” I O C :1 _ o r _ 40:— '1 t : 20 :— __ '1‘ a 1 5111 m TIPCIQI‘E-Ij "' ll|l 0 50 . 100 150 Figure 1124: The relative momentum spectrum for triton-a pairs (squares) and an estimate of the independent background (dashed line). The lower portion displays the counts above background (squares) and fits to the individual peaks (solid lines). 57 Table 11.3: The temperatures extracted from this study. System Pair Temperature (MeV) 500 MeV p+Ag p~t 5.6 :l: 1.0 500 MeV p+Ag d—t 4.8 d: 1.0 500 MeV p+Ag d—a 3.5 :I: 1.0 500 MeV p+Ag t—a 3 :t 2 500 MeV p+Be d—a < 4 contribution to the independent background been included, a different temperature would have been extracted. Errors are also introduced by the estimates of the branch- ing ratios for given excited states. Several states are open to neutron emission as well as particle emission, and an accurate estimate of the branching ratios must be used. The average temperature extracted in this study is about 4 MeV. This value can be compared to the slope parameters of the kinetic energy spectra. These spectra were measured in an earlier experiment by Green and Korteling [Gree80]. The values derived from that study were from 15 to 20 MeV depending on the specific fragment species. As with the heavy-ion induced reactions, the temperature extracted from the excited state populations was substantially lower than that'suggested by the kinetic energy spectra. E Conclusions It is evident from the data presented in this chapter that the determination of the dimensions of the source of energetic fragments is not a straight-forward task. The effective source for fragments of a given type is not the same as that for a differing pair. One might expect the lighter pairs to come from predominately smaller sources, because fewer nucleons are required to form such pairs. This is not observed, instead the evidence from the 500 MeV p+Ag study suggests that only states that do not 58 contain nuclear resonances can be used for estimates of the size of the thermalized source. The pairs that contain resonances are dominated by the decays of these fragments which masks the contribution from the independent emission from the thermalized source. From pairs with nuclear resonances one learns mostly about the size of the parent nuclear fragment and not about the thermalized source. The pairs without resonances all yield high values for the source radius. This is taken to be an unbiased estimate of the size of the source. It has been tacitly assumed in the introduction to this work that the nuclear freeze-out took place at a specific nuclear density. One of the goals of this work was to measure the source size, from which the density could then be estimated. The data presented in this thesis suggest a freeze-out density of 0.171333%, which is derived from the d—d correlation function. A major weaknesses of this method is that it assumes simultaneous emission from a static surface. However, the system is expanding prior to fragment formation at freeze-out. If the system had just enough energy to reach freeze-out, then the fragments would be emitted from a region that had reached the limits of its expansion. However, if the system had excess energy, it could still be expanding radially at the time of fragment formation. This would distort the perceived source radius, because the emitted fragments should also contain a. radial momentum component. The temporal extent of the source can not be separated from the spatial extent using this method of analysis. Thus, the extracted source radius will be an overestimate for any system with a non-negligible life time. As it turns out, the data can also be understood in terms of a specific temperature for freeze-out. This casts further doubt on the concept of a well-defined freeze-out density. The extracted temperature from the 500 MeV p+Ag study is 4 :I:1 MeV. This value agrees with the temperature estimates from several heavy—ion induced reactions [Poc1185c, Chen87, Bloc87, Fox88]. As with the heavy-ion studies, the tem- 59 perature extracted from the distributions of excited states is not in agreement with that suggested by the slope parameters of the fragment kinetic energy spectra. The excited state populations are providing information about the final stages of the reac- tion (fragmentation and final-state interactions), while the fragment spectra seem to be established at an earlier stage. The temperatures extracted using kinetic energy spectra are dominated by the data that make up the high energy tail. Thus, the portion of the spectrum that contains the fewest counts will have the greatest effect on the extracted slope parameter. These high energy particles may be emitted at the earliest/hottest stages of the reaction. The temperature measured using the popula— tions of states weights particles of all energies equally, and is therefore dominated by the data from the low end of the kinetic energy spectrum, which contains the largest number of counts. Thus the temperature extracted using the populations of states is dominated by the stage of the reaction in which the largest number of fragments is formed. Additionally, if the freeze-out occurs While the system is still undergoing an expansion, the energy spectra would not agree with the chemical temperatures. The thermal models assume emission is from an equilibrated source. A comparison of this work to the heavy-ion induced studies suggests that the major features previously observed are still present and are therefore not artifacts of the reaction dynamics. The source size analysis is clearer in the p+Ag study suggesting that the thermalized region which is formed is more well defined than that created during a heavy-ion induced reaction. The temperature estimates agree with the earlier studies. Chapter III The Search for Mult i-Fragmentation A Introduction ~ Multi—fragmentation is the simultaneous disassembly of a hot nuclear system into several intermediate mass fragments (IMFs). It is a disassembly mechanism that has been proposed to occur when the excitation energy of a nuclear system is sufficient to allow the system to expand into a region of mechanical instability. Theoretical studies have suggested that at excitation energies around 4 MeV/nucleon this mech- anism should become the dominant decay process [Bond85a, Bond85b, Gr0586]. At lower excitation energies, the de-excitation of a nucleus is accurately described by the Weisskopf model [Blat79]. This model considers the formation of a compound nucleus and its subsequent statistical decay through emission of protons, neutrons, and alpha particles. For heavy nuclei (A2100) a binary fission is also a possible de-excitation mechanism. The multi-fragmentation models propose that for nuclei with large amounts of excitation energy, the slow disassembly process depicted by the Weisskopf model is insufficient to cool the rapidly expanding system. The fluctu- ations in temperature and density grow fast enough (% 10 Fm/c) to tear the nucleus apart. (Refer to Figure 1.2 for a schematic diagram of the a reaction and subsequent 60 61 decay of the highly excited nuclear system.) Cracks are formed as nuclear bonds are broken within the nucleus. These cracks expand to form bubbles. At this point, the nucleus contains regions of high and low density. The high density regions condense to form droplets which mutually repel one another clue to the Coulomb force. This proposed disassembly process represents a completely different sort of de—excitation than the sequential emission that is observed at lower energies. For this work, multi-fragmentation shall refer exclusively to the simultaneous dis— assembly of a thermalized nuclear system into several intermediate mass fragments (IMFs, defined as fragments with A 2 4). Multi—fragment emission, on the other hand, will be used to indicate an observed emission of multiple IMF s without refer- ence to the mechanism of decay. Observation of multi-fragment emission therefore is not sufficient evidence to conclude a multi-fragmentation process is present. Specifi- cally, modifications of the Weisskopf model have extended the theory to allow decay by emission of all species of particles. This change allows a sequential binary decay model to produce events with multiple IMFs. A signature for identification of multi- fragmentation will require sensitivity to the time scale of the disassembly process. There is a great deal of interest in the multi-fragmentation process because it is a rapid disassembly process and can thus yield information about the early stages of nuclear reactions. Identification of its existence and determination of its charac- teristics will allow one to study features of the equation of state of nuclear matter. The maximum amount of internal excitation energy that a nucleus can support is indicated by the onset energy of the multi-fragmentation decay process. This onset energy will vary as a function of the compressibility of nuclear matter (76), because the growth rate of the density fluctuations depends upon 16. The onset will occur at a lower energy for a stiff EOS (K. = 380 MeV) than for a soft EOS (K. = 200 MeV) [Sura89] . 62 The first experimental evidence for the existence of a multi-fragmentation decay mechanism was presented by a group from Purdue. The data were IMF production cross-sections from the reaction p + Xe at 80 to 350 GeV. The data that were pre- sented are displayed in Figure 111.1 IFinn82]. The mass yields are fit by a power law, do/dA o< 111—2.64. A multi-fragmentation process was suggested, even though the results were for single particle inclusive data, because the simple models available at the time could not account for the observed shape of the mass yield spectrum. The Weisskopf model allowed for decay through emission of protons, neutrons, and alphas, but not IMFs. Total disassociation of the constituent nucleons would have left only free neutrons and protons. It was suggested that a nucleon vapor had been formed. This vapor then coalesced into droplets or IMFs. The Fisher droplet model [Fish67] for a vapor predicts a mass distribution of the form do/dA (X A" where -r is the critical exponent. 7' values between 2.0 and 3.0 are consistent the concept of nuclear vapor formation and subsequent particle condensation. This droplet model is a 3-dimensional percolation model that was developed as a simple way to understand vapor formation. The model considers interactions only between particles (or lattice sites) that are contained within the same cluster or droplet. Interactions between droplets are assumed to be negligible. The critical point is marked by the appearance of large clusters and is manifested as a critical opalescence in macroscopic systems. The phase transitions that are produced by the Fisher droplet model are instructive for the study of nuclear phase transitions, however, since the interactions between particles in nuclear matter are not the same as the simple interactions used in per- colation models, there is no guarantee that the phase transitions will have the same cluster properties [Stau79]. These data prompted many theoretical and experimental studies focusing on the multi-fragmentation process and phase-transitions in nuclear matter. 63 IOB : I 1 r I I I : : P + Xe - Af + X :1 107 E = : -as4 : a _ _ 11.1 5 _ _ >‘- '0 E E '05 E E '04 1 l 1 4_ 1 ° 0 5 IO 15 20 25 30 35 Figure 111.1: The measured mass yields from the reaction p+Xe. The energies of the proton beam varied from 80 to 350 GeV. The exponent of the solid curve is fit to the data. This analysis was inspired by the Fisher droplet model which predicts dU/dA o< A" [Finn82]. 64 There are three main types of models that have been employed to study multi- fragmentation. The categories are Statistical, Transition-State, and Geometrical. The different approaches yield similar results and are capable of reproducing the basic multi-fragment emission observables. In general, the models described below predict that multi-fragmentation becomes the dominant decay mode for thermalized systems with greater then 4 MeV/ nucleon excitation energy. B Theoretical Studies 1 Statistical Multi—fragmentation The statistical models for multi-fragmentation have three major assumptions. First,’ the different phase space states are all given the same weight. Second, one character- izes a given configuration with a total weight which is the summation of the weights of all of the phase space states that lead to that configuration. Lastly, it is assumed that the system is in thermodynamic equilibrium at the time of the breakup. Within the statistical studies, there is a further grouping based upon the type of ensembles used to evaluate the results. The simplest approach is a grand canonical ensemble which allows both particle number and energy to fluctuate (open system). This ap- proach is advocated by Mekjian because of its simplicity [DeAn89]. The Copenhagen group uses a canonical approach allowing only thermal fluctuations (closed but not isolated) [Bond85a, Bond85b]. The Berlin group prefers to use a micro-canonical ensemble which conserves mass, charge, and energy explicitly (closed and isolated) [Gr0886, Zhan87, GI0887I. In order to keep the calculations manageable, they em- ploy a Metropolis sampling [Metr53] which is an intelligent method of getting a good sample of the representative states of a complex system. Predictions from Bondorf’s calculations are displayed in Figures 111.2 and 111.3. 65 These figures show the average IMF multiplicity and average temperature as a func- tion of the internal excitation energy of the nucleus. This model predicts that at 3 MeV/nucleon the onset of multi—fragmentation occurs. That excitation energy cor- responds to a temperature a little above 5 MeV. Bondorf’s model predicts that the energy fluctuations will be large enough at this point to start the formation of cracks within the nucleus. These cracks represent the breaking of internal nuclear bonds. This cracking then leads to multi-fragment emission, which is why the average mul- tiplicity starts to rise at 3 MeV/nucleon. The temperature levels off after crack formation, because at this point the system has entered the region of the liquid-gas co—existence as depicted in Figure 1.1. As the system traverses this region, the tem- perature remains constant until the excitation energy is great enough to vaporize completely the entire system. Beyond that point, the temperature curve follows the free nucleon gas prediction. The micro—canonical models of the Berlin group make similar predictions as those of the Copenhagen group. The relative probabilities for three different decay processes are displayed in Figure 111.4 [Gr0587]. There are two noticeable transitions. At E“ = 400 MeV (3.0 MeV/nucleon) there is a marked decrease in the evaporation yield complemented by an increase in the fission yield. At E“ = 650 MeV (5.0 MeV/nucleon), the cracking mode becomes the dominant decay process. These two transitions can also be seen in plots of the temperature against the excitation energy as in Figure 111.5. The temperature levels off at 3 MeV for the evaporation/fission transition, and at 5 MeV for the cracking transition. 2 Transition-State Multi—fragmentation The transition-state approach to multi-fragmentation is an extension of Bohr and Wheeler’s relative potential model for nuclear fission [Bohr39]. Bohr and Wheeler’s .1 Lu P a 1 \ e - , 1 . .st. 66 30- i I , AVERAGE MULTIPLICITY M 26.. ,{Il _. l 7322.31.11... “W I l . EXCITATION ENERGY E'/Ao [MeV] Figure 111.2: Predictions of a statistical multi~fragmentation model. The mean multi- plicity and its dispersion as a function of the excitation energy are shown [Bond85b]. 67 1 r r I; I6 ~-—— Crilicol lomperoluro Ag 3 I00 :4}: 15 ' . II— I Monlc Carlo calculations IJJ / 0:: —- Compound nucleus - /, I" --- Free nucleon gas ’1 4 ,1 1 Eli '0 "' x ,l' 4 // f % onsol ol // z 1’ 11.1 lroqmenlal' ’ i' I 509’ ’I/I . ’ ./ I-U crack ’4 g lompcralufl" I f’ll .. 5- 1111111111 1 E / . < // . / / O 1 1 . 5 10 15 20 EXCITATION ENERGY E'/A.(MeVI Figure 1113; The average temperature T as a function of the excitation energy E'/Ao' The onset of fragmentation occurs when the energy fluctuations are large enough to break internal bonds (crack formation). The dashed line illustrates the temperature of a free nucleon gas [Bond85b]. 68 "5’ .. 2\_' /.‘. I... \ .. /° \ E /' .9 F ./‘ >~ ’- / Q! 50 L’ /' .Z.‘ . 4- I- /’-~~~~ g - I /. s..- ‘0.)- _ /’ /. ~ I l" / g. .4' Q‘ ' I ,__1__.1==_<1..4 ’ i 1 1 1 00 200 1.00 600 800 1000 1200 E”ll~1eVl Figure 111.4: Predictions from a micro-canonical statistical multi-fragmentation model. The relative probabilities of evaporation (E, solid), fission (F, dashed), and cracking (C, dot-dash) like events for 131Xe are shown [Gros87]. model first determines a relative potential between two pseudo-nuclei as a function of the relative separation. The potential is zero at zero relative separation, peaks at a separation comparable in size to the nuclear radius, and then falls at large separations. This potential creates a fission barrier that must be overcome in order for a decay to proceed, To determine the fission widths (Ffiugm), one considers all possible states of relative momenta and separations. A determination is made as to whether each state decays or is stable. 1‘ fau'm is then the ratio of the number of unstable states to the total number of states. In order to generalize this technique to multi-fragmentation, two new quantities are defined: 1 N 9% = Z mnrf, (III.1) m0 11:1 1 N PF : I]; pnrn (111.2) n=1 where the mn, pn and the rn are the mass, momenta and positions of the various fragments and mo is the mass of a nucleon [Fai89]. As with the two-body model, 69 E” E’IA IHIIVI IZOO‘_9 ° -B- lilxe . 1000'1_8 _ E'=T2A/B . __7 .... calculated results 800‘_6 _ ' .—5 o 600% ° :4 LOO'PB -‘_2 ..O 200- .' _1_1 . . 1 . L 1 1 1 1 1: O 0 2 A 6 8 TIMeVI Figure 111.5: The excitation energy as a function of the internal temperature calcu- lated from a statistical multi-fragmentation IGrosS7I. 70 the potential is determined as a function of the relative co-ordinants. All possible states are considered to calculate the decay Widths, FN, for breakup into N large fragments. Figure 111.6 displays the relative FN as a function of the excitation energy of the nucleus. This figure demonstrates that for E’ above 3 MeV/nucleon the decay widths for multi-fragmentation channels are greater than widths for binary fission [L6pe89b, L6pe89c]. Lopez and Randrup have analysed the results of their multi—fragmentation model in an effort to determine which observables could be used to distinguish a rapid fragmentation from a sequential binary decay [L6pe89a]. Figure 111.7 displays the predicted relative velocities of various types of fragments. The clearest separation be- tween fission and fragmentation can be seen in the relative velocity spectrum between two heavy fragments. The events generated from a model simulating a sequential fis- sion process display a larger relative velocity on average. The fission fragments expe- rience a Coulomb repulsion which corresponds to the full size of the parent nucleus, while the heavy fragments produced in multi-fragmentation events have a reduced Coulomb effect because the field of the emitting system is strongly modified by the simultaneous emission of multiple charged-particles and of the fragment that is being studied. Figure 111.8 shows the results of a shape analysis which compares predictions from a fission model to those from a multi-fragmentation model. The fission events appear elongated with respect to some primary axis While the fragmentation events display a more spherical shape. These two observables have been used in experimental studies that have attempted to demonstrate the existence of a multi-fragmentation process [Klot89, Trau89, Boug89b, Poul89, Harm90, Cebr90]. 71 10000 5 ‘ BREAKUP 0F A=120 INTO 4 N HEAVY FRAGMENTS 1°°° (ALL MASS PARTITIONS) 3 A 100 / > a.) 2 5 V 10 _ z I" : 1 E' 0.1 g 0.01 '- L I l I L 0 1 2 3 4 5 6 E; (MeV per nucleon) Figure 111.6: The partial widths I‘N for the breakup of 120877. into N fragments with mass numbers A>10 as a function of the excitation energy of the source. The curves are labeled by the value of N ILépe89bI. 72 I t i -— nssrou z 11.. t 3 ----- FRAGMENTATION .9. '- Al ALL FRAGMENTS < '. —l c g 0.0 - 1. a: 0 O ‘. ° ‘. 2: 0.4 t 6 o .4 m > 0.2 - 0 a 1. 0.0: 0.10 0.x: 0.20 —- FlSSlON z 0.1 ..... FRAGMENTATION g ------- a) CHARGED FRAGMENTS f. 3": 0.1 c: o o E 0.4 o o .4 In > 0.2 0 0.05 0.10 0.1: 0.20 2 —— FlSSlON o 0.0 ----- FRAGMENTATION 13 C) HEAVY FRAGMENTS .1 u: a: 0.0 c: O 0 E 0.4 ~ 0 0 d .o > 0.2 * .0. c.. o A‘.‘o . .°-1h . 0.00 0.0: 0.10 0.15 0.20 VELOCITY DIFFERENCE I c Figure 111.7: Fragment velocity correlation functions for (a) all fragments (b) charged fragments only and (c) heavy fragments (A>4) only. A2150, Z=62, and E‘=5 MeV/ nucleon [L6pe89a]. 73 - FISSION o FRAGMENTATION COPLANARITY o 0.2 0.4 0.6 0.8 1 SPHERICITY Figure III.8: Superposition of 200 events in the sphericity—coplanarity plane for A2150, Z=62, and E‘=5 MeV/nucleon [L6pe89a]. 3 Geometrical Multi—fragmentation This modeling technique attempts to describe the observed experimental features through purely geometrical arguments. The nucleus is abstractly considered as a lat- tice of nucleons with bonds between the nearest neighbors. The variables of interest for this analysis are the numbers of occupied or vacant lattice sites and the number of complete or broken bonds. In geometrical or percolation models, the excitation energy is modeled by a parameter corresponding to the probability that a given bond will break. An increase in excitation energy is modeled as an increase in this decom- position parameter. In order to determine the distribution of clusters that has been formed during the decay of an excited nucleus, the model considers which bonds have been broken. The aggregations of lattice sites (nucleons) that have been isolated are considered to form fragments without regard to how tenuous the interconnections may be. Though this analysis technique is simplistic, it can reproduce power-law shape of the production cross-section distribution and the value critical exponent observed in the Purdue data [Baue86]. Models of a similar nature can be made more sophisti- 74 cated through additions of post-breakup de-excitations and pre-breakup expansions [Néme89, Ng689], but the general concept remains the same. These models provide a mechanism that will produce multi-fragment events through a purely statistical process. C Experimental Studies The Plastic Ball group at LBL provided some of the first evidence that reaction chan- nels consisting of multiple IMFs were populated during nuclear interactions. The evi- dence presented was in the form of simple multiplicity distributions from the reaction 200 MeV/nucleon Au+Au. The device used for these observations was the Plastic Ball/wall [Bade82], which consists of one thousand phoswich detectors covering al- most all of 47r. The beams were provided by the Bevalac accelerator facility. A high granularity device is appropriate for measuring the Charged particle multiplicities, because the probability of non-detection due to multiple hits is reduced. Figure 111.9 displays the observed IMF multiplicity, AIIMF, distributions for several different trig- ger requirements [Harr87, Doss87]. Though these studies conclusively demonstrated that reaction channels with MIMF up to ten are populated and that in central colli- sions the maximum probability is a decay into several fragments, they did not isolate the decay mechanism. Proof of multi-fragment emission does not necessarily confirm M ulti- fragmentation. Another experimental group at LBL has been making an effort to determine the reaction process that leads to multi-fragment emission. This group works at both the 88-inch Cyclotron and at the Bevalac. They have carried out an exhaustive set of measurements employing reverse kinematics (heavy beams on light targets) [50b083, 30b084, Char86, Char88a, Char88b7 Ha1189, Plag89]. The energy of the beam par- 75 s at... :11 > tiplicity 1d W I ' I ' fiL-‘l E 89. r r x mm — .0 ‘. m 10‘ . -;'£=,\ “[313“ .‘.-- ’ 't. \ HUM —.- ';/ '3... m __ .’. —-¢\ 10’ / j" \{k \\ a: ' / "\ \ / \ \ 10’ a,“ . _ \, 1*. Id mm m L . 1 n v m L_ v r n , z 4 0 0 10 12 14 II II' I 'I Figure 111.9: The observed IMF (Z>2) multiplicity distributions from 200 MeV/ nucleon Au+Au. The five curves represent five participant charge multiplic- ity bins [DossS7, Harr87]. 76 ticles varied from 10 to 100 MeV/ nucleon. To observe the reaction products, a set of silicon—silicon—plastic telescopes has been employed in a variety of configurations. The first data presented from this group were inclusive IMF production cross-sections [80b083, Char86]. Data of this sort were compared to predictions from a modified se- quential binary decay code (GEMINI [Char88b]), and the agreement between the data and the calculations was taken as evidence against multi-fragmentation at this exci- tation energy. Figure 111.10 displays an example of this sort of comparison [Wozn88]. A more exclusive analysis plotted the charge observed in one detector against the charge observed in another [(301089]. Figure 111.11 is an example of this analysis technique [Bowm87]. The strong clustering of events around the Z1 + Z; = 60 was taken as evidence that the dominant decay mode was a single fission. The conclusion was challenged by Gross. IIe maintained that his micro-canonical statistical multi- fragmentation code could reproduce the experimental results. Figure 111.12 shows Gross’s calculations [Gr0588]. Though Gross may challenge the assignment of fission as the most probable decay mechanism, he can not rule it out. A group from Darmstadt has performed a series of studies designed to address the question of multi-fragmentation. The detection system they employ consists of two detectors, a high resolution silicon/time-of-flight (TOF) detector at 90° and a large parallel-plate avalanche counter (PPAC) on the opposite side of the target. The ex- periments were carried out at the Saturne II facility at Saclay using 800 MeV/ nucleon a beams on a gold target. The trigger condition is an IMF / heavy fragment in each of the two detectors. The observable is the relative velocity between the two fragments. One expects an enhancement in the relative velocity between two back-to-back frag- ments produced by a sequential process as compared to the relative velocity expected from a simultaneous multi-fragmentation. Calculations by Gross suggest that the observed relative velocity spectrum was produced by a multi-fragmentation process 77 Nb+Be E/A =-'- 14} Mei/ E’ = 131Mev E/A =' 114701.31} 3 E’ = 102 MeV 'E/Aé 18.0 Mév E' = 158 MeV Jmax =' 40 h . Jmax =- 4111 J"... = 44 n :f . - 102 O‘.‘ f‘. 0". 23 .-: ' a! g 5 '1 10' ,4 § -." § ‘ 1' .i . m ' I s 'i 3 v o 'a ‘ ‘ 9: '5 ‘ b 10 I?! g. o: x' 2:. I I 1: ‘ r w 5 i I I ° 2 . 01 I I : 3 104 o It I . I 3 10-2 I 0 20 4o 0 20 40 0 L 20 L 423 Figure 111.10: Inclusive production cross-sections as a function of the fragment charge from the reaction Nb+Be at 11.4, 14.7, and 18.0 MeV/nucleon [W021188, Char88b]. 78 SO-Mev/u La + C I 30 60 13 Z '- 0 L L L O 15 ° 30 45 . 60 Figure 111.11: A plot of the charge observed in the first detector against the charge observed in the second detector. The upper right corner shows a spectrum of Z1 + Z2 [Bowm87]. 79 Figure 111.12: A plot of Z1 against Z2, where Z1 and Z; are the charges of the two heaviest fragments produced by a microcanonical multi—fragmentation simulation of the decay of 146Nd' [Gr0588]. 80 ' I I I I r 800 MeV/u a+Au ‘ 100 - IMF-IMF (D F- (A640, 22(15) ” \\ % EXP. ’ \‘1 - “‘ seq. amiss. ‘ 8 5 0 - - W I “1 - o 1 2 3 I. 5 s v,,: (cm/n5) Figure 111.13: Relative velocity distributions between IMFs. The solid dots are the ex- perimental data with statistical errors [Klot89], the dashed line is the prediction from a sequential decay code, and the solid curve is the calculation from a simultaneous multi-fragmentation [Gr0589]. [Gr0589]. The observed spectrum and the predictions are displayed in Figure 111.13 [Klot89, Cass89]. This analysis has been challenged by Pochodzalla [Poch89b]. He suggests that Gross’s analysis failed to consider the pre-equilibrium evaporation of light particles. Pochodzalla’s estimates of this early particle emission reduce the size of the fragmenting system from 184 to 156. This reduction in the mass (and charge) of the system, lowers the relative velocities of the outgoing fragments because the Coulomb repulsion has been reduced. Figure 111.14 displays the predicted mass re- duction as a function of excitation energy and the subsequent reduction in the centroid of the relative velocity distribution. Pochodzalla’s calculations were based on the se- quential decay code GEMINI and reproduced the experimental spectrum. Therefore, 81 200-1 1 l I l ”04.19%“ __ IT 1.! I l_ +197AU -.... 0 .. , .. 0": h " ----.--.::':-..--...- i '- 3150— y -- - < p .. , -1— .. _. :' '----:_e. ------ 4 " g : ‘x "‘ seq. amiss. .- hllljl 11111111111111-1 00 ' 200 600 100 A 5 EXIMeV) (crn/ns) Figure 111.14: Calculations for two systems (180 +197 An, and a +197 A11). The left side shows the effective mass (A3”) of the compound system as a function of the original excitation energy; the right side displays the centroid of the relative velocity distribution [Poch89b]. a sequential process can not be ruled out. A group from GSI has been using a detection system consisting of twelve PPACs and seventy phoswichs and has been studying multi-fragmentation. Studies have been performed using oxygen and argon beams from the CERN synchrocyclotron and the SARA facility at Grenoble. The beam energies ranged from 30 to 84 MeV/ nucleon. The targets were silver and gold. The original analysis of the data presented only IMF multiplicities as a function of excitation energy (Figure 111.15) [Troc87, Poch88]. These data could not be used to resolve between a sequential and a multi-fragmentation process. The data were later analyzed in terms of the relative velocity at small rela— tive angle. For this analysis, one expects that a simultaneous process will display a depletion at small relative velocity due to the Coulomb repulsion of particles produced 82 1.5 - 1.0 1.5 - 1.0 o 200 £00 soo 800 1000 (MeV) Figure 111.15: A plot of the average IMF multiplicity as a function of excitation energy for various beams on silver (top) and gold (bottom) targets [Trau89]. 83 close together in phase space. A model simulating a decay process with an adjustable decay time constant was fit to the data; Figure 111.16 displays the results [Poch89a, Tr0089, Trau89]. From this analysis one concludes that the decay process must have a time constant greater than 1000 Fm/c. An additional analysis of relative velocity of IMF s at large relative angle was conducted by the same authors. The conclusion from this study is that a sequential process dominates at excitation energies up to 6 MeV / nu cleon. A group from Caen has performed a set of experiments at GANIL [Rudo88, Hage89, Boug89a, Boug89b]. These experiments are similar in design to those of the G31 group. The detector apparatus consists of two arrays of PPACs, DELF [Boug87] and XYZt [Rudo86], and a wall of plastic scintillator, MUR [Biza86]. The system is sensitive to IMF s over 55% of 47r and to light particles in the forward direction. A variety of beams and targets have been employed and the data has been analysed for evidence of compound nucleus formation [Hage89, Rudo88], collective motion, side- splash effects, event shape, and fragmentation time scale [Boug89a, Boug89b]. An example of their time scale studies is illustrated in Figure 111.17. This figure displays the relative angles and velocities between three IMFs detected in events from the reaction 60 MeV/ nucleon Ne+Au. The data are compared to predictions based on either a sequential, intermediate, or simultaneous decay process. The model con- siders an excited compound nucleus which decays into three fragments. The second decay occurs at a time delayed by either 00, 300, or 0 F m/ c. From this analysis, the simultaneous process is ruled out, and a lower limit is set on the time scale for the disassembly process. Figure 111.18 displays the results of an event shape analysis performed by the Caen group on data from the reaction 43 MeV/ nucleon Kr + Au, Ag, Th. Though the event shape is analysed, it is not compared to simulations and therefore can not be used to determine the decay process. 84 _ a 1 T 1 1 1 .- 1.5—10+197AUIHL E/A=84Me 10— -‘°°i--‘“..’-‘ i —- 10001‘m/c "’ — / 1000013711: -1 ‘2' . 'é‘ Vi?- 15 M“ I- . ‘T‘ 100 fm/c 05- —-— 101mm: - - oi‘x’ .. ----- 10000fm/c‘ 0.0,.ll,j L1 ._1 . 1. o 1 2 3 1. s vredcmlns) Figure 111.16: The relative velocity correlation functions from 84 MeV/ nucleon 18O on 197/111 (top) and "“‘Ag (bottom). The three curves are calculations from a sequential decay code with an adjustable decay time scale [Trau89]. 85 61] (deg) Vij (cm/r15) Figure 111.17: The relative angles (left) and relative velocities (right) between pairs of IMFs from the reaction Ne + Au at 60 MeV/nucleon. The particles are ranked from heaviest to lightest [Boug89b]. 86 1 1 1 Au ND=4 ‘ A11ND=5 >1 .1: C5 2 a: 0.5 O A 0 :im ;,'/ -'// t ' .~ . -yl / ( )\ . .////"i([ "'(.}}(L§ '/ 425 9e; 0 0.5 ~ 1 1 ‘ srmzmcrrr srurmcmr 1 1 - Ag ND = 4 Th ND = 14 m < 0.5 2 < J D- o u SPIIERICITY SPHERICITY Figure 111.18: Shape analysis for data from the reaction 43 MeV/ nucleon Kr + Au, Ag, and Th. The ND value represents the number of IMFs detected [Boug89a]. 87 The last group that is actively studying the question of the time scale of nuclear disassembly is another group from LBL. This group utilizes a forty-eight element phoswich array to study projectile fragmentation. The array is assembled in a seven- by-seven configuration with the center element removed to allow the beam to exit. The 88-inch cyclotron at LBL provides C, N, O, and Ne beams which are incident on Au targets at 32.5 MeV/ nucleon. Only events in which the sum of the charges near beam velocity is equal to the charge of the projectile are analysed. These events are sorted by breakup channel and identified with an excitation energy, which is de- termined by the relative energy between the fragments. The researchers compared the production cross-sections for the various channels and the excitation energy as a function of the Q-value. From this analysis they suggest that the projectile and target each receive the same amount of excitation energy and do not reach a thermodynamic equilibrium. The excitation energy of the projectile varies from 1 to 4 MeV/ nucleon and should span the regime where multi-fragmentation has been predicted to occur. The data are analysed using an event shape analysis in order to determine the decay process. Figure 111.19 displays the data for a single breakup channel compared to predictions of both a sequential decay and a multi-fragmentation [Pou189]. The re- searchers find that the sequential decay more accurately reproduces all of the observed breakup channels. D Conclusions Multi-fragmentation is expected to appear in reactions for which the internal ex- citation exceeds 4 MeV/ nucleon. The theoretical calculations predict that at this excitation energy the nucleus should reach a region of instability. The fluctuations in energy and in density will cause bonds within the nucleus to be broken. The breaking of internal bonds leads to crack formation and then to fragmentation. The (I) oo 0'5 rv—Tf'lvv'vfr‘Y—V'IVVYrI'Yfi 0.5 ,rr"|'rfi'FrV—Vwirrvrrvrrw l; a) Multtfragmentation : b) Sequential 0.4 '- -1 0.4 5» . i 3:- .1: 0.3 " .0 ‘1 a 0.3 g . O . 1 a :9. 0.2 I- . . - :2 0.2 8‘ : o. 1 8. 001- 4,’ .11 ° 7001 0.0 4.1--an 1‘ 11 0.0. . 0.2 0.4 0.6 0.8 1 :0) Data Sphericity 0.4 :- 1 33‘ : . 3 a 0.3 C" '1 a z : g 0.2 I- -j 8‘ I o I o 0.1 E- go - ° : 11911.1.1m11.1..m111n11 0 0.2 0.4 0.6 0.8 1 Sphericity Figure 111.19: Shape analysis plots for the breakup of 160 into four as (c). For comparison, the predictions from a multi-fragmentation (a) and a sequential decay (b) are shown [Pou189]. theories predict that there should be a dramatic rise in the IMF multiplicity as well as temperature anomalies associated with the onset of multi-fragmentation. These anomalies correspond to increases in the excitation energy of a system without ap- parent increases in the temperature. Since other processes may also lead to increased IMF multiplicities and nuclear temperatures can not be measured directly, the search for evidence for the existence of a multi-fragmentation process has concentrated on observables associated with the time scale of the disassembly process. These observ- ables include relative velocity distributions for back-to-back fragments, small angle velocity correlations, and event shape distributions. The experimental studies have concentrated on the region of excitation energy from 1 to 6 MeV/ nucleon, which en- compasses all of the predicted threshold energies. Though a rise in IMF multiplicities is observed in this energy region, no experiment has yet been able to exclude the pos- 89 sibility that a sequential decay process is the preferred decay mechanism. Attempts to estimate the time scale of the decay process have ranged from 300 to 1000 F m/ c. A simultaneous process should show a lifetime of under 50 F m/ c. Therefore, the existence of a simultaneous multi-fragmentation decay mechanism at intermediate energies has not yet been confirmed. Chapter IV Multi-Particle Correlations A Introduction to Multi-particle Observables The study of multi-particle observables in heavy ion nuclear physics was introduced by one of the early large detector array systems for nuclear physics, the Plastic 3011/ Wall [Bade82]. The observables that have received the most attention are collective flow and ratios of production cross-sections for various particle types as a function of the observed multiplicity [Gutb89]. These observables probe questions concerning the equation of state of nuclear matter. A question that has received a great deal of theoretical attention recently has been the existence of a' multi-fragmentation de- cay mechanism that occurs as the nuclear system enters a region of mechanical in- stability [Bert83]. This multi-fragment reaction mechanism is expected to become dominant at bombarding energies above the Fermi energy [Rand81, Fai83]. This process will be characterized by a simultaneous disassociation of the system into many fragments. Early attempts to identify this mechanism relied only on multi- plicity distributions [DossB7, Klot87, Harr87]. It has, however, been demonstrated that information from such distributions or other inclusive observables (requiring the detection of only a single fragment) alone is insufficient to identify conclusively multi-fragmentation [More88, L6pe89a]. Models incorporating other processes also 90 91 can produce these inclusive results. A definitive identification of this process requires study of an observable which is sensitive to the time scale of the disassembly and not merely to the final distributions of fragments and their energies [L6pe89a, Jaca88]. Recently, experimental studies attempting to identify evidence of this process have been conducted [Boug88, Tr0087, Klot89, P001188, Pou189, Harm89]. .Only one of these studies has claimed to see evidence of a multi-fragmentation process [Klot89], and even that result is disputed [Poch89a]. These studies have surveyed the range of excitation energies from 1 to 6 MeV / nucleon, which is where the theoretical analyses had predicted that the onset of multi-fragmentation would occur. The work that is presented in this chapter searches for evidence of multi-fragmentation on central col- lisions from the system Ar + V at bombarding energies of 35 to 85 MeV/ nucleon. If one assumes complete fusion, the excitation energies attained during these collisions would be higher than than those that have been previously examined. Evidence for the onset of multi-fragmentation is found within this bombarding energy range for ‘symmetric systems’. In this chapter we use an observable that has recently been proposed [L6pe89a] to be sensitive to the time scale of the fragmentation process. This observable is the spheroidal shape of the envelope of the energy flow from the reaction, which is defined as the event shape. This shape will be influenced by the dynamics of the break-up. The emission of fragments from a simultaneous multi-fragmentation process will be isotropic in the center-of-mass frame, and therefore the events should be spherical in shape in the limit of infinite multiplicity. Conversely, sequential emission should lead to an event shape that is elongated along one axis due to the kinematical constraints of the binary decays and the time ordered emission of the fragments. 1n the following analysis, 1 shall compare the average event shapes extracted from my experimental data to predictions from simulations of both a sequential decay and a simultaneous 92 multi-fragmentation. B Experimental Details All the results described in this section were derived from experimental data acquired using the MSU 47r Array [West85]. I have studied the 40Ar +51 V system with beams provided by the K500 and K1200 cyclotrons. Incident energies of 35, 45, 55, 65, 75, and 85 MeV / nucleon Were studied. For the experiments described in this work, the array was instrumented with 215 phoswich [Wilk52] detectors. The 170 main ball detectors covered an angular range from 20° to 160°, while the 45 detectors of the forward array extended the coverage in from 20° to 7°. The minimum energy required for a charged particle to be detected was determined by the pulse height necessary to fire the discriminators. This is related to the light generated by the stopping par- ticle. In the ball detectors, this energy is approximately 4 MeV/ Z, where Z is the charge of a given fragment (Appendix B describes in greater detail the nature of the light response of the scintillators). However, in order for a fragment to be identi- fied it had to penetrate into the stopping scintillator, which required approximately 20 MeV/ nucleon. For identified particles, we had isotopic resolution for Z=1 frag- ments and charge resolution up to Z26. Particles with energies from 4 MeV/ Z to 20 MeV / nucleon were stopped in the AE scintillator and were assigned an estimated charge and energy for this analysis. In the forward array detectors, the minimum energies required were 3 MeV / Z for detection and 12 MeV / nucleon for identification. For identified particles, we had isot0pic resolution for Z==1 fragments and charge res- olution up to Z218. Particles that were stopped in the AE scintillator were assigned an estimated charge and energy. Further details of the detector specifications, the energy calibration of the scintillators, and the pre-analysis of the raw data are given in the appendices of this thesis. 93 The multi-particle analysis that was performed on the events from these data assumed only a single source for the emitted charged-particles. Therefore, an effort was made to minimize the contributions from both the target and the projectile remnants. The minimum energy thresholds effectively eliminated the target velocity data, however, a significant projectile component could be identified in the forward array detectors. Therefore, the data from the forward array were used only for the determination of the impact parameter and not in the event shape analysis. This suppressed the contribution of projectile-like spectator matter. 1 Exp 87008A The first experiment using the MSU 47r Array was performed in April of 1988. For this experiment, the K500 provided a 35 MeV/ nucleon 40Ar beam of 70—100 electrical picoamps of current which was incident on targets of 51V (3.34 mg/cmz) and 197Au (7.02 mg/cm’). In addition, experimental runs were taken with a blank target frame and with no beam in order to measure the background due to beam halo hitting the aluminum target frame and due to cosmic rays induced events. The triggering requirement for these runs was an adjustable number of ball detectors (Mb) firing in coincidence. The levels were set at Mb greater or equal to one, two, five, seven, and ten. 2 Exp 87008B The second experiment using the MSU 47r Array was performed in June of 1988. For this experiment, the K500 provided a 50 MeV/ nucleon 1""C of approximately 20~ 60 electrical picoamps of current. This beam was incident on targets of 12C (2.00 mg/cmz), and 197Au. The triggering requirement for this run was either 1111,, or an adjustable number of forward array detectors (M FA). The levels were set at Mb 94 greater or equal to one, two, or five, and MFA greater or equal to one or two. 3 Exp 88012 The third and fourth experiments using the MSU 41r facility were collaborations with Hope College and Chalk River National Laboratory. The results from these experiments are not reported in this thesis. Following these experiments the detector array was moved to an interim vault location. The fifth experiment was performed in April of 1989. For this experiment, the K1200 provided beams of 45, 55, 65, 75, and 85 MeV/ nucleon 4”Ar of approximately 100 electrical picoamps of current. The energy systematics were extended in January of 1990 to include a 100 MeV/ nucleon beam. These beams were incident upon a 51V target. For this experiment the trigger units had been improved so that M1, and [MFA could be added together to create an overall system multiplicity signal (M5). The triggering levels were set at MS greater or equal to one, two, and five. C Event Characterization One of the most powerful techniques available to an experimenter who is working with a semi-exclusive data set is the ability to characterize individual events and make selections based on the observed features. The MSU 47r Array provides researchers with this type of data set. First, one must decide what parameters one wishes to. use for categorizing the events and then determine which observables will provide information concerning those parameters. The parameter that was chosen to characterize the events for this analysis was the magnitude of the impact parameter. This determines the degree of overlap between the target and the projectile nuclei and is related to the violence of the collisions 95 and the number of constituents that were exposed to the full force of the impact. The matter on the fringes of the colliding nuclei provides little information about the hottest and most dense phase of the interaction, because the regions that do not overlap do not interact strongly [West76]. The magnitude of the impact parameter is of great importance for separating effects which arise in the most violent collisions from effects that are common in the peripheral interactions. The approach that we have taken for measuring the magnitude of the impact pa- rameter on an event-by-event basis is to place cuts on the number of charges measured in the mid~rapidity region [Ogi189a]. We measure the total detected charge because our detector is sensitive to charge and not to mass, and we assume that the distribu- tion of neutrons leaving the reaction will be similar to the measured distribution of protons. Only the charge detected in the mid-rapidity range (0.75 ymget < y fragment < 0.75 gym-edge) is counted, because it is assumed that the matter traveling at rapidities either higher or lower than those limits will be associated either with the disassembly of the projectile or the target remnant. This observable, therefore, is a measure of the size of the interaction region, which is determined by the impact parameter. The reliability of this observable as a measure of the impact parameter has been studied using the event simulator FREESCO [Fai86]. This code uses statistical emission from three thermal moving sources of nuclear fragments. The fireball ge- ometry [West76] is used to assign sizes and velocities for the different sources. The code conserves energy and momentum. We have used this code to produce events simulating the systems that we have studied experimentally. These events were then filtered through a software model of the MSU 47r Array [Wils89]. This filter contains the minimum energy thresholds for detection and identification, the finite granularity, the regions which are not covered by detectors (either due to the finite thickness of the detector walls or due to gaps left to allow the beam to enter and exit the array), 96 the rejection of neutral particles, and the shadowing due to the target frame. We are therefore able to take a simulated event and determine exactly how it would have been seen by the 47r Array. The mid-rapidity charge detected has been studied as a function of known impact parameter for these simulated events. Figure IV.1 dis— plays a plot of the detected mid-rapidity charge versus the known impact parameter. The data are from a FREESCO simulation of Ca+Ca reactions at 40, 70, and 100 MeV/ nucleon. The data points represent the average charge detected in the mid- rapidity region, while the error bars correspond to the width of the distributions. It should be noted that though the centroid of the mid-rapidity charge decreases as a function of impact parameter, the width of the distribution is quite large. Therefore, gating on the detected charge will yield a measure of the impact parameter (b) that is accurate only to :l: 0.3 0",“, where 0mm, is defined as Rtarg¢¢+R1,,-ojecta¢ (radii of the target and the projectile). We have chosen to identify four impact parameter bins. Details of these bins are given in table IV.1. Figure IV.2 displays the percentages that various impact parameters contribute to each bin (the data are from F REESCO simulations). D Event Shape Analysis The data from these experiments have been studied using an event shape analysis technique. The event shape parameters allow the experimenter a quantitative measure of the general shape of the distribution of outgoing fragments from the reaction. The shape analysis is used to determine if the fragment distribution contains some amount of anisotropy. The analysis technique requires the observed event to be transformed into the center-of—mass reference frame. Figure 1V.3a displays a typical event in the laboratory frame while Figure 1V.3b shows the same event in the rest frame. Having transformed an event into the center-of—mass frame, one then constructs a kinetic flow 97 40 MeV/nucl. '70 MeV/nucl. 100 MeV/nucl. so 0 : . M 40”— E— Z— 3 311110. 311111 311111 g; 307 t— L" o : : t 20;- :— E— 105— . E— E- v o r....1....1....1 1.. 1....1....1..n1-.. = ....1....1.111.... Q) 1.. 20~_ 3 C c : E ‘5: ii iii 01 Z .. 1011111 . l . . {a ; t + Z + :1 se- 11 5— + 5- 1 ° 5,,,§,,5 + =.,’,+,, 0 MM " 0 0.2 0.4 0.0 0.0 1 0 0.2 0.4 0.11 0.0 b/bmax. b/bmax. Figure IV.1: The mid-rapidity charge as a function of the known impact parameter. Both the total charge and the quantity that passed the filtering requirements are displayed [Ogil89 Percentage Figure IV.2: The relative contributions that different impact parameters make to each of the bins [Ogil89a]. a]. l O 0.2 0.4 0.6 0.8 b/bmax. 60 20 . L. l---— . 1. l. . 4GP . l. . _. . 1- 0 0.2 0.4 b/bmax 98 Table IV.1: The mid-rapidity charge gates for the various impact parameter bins for the various systems that were studied. System Incident Energy Bin 4 Bin 3 Bin 2 Bin 1 peripheral mid- mid— central peripheral central 4”Ar +51 V 35 MeV/n 0-2 3-6 7-11 >12 “Ar +51 V 45 MeV/n 0-2 3-6 7—11 >12 40Ar +51 V 55 MeV/n 0-2 3-7 8-12 >13 “Ar +51 V 65 MeV/n 0-2 3-7 8-12 >13 40Ar +51 V 75 MeV/n 0-2 3-7 8-12 >13 40Ar +51 V 85 MeV/n 0—2 3—7 8—12 >13 “Ar +51 V 100 MeV/n 0-2 3-7 8-12 >13 “Ar +197 Au 35 MeV/n 0-2 3-7 8-13 >14 ”0 +12 C 50 MeV/n 0-1 2-4 57 >8 tensor by summing over the fragments detected in a given event as follows: F = Z Fij (IV.1) 1.7 ' where M n n. 1’in 1.. : __ V. F, :4; 2m. (1 2) where p? denotes the ith momentum component of the nth particle, mn denotes the particle’s rest mass, and M corresponds to the multiplicity of the event. Since this analysis is done over different fragment masses for each event, the 57,1: weighting must be used to ensure that heavy fragments do not dominate the kinetic flow tensor. In many physical problems involving tensors, it is desirable to carry out an or- thogonal similarity transformation to reduce the matrix to a diagonal form, where all non-diagonal elements are equal to zero. For analysis of the kinetic flow tensor, the problem is to re-orient the co-ordinant axes in space such that the non-diagonal elements vanish. In the new co-ordinant system, the axes correspond to the principal axes of a 3-dimensional ellipsoid that approximates the kinetic flow distribution. The 99 Figure 1V.3: Illustration of a typical event. a) The lengths of the vectors corresponds to the velocity of the particles in the laboratory reference frame. b) The same event transformed to the center-of-mass reference frame. directions and lengths of the principal axes is given by the orthonormal eigenvectors (ui), and the eigenvalues (t,- of F given by the equation: F = tlulul1 + tzuzu; + t3u3ug. (1V.3) The eigenvalues, which correspond to the diagonal elements of the diagonalized tensor, satisfy the cubic equation: til-1- agt? + alt,- + do = 0 (IVA) where do = 171le23 + F22F123 + F33F122 — F11F22F33 —- 2F12F13F23 (1V.5) (11 = F11F22+F11F33+F22F33*Ffsz123~F223 (IV.6) a2 : —F11 —— F22 — F33. (IV.7) Using auxiliary variables p and r as defined below, the three solutions to the cubic equation can be written as follows: 1 . 2 t,- = —%02 + 2p cos[-3— arccos(r/p3) + (z — 1)§7r] (1V.8) 100 p = 311/111 — 3a. (IV-9) 3 r = $01.52 — 3a0) — 23;— (IV.10) Specifically, the three eigenvalues are: t1 = —a2/3 + 2pcos(% arccos(r/p3)) I (IV.11) t2 = —az/3 + 2p cos(% arccos(r/p3) + 27r/3) (IV.12) t3 : ——a2/3 + 2p cos(-:— arccos(r/p3) + 47r/3) (IV.13) The eigenvalues are then ordered by magnitude (t1 < t; < t3) and used to define three reduced quantities [Gyul82]: t? qi: 31 2' j=1tj (IV.14) In effect, the eigenvalues have been ‘normalized’ such that q1 + q; + q3 = 1 for all distributions. From these reduced quantities, the sphericity, S = 3(1 — q;;), and the coplanarity, C = %\/3(q2 — QI) parameters as defined by Fai and Randrup [Fai83] and applied to this problem by Lopez and Randrup [L6pe89a] are determined. The orthonormal eigenvectors of the tensor can be evaluated in spherical co- ordinants using the following calculations: (F11 _t1')(F33 —* ti) — F123 -: IV.15 Q F12F13 — F23(F11 — ti) ( ) 0, :: arccos(1/(fi + C? + (it; ~ F33 - Fzgc,)2/F123) (IV.16) 9151' = GTCtanfciFia/(ti — F33 ‘- F2301)) (IV.17) The eigenvectors, u; : (1,0,,¢,), of the kinetic flow tensor represent the direction of the axes that best approximate a spheroidal envelope of the outgoing energy flow vectors of the fragments. The eigenvalues are related to the lengths of these principal 101 axes. The sphericity parameter represents the relative strength of the third or major axis with respect to the two other axes and is a measure of the amount of elongation. (1011-0 A spherical distribution would have three equal axes therefore, q1 = q; = q3 = , which results in a sphericity equal to 1.0. The coplanarity parameter represents the asymmetry between the two minor axes, and is a measure of the degree of flattening of the spheroid. A spherical distribution would have a coplanarity of 0.0, while a flattened distribution would have an extremely small ql axis, and therefore, the coplanarity parameter will be large. The data presented in this thesis are in the form of sphericity versus coplanarity (S-C) distributions and the centroids of these distributions, and conclusions will be based upon comparisons between experimental data and predictions from simulations. S The sphericity parameter can vary from 0 to 1 while the coplanarity can vary from 0 to x/3/ 4. Figure 1V.4 displays the S-C space. The triangular region formed by the lines connecting the points at (0,0), (0.75, 0.413), and (1,0) defines the limits of the space. The different regions of the space correspond to different distribution shapes. The origin (0,0) corresponds to a 1-dimensional object. The line from (0,0) to (.75,\/3 / 4) contains the possible 2-dimensional ovals. The rest of the space is filled by 3-dimensional spheroids. The importance of transforming the event into the reference frame of the emitting source is demonstrated by Figure IV.5. This figure displays the average sphericity parameter extracted from a set of La+La streamer chamber data [Krof89]. The velocity of the reference to which the data are boosted is varied. The maximum average sphericity value is extracted for a boost velocity of 0.18c (VCM = 0.18 c). The sphericity value is depressed when an incorrect transformation is performed because this induces a perceived motion of the entire system, which produces an artificial elongation in the direction of the transformation. 102 Narrow Oval Hide Oval . [If 1, I”. V3 \ Sphericity vs. Coplanerity General Features of the Plots 1 10 O. Oblate Rod >5 .. p w _ ":1 ii "~ . . I I1 . ‘5 O? .' '3 »$<1 ' C o - «1'. *1 i g a .a‘ Q. '- ‘ ' ad's' .3 F1 ‘u-- . El... 0 . L. .I-‘ - '-" are .~'.‘. - CLJ. 0.3 0.?/ (177 0.9 Sphericity Prolate Tri-axial Sphere Figure 1V.4: A display of which areas of the S—C space correspond to which spheroidal shapes. 103 _ fff F r1 1 l 1 l 1 1 1 1 1 1 1 1 hr r X 0.3 — XX XX —J _. X X _ r- X -l q) - - E,“ 0.2 .—— __J 1; - >< ~ < V) _ _ X a 0.1— ‘ —+ _ >< - l- l l L l l L J L l J l_ l 1 LJ L l l J_ lLl l I l 0.0 0.1 0.2 0.3 0.4 0.5 Vboost (C) Figure IV.5: The average sphericity parameter extracted from a set of 70 MeV/ nucleon La+La data as a function of the velocity with which the data were transformed. 104 E Events Shape Distributions as a Function of Incident Energy The Ar+V system was studied for a range of beam energies from 35 to 85 MeV / nucleon. We have studied the event shape for central events from these reactions and the dis- tributions are displayed in Figures IV.6 t0 IV.11. For all of the beam energies the mid-rapidity charge cuts sampled events from similar impact parameters. The positions of the cuts changed because the detection efficiency of the array varies with the average energy of the exit fragments. The event shape distributions for all six of the beam energies are peaked along the 2-dimensional axis. The average event shape can be qualitatively categorized as a flattened prolate spheroid. The widths of the distributions get broader as the energy of the beam is increased. The centroids are listed in Table IV.2. Though the primary effect that causes this increase in the Table IV.2: The average values for the sphericity and coplanarity parameters from the analysis of central collisions from the system 4°Ar +51 V at six different beam energies. Incident Energy Sm,e Cave 35 MeV/n 0.2826 :1: 0.0009 0.1264 :1': 0.0004 45 MeV/n 0.3025 :1: 0.0005 0.1338 i- 0.0002 55 MeV/n 0.3211 :1: 0.0005 0.1384 :1: 0.0002 65 MeV/n 0.3452 :1: 0.0006 0.1433 :1: 0.0003 75 MeV/n 0.3496 :1: 0.0011 0.1439 i 0.0006 85 MeV/n 0.3826 :1: 0.0014 0.1492 i 0.0007 average sphericity is the increase in the average multiplicity as a function of beam energy, it shall be demonstrated that the lower energies exhibit an anisotropic emis- sion of fragments. This anisotropy in the average event shape can be attributed to the timescale of the decay mechanism. These experimental distributions can be naively compared to those calculated by 105 0.6 1 1 1* l l [i m l f a? 0.4— —- S... 10 C1 —- __ £5. 0.. o 0.2 __ u 0.0 - L 0.0 0.2 0.4 0.6 0.8 1.0 Sphericity Figure IV.6: A contour plot of the event shape distribution gated on central events from the system 35 MeV/ nucleon 40Ar +’51 V. 0.6 V [7 I j T If I I 1 1— A >\ :1 0.4 — s... S 2 a Q. o 0.2 __ O O 0.0 " ' fl“ 1 0.0 0.2 0.4 0.6 0.8 1.0 Sphericity Figure IV.7: A contour plot of the event shape distribution gated on central events from the system 45 MeV/ nucleon 4°Ar +51 V. 106 0.6 1 l— —I 1 I 1 1 1 r 0.4 5'“ i —- Coplanarity 0.0 0.2 0.4 0.6 0.8 1.0 Sphericity Figure 1V.8: A contour plot of the event shape ilistribution gated on central events from the system 55 MeV/ nucleon 40Ar -|~51 V. 0.6 r l 1 l 7 I r l l ,_ a >. _ :1 0.4 L. 10 a ‘1 .59 o. __ o 0.2 o n 0.0 0.2 0.4 1-0 Sphericity Figure IV.9: A contour plot of the event shape distribution gated on central events from the system 65 MeV/nucleon 40Ar +51 V. 107 0.6 1 I 1 I 1 T I 1 1 >. :3. 0.4 _ S... 10 L1 __ g 0.. o 0.2 __ o 0.0 -’ L 0.0 0.2 0.4 0.6 0.8 1.0 Sphericity Figure IV.10: A contour plot of the event shape distribution gated on central events from the system 75 MeV/ nucleon 40Ar +‘51 V. 0.6 1 1 1 1 1 1 m l —r .0 .1; Coplanarity O '10 0.0 0.2 0.4 0.6 0.8 1.0 Spherioity Figure IV.11: A contour plot of the event shape distribution gated on central events from the system 85 MeV/ nucleon 40Ar +51 V. 108 Lépez and Randrup (refer to Figure 111.8). One notes that the centroids for all six energies are below the prediction for the sequential decay. The theory has not been filtered, and therefore the average number of particles used in the event shape analysis is much greater than for the experimental data. It will be demonstrated that this multiplicity has a great effect on the extracted event shape. This comparison of the raw theory to the data highlights the importance of filtering the model predictions to account for the detection biases. F Comparison to Published Models Though the event shape distributions may be interesting, it is not until a comparison to model predictions is made that one can begin to draw conclusions about the nature of the disassembly mechanism. The distributions shall be compared to models sim- ulating a sequential emission of fragments and a simultaneous multi-fragmentation, which is. defined as a purely isotropic form of disassembly. 1 First Order Observables For the purpose of evaluating the overall performance of various models of nuclear disassembly a set of first order observables is introduced. These observables must be reproduced if one expects to be able to generate events that resemble those contained in the experimental data set. Discrepancies in these observables will cause differ- ences in the multi-particle observables that are unrelated to the effect of interest. The observables that must be properly reproduced are the total detected multiplic- ity of charged particles (neutrons and gammas are excluded because the detector is insensitive to these particles), the multiplicity of fully identified charged particles (full identification requires the detected particle to have sufficient energy to pene- trate into the slow scintillator), the mass distributions for hydrogen isotopes (the 109 only element for which isotopic resolution can be maintained after summing large numbers of detectors), the charge distribution of identified fragments, and the kinetic energy distributions for both protons and heliums. The total detected charged par- ticle multiplicity of an event has an extremely strong effect on the event shape. It is essential to reproduce this quantity accurately. The identified multiplicity is a check on the accuracy of the energy distributions of a model. If the model is producing incorrect energy spectra at low energies, the ratio of the multiplicity of fully identi- fied particles to total detected multiplicity will be incorrect. The tensor from which the sphericity and coplanarity are determined uses the momentum of each fragment, which is determined from the kinetic energy and mass of the fragment. It is there- fore also important to identify accurately the mass of light fragments. For heavy fragments, isotopic resolution is less important, however one must reproduce the ele- mental distribution. The heavy fragments contribute the largest momentum vectors to the calculation and dominate the measured shape of an individual event. Though the ratio of the identified multiplicity to the total multiplicity tests the low end of the energy spectra, the top end must be checked as well. High energy light particles will dominate the event shape in the same manner as heavy fragments. The energy spectra define the apparent temperatures for the disassembling system. Models that assume thermal emission should reproduce the energy spectra. Figure IV.12 displays the first order observables for the Ar + V reaction at 35 MeV/ nucleon. Section a) of the figure displays the identified charged particle mul- tiplicity. These particles must have more than the 20 MeV/ nucleon energy which is necessary to penetrate into the stopping (slow) scintillator. Section b) displays the total charged particle multiplicity. This includes particles in the energy range from 4 MeV/ nucleon to 20 MeV/ nucleon that stop in the AE (fast) scintillator. The mass distribution of light charged particles is shown in section 0) and the charge distribu- 0 5101 50123 110 40 50 100 150 200 0500?:TTIIIIIIIITII1::mllllfllIIlfllllillE-I:lIllITTIIIIITIIIIIl_, 0500 1—x X -— ——)5< 1 X X 0.100";- x j;— X xii-T —§ 0.100 0.050:— —E:— x I 2 —§ 0.050 _ >< __ x “X >11< . 0.010?- 1:5— (+10) ‘2';— >5< *g 0.010 E 00055—— X —E.%- —::— >1; ——_ 0.005 0; : a) :: C) :: e >1§< : m 0001 E-IJLI l L1)I<>S§S< —- 0.100 O 0.050;— X {— x —;§—>< xx ——3 0.050 U L. X x _: -p >§< .. X 0010:— x ‘3:— (110) ‘sa— ); -: 0.010 0.00573— -EE- ' X ——:~ >< ‘ 0.005 : b) X :: d) :: I"): “i 0001 Emma... 1.725.111111111155;1.1.. 1.0.11 @001 0 5 10 15 0 1 2 3 4 50 100 200 300 400 471 Data Ar+V 35 MeV/n Multlphclty Energy (MeV) Figure IV.12: The first order observables for the system 40Ar+51V at 35 meV / nucleon. a) The identified multiplicity, b) the total multiplicity, c) the mass distribution, d) the charge distribution, e) the proton kinetic energy spectrum, and f) the helium kinetic energy spectrum. 111 tion in section (1). The proton and helium kinetic energy spectrum are displayed in sections e) and f) respectively. The energy has been transformed to the center-of-mass reference frame and summed over all angles. This figure represents the benchmark that a model must meet before the multi—particle observables can be compared. 23 (IELIHNI GEMINI is a model for nuclear reactions that simulates a sequential binary fission process. It has been employed in several studies that have attempted to differentiate between sequential multi-fragment emission and simultaneous multi-fragmentation [Bowm87, Wozn88, C01089, Poch89a, Poch89b, Trau89]. It is an extension of the Weizsacker model nuclear decay. The model was developed from the asymmetric fission work by Moretto. The details of the model are reviewed in a paper by Charity [Char88]. Since this model had received such wide scale use in similar analyses, it was the first simulation employed to describe this data set. Figure IV.13 demonstrates that this model is not satisfactory, because it fails to reproduce the first order observables. The model tends to produce too many protons, while failing to generate a sufficient number of IMFs. 3 Lopez Lopez and Randrup have developed a simple model for sequential binary fission which they have employed as an alternative to multhragmentation in a paper that explored several possible analytical methods to differentiate experimentally the two processes [L6pe89a]. The model considers an excited nucleus in its rest frame. For a nucleus of mass A, A / 2 possible exit channels are considered. For the channels with an odd number of nucleons, the extra nucleon is randomly determined to be either a neu— tron or a proton, therefore, the emitted fragments can never vary from the Z=N line 112 0 5 10 15 0 1 2 3 4 0 50 100150 200 O500_rlfillrIIITITI[w1_Hilérnrlrflllilrl‘fi IifilerTlllillllllld 0500 ixx a) -' ’ a) ‘ "x 092 o -. «- X - 01005—71; i=5— X ® 55 —5 0.100 0.050:— “p —5:-— X x—EE—xfi‘ —: 0.050 : >< : 9 IX : 0.010% l -—__a_— (: 10) 03g— ——§ 0.010 *5 0.005 :— X l) ——:E— :3— x -—: 0.005 0 : a) 1 1 C) 2: >5. : > X La 0.001;- X 1:5“ >¥ 1‘ 0.001 \ 0,500:..+1ll%+41)a1+t}a+33 ”illfidTTaaIj:r+%+ln.+w:lLr1++j 0.500 m _ __ _. _ -+—9 * 3526320) " x ‘L 1 Q X 0.1005“ (1) X '25— 1r1§< —= 0.100 :3 = X 9 :3_ x is... >5< o 0050;— 4; ~:. 1- x 0.050 U :x x0 :: .: M : 1 ® x X 0.010 E‘ X if” (-10) ‘2?— XX “—5 0.010 0.005 :3— b —€E— id) x -EE- )XX —5 0.005 : l -: _- X . 0.001 é-.1Ll1_‘l ll_l l , 14' 11:: LLillJJilLllLlJJJLLlITEE—l 1i HELJLLiLll—ig 0.001 GEMINI 0 5 10 15 0 1 2 :3 4 50 100 200 300 400 Ar+V . , _ 35 MeV/n Multipholty Energy (MeV) Figure IV.13: The results of the GEMINI simulation (circles) compared to the exper- imental data (crosses). 113 by more than a single unit. Additionally, emission of both neutrons and protons is considered for all decay decisions (expect those for which emission of one of the two particles is forbidden - i.e. 2H8 decaying by neutron emission). The truncated list of exit channels is used to reduce the CPU time required by the simulation. For each exit channel considered, a fission barrier is calculated, and a relative probability determined for each channel based upon 1/lifetime(sec) for decay of the parent into a given exit channel. Having chosen a channel, an emission energy is determined. The directions are randomized, and the recoil determined to get the final momentum vectors for the two daughters. The daughters continue to decay until they either dis- assemble into their constituent nucleons or they no longer possess sufficient excitation energy to overcome the fission barriers. Using this routine, one is able to generate a set of events which can be compared to predictions from multi-fragmentation models. Unfortunately, for the purposes of this analysis, the results from Lopez’s sequential binary fission model failed to reproduce the basic observables from the 411' data set. Figure IV.14 displays the predictions from Lopez’s model against the experimental data at 35 MeV/ nucleon. The main problem with Lopez’s model is that it tends to overproduce single low energy neutrons and protons. The model fails to produce the IMFs that are characteristic of the data at this energy. To be fair to Lopez, the model was not designed to be used at these energies. The channel selection method may accurately describe a cold nuclear system, but when there is an abundance of excitation energy it does not reproduce the enhanced production of heavy clusters. G Other Effects That May Induce an Observed Elongation Lopez and Randrup suggested that a spherical average event shape could be used as a signature of multi-fragmentation [Lope89a]. But this does not necessarily prove 0 5 10150 1 2 3 40 50100150200 0,500 - Xllllfll[lllll I::TTTTl'llll]llllrriTllliT-Jllfrll[Tllllilljll: 0,500 Roxx " “ Q; " 0.100 _— x —55— 0 x 5;” >3 —_ 0.100 0.050 :—‘D 5E- X :2 a); —3 0.050 : x :: x ::>< x, : - :: -.-10 :: : E: 0.005 E— (3 X —I:_ ( ) —_:- >5; —: 0.005 EB : a) :: C) :: )g : m 0 001571..1..X.1..r1—*.1111"“11X15: g-ggg, > O 500 :Ggl jW—T—l lfiIT—F::}TTIIFTITIFTTTIFTFr[l—rrfllf:llT‘V‘llmlTIllI'le: , -+--J " QXXXX “" x T ~ g 0.100 5— X X 3“ ® 13.2% -: 0.100 O 0050:—~ 0 X €5— X -—:——>< X -—: 0.050 D :X X ‘7‘: I Xx : ' 0.01.05“ >< "Eg— (;1O)X "E? X: ‘§ 0.010 0.005 E1- —E:— ' X —.—:— X —3 0.005 — 10> -- d> -- 0x - 1.. X __ __ X _ 0.001 EulLllglJLllLlll[TE-‘l-lLllllliJJLliLlLJiLll'TEETI-TLlil¢1?1_llI'll; 0.001 LOPEZ 0 5 10 15 0 1 2 3 4 50 100 200 300 400 Ar+V 35 MeV/n Multiplicity Energy (MeV) Figure IV.14: The results of Lopez’s simulation (circles) compared to the experimental data (crosses). 115 that an elongated event shape, as is observed in all of the systems that we have studied, can be used as evidence that a sequential disassembly process is at work. Several other effects that were not considered in Lopez and Randrup’s static analysis can also induce an elongation in the event shape distributions. Examples of such effects include finite multiplicity, rotation of the interaction region, collective flow, and spectator matter. 1 Finite Multiplicity Effect The most trivial of these effects is an artifact of the limited multiplicity of the events used in this analysis. Since all of the available phase-space is not filled by an event with only a finite number of momenta vectors, a distortion is observed in the measured event shape. Only in the limit of infinite multiplicity will an event corresponding to isotropic emission have a purely spherical shape. Consider the case of only two fragments, the emission pattern may be isotropic, however the measured event shape must be two dimensional (the minor axis of the spheroid has to be zero). There is a substantial distortion of the measured event shape even for events with as many as 100 outgoing fragments. This finite multiplicity effect was studied using some simple simulations. The first of these studies allowed only the emission directions to vary. All emitted particles were arbitrarily chosen to be 50 MeV protons. 1000 events were generated for each multi- plicity up to 50. Figure IV.15 displays the trajectory of the average event shape from this simple model on Sphericity/ Coplanarity space (S-C space) as a function of multi- plicity. The multiplicity-one events all have a (0,0) event shape. The multiplicity-two events produce a distribution along the line from (0,0) to (0750,0413). The higher multiplicities begin to fill out the S—C space. As the multiplicity is increased, average event shape follows an arcing path towards the (1,0) limit. 116 030 I I I—fT I f I I l I I Ifi I I I I l I I I I : >< M220j 0.25 — a j 0 M=30j : U M=40j 3 0.20:— + M2507 .g : : g 0'15 E— 3 4- 5 6789 —: "g. i ,2 X o I C) 0.10:— 8.; -: l. 1 0.05 :— _: i Z r 1 000 P- 1 14 i i 1 m 1 l L #1 1 l i 14 4 mi; L_1 ; 0.0 0.2 0.4 0.6 0.8 1.0 Sphericity Figure IV.15: The trajectory of the S and C centroids as a function of multiplicity. #————-——-———— - —- ., - -w———— 117 In order to make this simulation a little more interesting, the energy of the frag- ments was allowed to vary, and the detection biases were added. This version of the simulation was given a temperature, velocity, and multiplicity. The only allowed particles were again arbitrarily chosen to be protons. However, their energies were assigned based on a thermal model using the temperature. The directions of emit- ted particles were chosen randomly (isotropic in the rest frame of the source). Each fragment was then individually transformed into the laboratory reference frame and filtered by the detection acceptances. If a particle was rejected by the filter, another was chosen. Protons continued to be produced until the required event multiplicity was achieved, at which point, the event was analyzed. This moving thermal model contains most of the features of the real events. The multiplicity distributions and proton kinetic energy spectrum could be constrained to agree with the experimental data. The simple simulation lacks the variation in fragment charge and mass. The added energy variations and detection biases tend to depress the measured sphericity compared to that expected from the simpler model at the same multiplicity. For sim- ulations constrained to multiplicities of 24, the simple finite multiplicity model yields an average sphericity of 0.65 while the filtered thermal version yields a value of 0.62. 2 Rotational Distortions A rotating system will emit particles the momenta vectors of which are enhanced in the plane of rotation. This sort of emission pattern should produce observable distortions in the measured event shape. A simulation has been performed to test the nature of this effect [Li89]. The model considered a rotating spherical source which emitted thermal protons from its surface. The primordial emission spectrum was considered to be isotropic. A rotational component was added to the momentum of the emitted proton based upon where on the surface the particle was produced. 118 This effect has been studied as a function of multiplicity. Figure IV.16 displays the trajectories in sphericity/coplanarity space of the centroids of the event shape distri- butions predicted for four different rotational frequencies. An extrapolation of the multiplicity to infinity will produce centroids that lie on the line representing oblate spheroids (from (0.75,0.413) to (1.0,0.0)). Figure IV.17 provides a detailed view of the rotational effect for multiplicities around eight. These multiplicity values cover the range of the measured experimental multiplicity distributions at the six beam energies. The experimental rotational frequency is estimated to lie between rw=0.0 and rw:0.1 [Tsan86]. For events with an average multiplicity of eight, we expect a measured elongation of approximately -0.01 units of sphericity and +0.001 units of coplanarity. Additionally, the rotational effects should be strongest for collisions with large impact parameters. For central events, there is little initial angular momen- tum. In this analysis we select exclusively central events which should minimize the rotational distortions. 3 Collective Elongations Reaction dynamics will also potentially add some collective effects. If the target nucle- ons and projectile nucleons retain some memory of their origin, their final trajectories may tend to form two jets. At low energies, the mean field attracts the projectile to the target and a partial orbit is carried out during the reaction. The fragments origi- nating from the target will be scattered to negative angles with respect to the impact parameter. At high energies the collision occurs on a more rapid time scale. Before the participants are able to orbit, the nuclei interact and form a highly compressed region. The target and projectile remnants recoil from this collision and the target fragments are preferentially scattered to a positive angle. At some intermediate en- ergy, one expects these two opposing effects to balance out to create a system which 119 0.5 : l l l l I l I f—l I l l l T—[ l l l T l I f l .— _ Angular Velocity : 0.4 L X rca=0.0 .1 f; : <> rw=0.1 : ‘g : CI m=0.2 : g 0-3 .— + rw=0.3 +,+ —- -— "+-,~’ _ co 2 ,.+"‘ £3 ; "a 0.2 ;— ”VI-"T EJEMBB’ T 8 : 4.5+ — "1...... .. -. .. : 0.1 :— ‘3 O O ; l l L l l l l 4 1 lg! I l l l l l J l i I l l i T 0.0 0.2 0.4 0.6 0.8 1.0 Sphericity Figure IV.16: The trajectories of the S and C centroids as a function of multiplicity for emission from a rotating source. 0.20 {>, r: 0.18 L. 2 .52 0.16 8* 0 o .14 ,.1 .u' .u- u' I ’ I I f o ’ 'w ’ .1 ’ _ f T f f lv'vj Tili I—l m /l 1 I r , - : Angular elocity : L X rw=0. 0 3%,." __j I 0 rw=0.1 : : Cl rw=0.2 .— + rco=0.’3,.«+’ ............. ‘1 I [fag/i, "/ ,--————.‘j_f'ffl£73. ....................................... 0.12:“4 l__J_l 1 1 L’J lnI 1 L l L1 1m 0. 25 0.30 0.35 0.40 0.45 Sphericity Figure IV.17: An expanded view of the trajectories from the rotational simulations in the region of the multiplicities that are measured for the experimental data. 120 is neutral with respect to preferential scattering angles. The question of where this transition occurs has recently been probed by several groups [Krof89, Zhan89, Ogi190]. In addition, the magnitude of the collective motion has been measured for the Ar+V system at 35 through 85 MeV/ nucleon [Ogi189b, Ogil89c, Ogil90]. Since the magnitude of this effect is known, the influence that it should have on the event shape analysis can be estimated. Figure IV.18 displays the predictions from various models. The simulations presented in this figure are all constrained to have a multiplicity of 24. Considered are rotational effects, collective motion effects, finite multiplicity effects, and the effect of the detector biases. The collective motion does not create a large elongation compared to that predicted by sequential decay. These calculations were not done for each of the beam energies, however, one can confidently say that the magnitude of the effect will diminish as a function of increased beam energy because it has been shown that the minimum collective motion is induced for reactions around 85 MeV / nucleon [OgilQO]. This is the energy at which the balance between the attractive scattering and the compression recoil occurs. As with the rotational effect, the collective motion should diminish for central events. H The Sequential Simulation A new sequential simulation has been developed which is able to match accurately the first order observables from the 47r data set. The simulations that had been developed by other groups had not been able to accomplish this feat, and as has been demonstrated, failure to reproduce the basic observables invalidates the comparison between the multi~ particle observables from the experimental data and the predictions from a simulation. By deveIOping a new model, we were able to control the algorithms 121 Comparison of Effects (M=24) 0,250 — I I I I—[ I I I I 5 I j I I I I T I I T I I I I l I V I I - t + - l X 2 : 0.225 e m 0'0 — ; 0 rco=0.l : : CI rw=0.2 : :>. 0.200 L— + rco=0.3 ‘— 44 I' - C r )1 Flow=0 - (U : “P Flow-10 C] : C“. u. - .... 2 0175: x Flow-:20 : 8* : as Flow=30 ; O 0.150 :— 0 F.M.S. ,3 x11, @ —: : # Thermal #, 0X I 5 @ Filtered O 1 0.100—141nl1114l1144L;11II1111lmm11-i 0.40 0.45 0.50 0.55 0.60 0.65 0.70 Sphericity Figure IV.18: An overview of the relative magnitudes of the various effects that induce elongations. 122 which determine how the simulated nucleus decayed. The simulation considers a. nucleus at rest which has a given amount of internal excitation energy. The nucleus is allowed to dissipate this energy only through the emission of energetic fragments. These fragments were allowed to range from indi- vidual nucleons up to nuclei the masses of which corresponded to symmetric fission. For decaying nuclei with fewer than 12 nucleons, every possible exit channel was con- sidered. To each channel, a barrier energy was assigned based on the rest masses of the two daughter fragments (reaction Q value). The binding energy per nucleon is highest for “Fe. Since this simulation models decay from heavy nuclei into light and intermediate mass fragments, most channels will have a positive barrier energy. However, if the decay chain leads to a nucleus that is far from the valley of stability, it is possible to assign a negative barrier energy to an exit channel. A relative weight for each channel was determined based on the temperature of the excited nucleus. Using the principles of thermodynamics and an estimate of the level density as a function of excitation energy, the temperature of the nuclear system can be deter- mined. The entropy (S) is related to the natural log of the level density (Q(E)): S = kin IKE) (IV.18) where k is Boltzmann’s constant. The temperature is then related to the partial derivative of the entropy with respect to energy: 1 85 .1: __ 5.17:, (IV.19) which leads to: 1 1 BIKE) Er- — “(El 8E (IV.20) The density of nuclear energy levels increases rapidly as a function of energy. If the nucleons are considered to be a degenerate fermion gas contained within a box the 123 size of a nucleus, the level density is proportional to e‘@ [Beth61]. This density of states yields 1/kT oc l/x/E. Blatt and Weisskopf propose a level density of the form 0(3) = Ce“E (IV.21) where C and a are constants [Blat52]. Solving equation IV.20 yields l/kT = W. Thus, the temperature of the parent nucleus can be calculated from the excitation energy (E ‘) using the formula T = \/E—*/—¢i. The constant a is the level density param- eter, which is assigned values between A/12 and A/8, where A is the mass number. Adjustments in 0. affected the calculated temperature and therefore influenced the ki- netic energy distributions of the final fragments and the rate at which the thermalized system was able to dissipate its excitation energy. From this temperature (T), a probability was assigned to each channel using the formula P = 6*” , where b is the barrier energy. A relative probability is assigned to each channel, and a selection is made based on this probability using Monte Carlo techniques. For systems with greater than 12 nucleons, the total number of exit channels that were considered was truncated to reduce the amount of computational time required by the simulation. Only one combination of neutrons and protons was considered for each possible exit mass. For even mass channels, the emitted fragment was assigned the same number of neutrons and protons. For odd mass channels, the extra nucleon was randomly assigned as either a proton or a neutron. This method of choosing an exit fragment tends to lead to a single remnant which is neutron rich. The initial system started with a neutron excess, and the early decays (while the mass of the parent system is above 12) emitted fragments with approximately equal neutron and proton numbers, leaving the initial neutron excess with a single remnant. Once the mass of the remnant falls below 12 it was allowed to dissipate these extra neutrons. It is possible that this method of channel selection produces an 124 enhancement of neutron rich light fragments. The 4x array does not have isotopic resolution for most elements, so this feature of the simulation can not be tested with a comparison to the experimental data. Having selected a decay channel, a separation energy is picked from a thermal distribution using the mass of the emitted fragment and the temperature of the sys- tem. To this thermal energy, a Coulomb repulsion is added assuming a separation distance at emission of 1.22(A:/3 + Aé/a) Fm. The recoil momenta of the two daugh- ter fragments are calculated from this separation energy assuming a random breakup direction. These recoil momenta are added to the momentum of the parent nucleus to get the total momenta of the daughters. In order to determine the internal excitation energy of the two outgoing frag- ments, the separation energy and the reaction Q-value are subtracted from the ex- citation energy of the parent nucleus. If neither daughter is an individual nucleon, the remaining excitation energy is shared assuming equal temperatures of the two daughters (E; = E‘fij, E‘ = E*%: where E“ is the excitation energy remaining after subtraction of the barrier energy and the separation energy from the original excitation energy, A1 is the mass of the first daughter fragment, A2 is the mass of the second fragment, and AP is the mass of the parent). If one of the outgoing fragments is a nucleon, then the other fragment keeps all of the remaining excitation energy. If both final fragments are nucleons, then the remaining excitation energy is added to the calculated separation energy. As the decay is followed through its stages, the average temperature of fragments falls, because internal excitation energy is being converted into kinetic energy and into a collection of less well bound nucleons. This simulation differs from the others that had been developed in several ways. First, the method of assigning probabilities to the various possible exit channels uses the final state mass differences and not fission barrier heights. For systems with ex- 125 citation energies several times higher than the barriers, these barriers were no longer meaningful. Secondly, the method of assigning a thermal energy to the outgoing frag- ments differed from other models. And finally, the number of exit channels considered for light systems was more complete than other models that were examined. In order to compare the results of the simulation to the experimental data it was necessary to run the simulated events through the 41r filter. Since the simulation was carried out in the rest frame of the decaying system, it was necessary to boost the event into the laboratory frame prior to filtering. For analysis of the event, it was then boosted back into its original rest frame. Figures IV.19 to IV.24 display the results from the simulations (circles) compared to the experimental data (crosses) for the six beam energies. The input parameters for the simulation are given in table IV.13. The fit to the first order parameters is generally good. There are a few weak areas that should be noted. First, for 35 MeV/nucleon, the simulation underpredicts the number‘of deuterons. This is not a problem for the other beam energies. Second, for the low energies the simulation underpredicts the production cross-sections of heavy elements. Finally, for high energies the simulation overpredicts the low end of the helium energy spectrum, and consequently overpredicts the number of heliums. These are minor shortcomings. Overall the fit is far superior to that of the other models discussed earlier. I The Simultaneous Simulation The event generator F REESCO [F ai86] creates events as if they were produced from a simultaneous break-up, however I chose to develop an original simultaneous simu- lation. This was necessary because it was essential that the simultaneous simulation and the sequential simulation agree with resect to the fist order observables. The 126 5 10 15 0 1 2 a 40 50 100150200 0500 "’I‘TFI"’T'jfi'W‘”'lrml'ml‘:‘1'I“"I”I'l”: 0500 Rugs “ "(a " 0.100? X —E___=" Q 955% —§ 0.100 0050?— (p Jif— X H:— % ‘75 0.050 r X T 0 a «X a — +) 0.010? GP €37 (+10) 2;.— —§ 0.010 C. '0.005E- X 7:— C) 7:— 8) —: 0.005 .. — a1 ._ ,_ _ m 0'00155.1.3811”.I.Tiéfml....l....l....lfii..l.... ”‘77 0001 E 0500 ””l‘mlm‘l‘:‘ml‘mlml'"l"”I‘Er""l”"l"' ”'E 0500 +9 7 “”9 I7 5 77 “33, 7 g 0100?: 59 —:_— 9 —§§-—x —§ 0.100 O 0.050:— —3;— —.:—X —. 0.050 U :5 x :: X 1: : 0010; ix 75:7 (+10) . 72:77 if? 7: 333g ”“051 b) I. i d), :E 0H 3 0.001—1117—n111f3—15mi‘lll573 00“ cm 0 5 10 150 1 2 3 4 50 100 200 300 400 Ar+V 35 MeV/n Multiplicity Energy (MeV) Figure IV.19: The results of the sequential simulation (circles) compared to the first order observables from the experimental data for the system Ar+V at 35 MeV/ nucleon (crosses) . 127 0 5000 5 10 15 0 1 2 3 4 0. 50 100150 200 _ _ III'IIIIIIIIIrI IjFIIITTIIIlImlIIIII LCIIIIP—IIIIIIrIlII : 0.500 0.1005 0 j;— x Ell—E? —§ 0.100 0.050E— X —:—- 5 6 ji— —: 0.050 I (bx " IX 1 0.010— —_e—- . ‘5:— —- 0.010 E :E 7710 :: E *5 0.005 :— ‘IX 1— ( ) —::- —. 0.005 a; t a) 1: C) i: : m 0001Ellli111i]l[Il—l—‘IE'E'TTllillllIlllllllll'j-E-f-Il I? 0001 \ 0500 f__IIII}IIIIIIIIIIII:FIIIIIIITIIIIIIPIIIIWIIIILJII II: 0500 B : Q3139 ._ 6 -C % - g 01005—3:B @615 :5— : jg- ‘g 0.100 c 0.050 E—x‘D X 1‘:— 23— 5% —: 0.050 0 I0 43.. -— >< 1— %, - 0.010 57x7 (i, jig—— (—10) x 7'55— —§ 0.010 0.005 X —:Z— €2— I} —1‘ 0.005 : .1 ¢ - -1 L b) T -1: d) ,1, xi f) ‘ 0.001E11Lillllll ElljflLkLLllilllllllUillllEE—ITJJilllll ll Ill-E 0001 cm 0 5 10 15 0 1 2 3 4 50 100 200 300 400 Ar+V 45 MeV/n Multiplicity Energy (MeV) Figure IV.20: The results of the sequential simulation (circles) compared to the first order observables from the experimental data for the system Ar+V at 45 MeV / nucleon (crosses) . l____ .. 128 o 15 o 1 2 40 50 100150 200 0.500 IIII IIII (III II IIIIIIIIIIIIIIIIIIIIII IIIIIIIII'IIII[ 0.500 ‘5 o 0.100 g 0.100 0.050 a 0.050 0.010 . 0.010 +10 E 0.005 ( ) 0.005 2 ail C) m 0001 II IIIIII>I , :III lllIIlIIII[II:E-ITTT$ I I I I:: ©[ l l : 4') - QQQ .. 9 fl _ g 0.100 g— m? g jg— X _§§_ X —§ 0 0.050:— X 55 —:E— :— —_ o : X x :: X :: : X X 0.010 g-X‘P 4” {=— (+10) an ’3? fix -—§ 0.005; x {E— d o ‘1— f) ‘3 _ b) I :: l q); _ IX x - XX 0001 :Ililjljilll IIESIIIJIllIllIIII'IIIIIIlI—l—EIliIIlllIIIILII 11% cm 0 5 1o 15 o 1 2 3 4 50 100 200 300 400 Ar+V . . 65 MeV/n Multiplic1ty Energy (MeV) 0.500 0.100 0.050 0.010 0.005 0.0 the OH 0.100 0.050 0.010 0.005 0.001 Figure IV.22: The results of the sequential simulation (circles) compared to the first order observables from the experimental data for the system Ar+V at 65 MeV/ nucleon (crosses) . —_L -— 130 O 5 10 15 O 1 2 3 4 O 50 100150 200 O_5OO_111I I_I—fI Illl TT_III_FIIIITTII I—WI lLIl-II irTr Illr III: 0500 1 I I _ I I I I _ I I I 1 b we —— g e -— 4 0.100 is“ 9 —— 5‘ a 355% 0.050 :—>< ’3 _ _:E_ ~23- :qu Q5 I :_X X ---.::~- '1:— .._1 0.010 E— 55 (+10) 25 0.005 if T?“ E a) X -- C) -- 93 j “I “ m 0.001 g— X “5';— —:§— } 0.500 :14+'I+++41I,.+1 14,3;155'Llaflalfilfie51Tuf w -I—-9 '- 000 -~ G) «- C: o 100 :— ‘DXXXQ —_=— X —__—— :5 ‘ 5 6 X 35 55 O 0.050 :— X 0X ‘TT‘ fr— 0 ' ch (P ‘5 X ‘r _ X i- -_ 0.010 5} x j;— (_._10) a) x fig— % “g 0.010 0.005 :—X— «I i: ' 0 if:- X“€ 0.005 : f b) TX :: d) : 1°) fig“ X 0.001 '%J I 1 I41 LII—ILLL Ix-Té:ILIILIII_IIJ_IILLIILIILI I E-I—LIJAI IilIJLLlLII VII 0,001 CEQ 0 5 10 15 0 1 2 3 4 50 100 200 300 400 Ar+V 75 MeV/n Multiplicity Energy (MeV) Figure IV.23: The results of the sequential simulation (circles) compared to the first order observables from the experimental data for the system Ar+V at 75 MeV / nucleon (crosses) . 131 o 5 1o 15 o 1 2 4 o 50 100 150 200 0500 Trill[IrrlrirnII1_-HJT;IIIllrrrlllfilri:1fliTTJIrlIlfllfirrl: 0500 r m0 “ " ‘ 0.100;- x6 x92 —-_:~ 3‘3 x—ggg . —_ 0.100 0.050 E-X‘D 92 —:5— —:5— “tr —: 0.050 .. g __. +X .. F q) a -- 1- .. o 010 j: >‘< 3— ——-— — o 010 ' E E —10 EE E ' *5 0.005 _ -—+—- ( ) —::.— 0.005 a; :x a) f i: C) 1: e) f Lil 0.001 : I L X] ‘35—“ I LL. IE:— | I 1 0.001 E 0500:11rTl‘TilTlr’FIfli :iITWI‘WIIITfl'WHlTTflI __lr1_I—lTTIFl’VIVFIITFI: 0500 _ _ G T _ +J 929992 ' x I m g 0.100 :I__— 556 ”a —__:5—‘ 7;— —__-. 0.100 O 0.050 E- X —:‘— i— —5 0.050 U h (I) 6 :. x I : l 4 I 0.010 :— E -—.J:..—— (410) o ‘32“ (We; 0.010 0.005 E—X X-Er— ' X—I— 0.005 *- b) X r d) “D" f) _ x j “I- 0001 :1+1LJ_IJ_I1414%[:iuIUIJIuIquJLuLEEI—JHIJJHIIIIIILIII OiOOl cm 0 5 10 15 o 1 2 3 4 50 100 200 300 400 Ar+V . . . 85 MeV/n Multiplicnty Energy (MeV) Figure IV.24: The results of the sequential simulation (circles) compared to the first order observables from the experimental data for the system Ar+V at 85 MeV / nucleon (crosses) . m“ om" 44>" -.. o..a.-. ~.- ~. -. ' _.. .._\_ _ .. '... 132 Table IV.3: The input parameters for the simulation for the six beam energies that were studied. The simulation attempted to reproduce central events from the reaction 4"Ar +51 V. Incident Energy A0 Z0 Excitation VCM (MeV/ nucleon) ( MeV/ nucleon) (c) 35 76 33 8.0 0.1200 45 76 33 10.3 0.1370 55 73 31 12.6 0.1516 65 70 29 14.8 0.1648 75 68 28 17.1 0.1771 85 70 29 19.4 0.1887 multi-particle observables have been demonstrated to be sensitive to variances in the basic observables. In order to ensure that the simultaneous simulation agrees with the predictions of the sequential model for the first order observables, events that had been generated from the sequential code were modified to create pseudo-simultaneous events as discussed below. The sequential simulation has already been shown to agree with the experimental data, and it was determined that it would be easier to create pseudo-simultaneous events from events from a sequential model than vice versa. It is easier to remove the kinetic correlations contained within the sequential events than to try to create some correlations in an event that had been generated with a simultaneous model. Several different methods of modifying the sequential events were considered with varying degrees of success. 1 Randomized Directions, Energy and Momentum Con- served The first attempt to create simultaneous events used complete unfiltered events that had been generated from the sequential simulation. The final fragments were then all randomly assigned new emission directions. In order to ensure momentum conser- 133 vation, the recoil momentum of the entire new event was determined, and then the event was boosted into a frame where it was at rest. This affected the overall en- ergy conservation, therefore kinetic energy values for all of the fragments were scaled so that the sum of the new kinetic energies equaled the sum of the original ones. This randomization of emission directions removed any correlations in the fragment distributions generating an isotropic event. The new events were then transformed to the laboratory reference frame, filtered, transformed back to the rest frame of the decaying nucleus, and analysed with respect to the event shape. Unfortunately, though the technique seemed quite elegant because it conserved momentum and energy, the removal of the intrinsic correlations between the fragments resulted in larger average relative angles between fragments. There was no longer any primary axis with preferential emission to the poles. This resulted in fewer fragments being rejected by the filter code due to multiple hits on single detectors. The net effect was an increase in the average multiplicity of the events in this simulation as compared to the previous one. As has been demonstrated, the event shape parameters are extremely sensitive to multiplicity, therefore, another method of producing simultaneous events had to be developed. 2 Randomized Directions, Multiplicity Constrained I continued using the events which had been generated by the sequential simulation because that method introduced the fewest simulation dependent changes. However, the multiplicity had to be artificially constrained, otherwise the difference between the average event shapes from the two simulations would contain a significant multiplicity induced contribution. This second version of the simulation started with events from the sequential simulation, randomly chose fragments and redirected them one at a time. For each re-oriented particle, a check was made as to whether it was accepted 134 or rejected by the filter. The routine would continue to pick particles randomly and randomize their emission directions until the accepted number was equal to the number that had been accepted for the sequential event. Therefore, the multiplicities and particle make~up of the two events were identical, the only differences had to be related to the emission directions and resultant kinematic correlations. 3 Coulomb Trajectories Selecting the emission directions completely randomly does not fill phase space prop- erly, because it does not discriminate against production of pairs with low relative momentum. The Coulomb interaction between fragments causes pairs of particles which are emitted close in phase space to repel one another. This reduces the relative production of low relative momentum pairs. An approach that accounts for this pro- cess starts by distributing the final fragments in the sphere of a given radius, and then follows the Coulomb trajectories in order to get the final directions and energies of the particles. This technique reduces the probability of finding two particles emitted with a very small relative velocity and should therefore reduce the fraction of particles that are rejected due to multiple hits. This method suffers from the same problems as the simple directional randomization without multiplicity constraints; the accepted multiplicity is larger than that from the sequential simulation. This complicates the comparison between these two simulations, however, an informative comparison can be made between this simulation and the simple directional randomization. The sim- ilarity between those two cases suggests that the directional randomization technique is as valid as the more detailed trajectory calculations. Therefore, I did not use the Coulomb trajectory method in our analysis because it was not necessary and could not control the multiplicity. 135 J Determination of the Reaction Mechanism The goal of this study has been to differentiate between two possible descriptions of nuclear disassembly. The first mechanism is slow sequential binary decay. This pro- cess is an extrapolation of a model that has worked extremely well at lower energies. Though the constraints of the model must be relaxed in order for it to be able to reproduce the basic observables, it does not represent a new decay mode. The simul- taneous multi-fragmentation is a rapid disassembly of the nucleus. It is an emission process that is qualitatively different from the emission processes at lower energies. The onset energy for multi-fragmentation can provide information about the equation of state of nuclear matter [Sura89]. In order to demonstrate that the event shape analysis technique is capable of resolving between the two alternate models for nuclear disassembly, the predicted shape distributions from the two simulations are plotted together in Figure IV.25. For clarity, the sphericity and coplanarity parameters for only one hundred events from each model are displayed. The large X’s correspond to the centroids of the distributions. The uncertainties of the centroids are smaller than the width of the lines of the X’s when calculated using the full 40,000 events generated at each energy by the simulations. There is a clear separation between the two centroids which indicates that this technique can resolve between these two models under experimental circumstances. It has been suggested that correlations in velocity and emission angles can effec- tively be used to differentiate between these two possible decay mechanisms [Lope89a, Gr0589]. Several exPerimental groups have employed this type of analysis to study the question of the decay time-scale [Troc89, Boug89b, Pou189, Klot89]. Unfortu- nately, the velocity correlations are most sensitive when two heavy fragments are 136 0.6 I I I I I I I I I T—I Ifi I T I l I 771 I I T I : : + Sequential O Simultaneous : 0'5 ”:— Average S=O.284 Average S=O.332 "E 3‘ : Average C=O.127 Average C=O.137 ; g 0.4 _— cm fl G E 00+ 0 E g 0 3 7’ t+fjo+o __. 1-—-1 : 0821.0 3; + O : Q1 - 1. + - O 02 ..— Oggfig + o O __ U : 0 RP: 03-1-00 : .. +Oigg O 0 0+ _ ._ + 00 O _ 0.1 __ #42?" +0 +o+ O O O _ _'_ @431” 0+ 0 g : O O m 1 1 f 1 n 101 T 1 1 1 C1) 1 1 1 1 1 l 1 1 1 1 0.0 0.2 0.4 0.6 0.8 1.0 Sphericity Figure IV.25: A scatter plot showing the sphericity and coplanarity values for 100 events generated from the sequential simulation (crosses) and for 100 events from the simultaneous simulation (circles). compared, and the MSU 4w array had high detection thresholds at the time of these experiments. Figure IV.26 displays the relative velocity and relative angle spectra for the experimental data and for the two simulations. The solid line corresponds to the experimental data, the circles correspond to the sequential simulation, and the x’s cor- respond to the simultaneous simulation. There is little to distinguish the three curves in this data set. This technique works well for arrays designed for heavy fragments, but given the status of the MSU 411' for these experiments, the event shape analysis was the best analysis technique for studying questions related to the time-scale of the disassembly. Contour plots of the predicted shape distributions for the simulated sequen- tial break-up process and the simultaneous multi-fragmentation are shown in Fig- ures IV.27 through IV.32 for the six different beam energies. The shape distribution for the sequential events from the 35 MeV/ nucleon case (Figure IV.27a) is that of a 1500 1250 1000 750 Counts 500 250 500 400 300 Counts 200 100 137 Relative Velocity (c) .0 0.2 0.4 0.6 0.8 1.0 _ I 1 1 I 1 1 1 1 I—T r T 1 I r 1 1—1 I 1 1 1 Si L o°°°°o° j : o "x 00 j 1: 01¢ x 00 i E .' °9° 3 —- . i : x °' —:l L 09: «l : c"'09-. 43 I: 3. ‘38-. h-‘JTI—l .4 '1 Jfil’l ’1 i 1 l ' l ' |1 L1, 1L I1 1... .......... A _ l 1 3 @— .IW.. . fan-1;; '..... .. ’é "‘ —_i : I if?“ ”Ivy-saw M“ ff“ '. :11: x i E.— . ’. 4;" art: xx x 1" xxx" ”2‘ x l v 3.0-.51"ch —:~I : .1." x " "x " I? x - A ,. as, .3 C— . ‘7‘. v xd‘ x ’1‘ xxx ‘l I: me I:- ,"l V" . : ..."f" 2‘. . 3". 1 A 1_ I l l I i J_ 4 I ml i I O 50 100 150 Theta(re1ative) Figure IV.26: Distributions of the relative velocity and angle between any two parti- cles from central events from the reaction 100 MeV / nucleon Ar+V. 138 0.4 0.2 0.0 0.4 0.2 Coplanarity 0.0 0.4 0.2 a _l 1 I L 0.0 ’ '- 0.0 0.2 0.4 0.6 0.8 1.0 Sphericrty Figure IV.27: A comparison of the predicted event shape distributions from the a) sequential and c) simultaneous simulations to that measured for the b) experimental data for the reaction Ar+V at 35 MeV/ nucleon. 139 0.4 0.2 0.0 0.4 0.2 Coplanarity 0.0 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Sphericity Figure IV.28: A comparison of the predicted event shape distributions from the a) sequential and c) simultaneous simulations to that measured for the b) experimental data for the reaction Ar+V at 45 MeV/ nucleon. 140 0.4 0.2 0.0 0.4 0.2 Coplanarity 0.0 0.4 0.2 0.0 ~ 1 - ‘ '- . - 0.0 0.2 0.4 0.6 0.8 1.0 Sphericity Figure IV.29: A comparison of the predicted event shape distributions from the a) sequential and c) simultaneous simulations to that measured for the b) experimental data for the reaction Ar+V at 55 MeV/ nucleon. 141 0.4 0.2 0.0 0.4 0.2 Coplanarity 0.0 0.4 0.2 0.0 ' ‘ ' ' 0.0 0.2 0.4 0.6 0.8 1.0 Sphericity Figure IV.30: A comparison of the predicted event shape distributions from the a) sequential and c) simultaneous simulations to that measured for the b) experimental data for the reaction Ar+V at 65 MeV/ nucleon. 0.2 0.0 0.4 0.2 Coplanarity 0.0 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Sphericity Figure IV.31: A comparison of the predicted event shape distributions from the a) sequential and c) simultaneous simulations to that measured for the b) experimental data for the reaction Ar+V at 75 MeV/ nucleon. 0.2 0.0 >5 fl .1 {g 0.4 ~ 1: . .2 o, 0.2 — O .1 U 0.0 0.4 a 0.2 - 1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Sphericity Figure 1V.32: A comparison of the predicted event shape distributions from the a) sequential and c) simultaneous simulations to that measured for the b) experimental data for the reaction Ar+V at 85 MeV/ nucleon. 144 flattened prolate spheroid (a long primary axis and an extremely short tertiary axis). This interpretation of the shape distribution is made by observing the steepness of the contours with respect to the line from (0,0) to (0.75,0.43), which corresponds to two dimensional shapes. The elongation of the primary axis suggests a strong kinematic constraint caused by the initial decays of the system. These earliest decays occur when the system is maximally heated, thus these decays carry off the most energy and there is a large relative momentum between the two daughters. The later decays occur after the system has cooled and are less likely to define the principal axes. The simultaneous process does not contain the cooling through particle emission or the emission-by-emission momentum conservation that would lead to elongation in one or two principal axes. It should produce spherical event shapes in the limit of infinite multiplicity. The event shape distribution displayed in Figure IV.27c is on average less prolate (a larger sphericity) than the previous simulation. The same comparison can be made between the two distributions generated by the two simulations for each of the other five beam energies (Figures IV.28 to IV.32). The contour plots provide qualitative information concerning the degree to which the two simulations are able to reproduce the event shape distributions of the exper- imental data. However, it is easier to quantify the results of these comparisons by studying the centroids of the distributions. Figures IV.33, IV.34, and IV.35 provide three alternate displays of the sphericity and coplanarity centroids as a function of beam energy for the experimental data and for the two simulations. For the 35 MeV / nucleon case, the experimental centroids are almost identical to those predicted for sequential decay. For energies greater than 35 MeV/ nucleon the experimental centroids fall between the two predicted values indicating that the sequential model has failed for these energies. Figures IV.34 and IV.35 suggest that from 35 to 65 MeV/nucleon there is a progression from the sequential extreme to the simultaneous 145 0.16111'T'r'31" : ............ SEQUENTIAL _ —_EXPERIMENTAL ‘ - ----- SIMULTAN 0113 0.15 E ”3,69 l l L l >‘ -1 .1.) ,® - i: 4 g - 2 0.14 —- Cl. _ O . U r 0.13 ‘- 012 1 m 1 1 l 1 1 1_ n l L .1 1 1 i 1 0.25 0.30 0.35 0.40 Sphericity Ar+V Beam Energy 35 MeV/n 45 MeV/n 55 MeV/n 65 MeV/n '75 MeV/n 85 MeV/n OKXUOX Figure IV.33: The trajectories of the sphericity and coplanarity centroids as the beam energy is increased from 35 to 85 MeV / nucleon. 146 Ar + V ——— Central Events 0 Sequential D Experimental 0 Simultaneous 0.20 0.30 0.32 0.20 0.30 0.32 0.30 0.32 0.34 III—III I [IT I I l L1 I‘IIIII’IIIIITU TI? I [WITIIII :; : : 0'18- 35 MeV/n ‘3?45 MeV/n ‘EE‘55 MeV/n ‘E 0‘15 0.16E— —E[— {E— —: 0.16 _ -0 _: : 0.1415- G ——:E— e E] 6.3—5:— 0 13 9—2 0.14 g 5 5 SE is a - a: “25‘ “:i— ‘5:— 5 W 5... ” ‘: it 2 (U 0 10 L1: l l1 IlIl ILTlTIT I I‘f—rl Ii; ALT IlTl Il Illll III] 1I 71 Eli—:1 JITI III' [171 i 1 if lTIl l—— 010 C. : I I I I 2; I I I I 3: I I "I 4 _c_q 0.13_— ——_;— —_:—- -d 010 Q. E E E: f 8 0.16:- 111:— —_1__- 6 694 0.16 : i: <9 G) 1: E] 2 0149-0 I3 9—1— E] —E— j 014 : II: I: -1 0.12}— —€E— —:—— ——j 012 5 65 MeV/n 5 75 MeV n i: 85 MeV n ‘ O-IOT’ n1_1_l114_111_11l "7111111 11m?— l1nul11n1mm11 0-10 .30 0.32 0.34 0.36 0.32 0.34 0.36 0.38 0.34 0.36 0.38 0.40 O Sphericity Figure IV.34: SAvuag, and Chewy, for the two simulations and for the experimental data at each of the six beam energies. Ar + V -—-— Central Events 0 Sequential D Experimental 0 Simultaneous O_4-5 ‘- 1 I I 1M If I 1 I I If “If 7 7f M I —D ——I I ’—T d i 1 0.40 - @ — >4 " @ E J a: : g g f-3 0.354 a a J 02 I “ E 1- © .1 O... t @ e — If} r g ‘ 030 ~— 5 8 - ' g ‘9 .1 1- I d 025 1 m . 1 d 1 #1 n 1 l 1 m1 1 I 1_J_ L m 20 4O 60 80 100 Beam Energy (MeV/n) Figure 1V.35: A plot of SAvcmgc as a function of the incident energy for the two Simulations and for the experimental data. '2- ‘5M’J..— r 147 extreme. This progression is less clear when the data are plotted as in Figure IV.33. This figure plots the trajectories of the centroids in S-C Space as the beam energy is increased. The average sphericity and coplanarity values increase with beam energy. This effect is due to the increased multiplicity of detected charged particles. The multiplicity increase is a function both of the increased energy available for particle production and the increased efficiency for detection. The trajectory of the simultane- ous simulation closely follows the trajectory predicted by the simple finite multiplicity simulations as one would expect. Regardless of how the data are plotted, the experi- mental distributions never reach the predictions of the simultaneous simulation. This may be because the effects of other physical processes have not been considered (col- lective flow, rotation, spectator matter, etc.). Additionally, for the purposes of this analysis, simultaneous multi~fragmentation has been defined as isotr0pic emission, which may be an unrealistic requirement. This assumption was made to eliminate any spurious effects that may have arisen had the first order observables not been con— strained to be identical between the two simulations and as similar as possible to the experimental data. It was decided to start with a model of sequential decay and then generate an approximation of multi-fragmentation. It would not have been possible to start with a reasonable model of multi-fragmentation and then try to insert the kinematic correlation that would have been manifested in a sequential decay. There- fore, the failure of the sequential model for beam energies above 35 MeV/ nucleon is of greater significance than the failure of the isotropic simulation for the higher beam energies. K Conclusions I conclude from this analysis that the kinematical constraints and the time ordered emission implicit in a sequential decay process create an observable elongation in 148 the predicted event shape. There is clear separation between the centroids of the event shape distributions predicted'for a sequential decay and a simultaneous multi— fragmentation, indicating that this multi-particle observable is sensitive to the break— up dynamics. At 35 MeV/ nucleon, the event shape distribution determined from our experimental data exhibits an elongation beyond that expected from an isotropic simultaneous simulation. The observed distribution agrees with that predicted for the simple sequential decay. This suggests that the decay dynamics of this system at excitation energies of approximately 8 MeV/ nucleon are dominated by the same processes which have been observed at excitation energies below 5 MeV / nucleon. At 35 MeV/ nucleon, the center-of—mass velocity is well below the Fermi velocity and mean field effects are expected to be strong. Sequential decay is expected to play an important role in the de-excitation process. At higher bombarding energies, the sequential model is no longer able to reproduce the observed distributions. This sug- gests that the expected change to a multi-fragmentation reaction mechanism may have occurred. The predictions of the simultaneous simulation never reach the ex- perimental values even for the highest energies. This may suggest that the region above 35 MeV/ nucleon, which corresponds to available excitation energies of 8 to 20 MeV/ nucleon, is a transition region. There may be a mixing of the two processes, a mixing of events that could be individually characterized as one or the other process, or a separate transitory process. The failure of the simultaneous simulation may also be an artifact of the unrealistic requirement of isotropic emission, the sampling of im- pact parameters, the failure to account for rotation and collective flow, or the failure to exclude completely spectator matter for the cold target and projectile remnants. For the purposes of work, the onset of multi-fragmentation is defined as the point at which the sequential model fails, which is above 35 MeV/ nucleon. Chapter V Conclusions The studies described in this thesis attempt to address questions related to the equa- tion of state of nuclear matter. The specific questions that were addressed included: at what density does a nuclear freeze-out occur, what is the temperature at which this freeze~out takes place, and at what excitation energy can the system be considered to go beyond the limits of the stability of normal nuclear matter and undergo a phase transition. All of these questions are concerned with the point during the evolution of a nuclear reaction 'that fragment formation occurs. ’In the expansion/freeze—out view of a reaction, the system initially converts excitation energy into mechanical work (expansion). This energy is later used to break nuclear bonds at the final stage of the reaction (freeze-out). The multi-fragmentation study was concerned with de- termining the amount of excitation energy necessary to allow the system to enter a region of mechanical instability. Upon entering this region, the system disassembles into fragments. The nuclear density was estimated by observing the correlation between pairs of particles that were emitted independently from the thermalized source created during the interaction of a 500 MeV proton with a silver nucleus. The shape of the correlation functions were compared to model predictions which allowed the spatial dimensions (source radius) of the source to be determined. A rough estimate of the mass of 149 _; 150 the source can be made by assuming that the target is completely thermalized and equilibrated prior to emission of fragments. This estimate does not allow for pre- equilibrium emission of particles or a separation of the thermalized zone and the target remnant. For the p+Ag reaction, one expects that the thermalized region would be completely contained within the target remnant. The excitation energy which was deposited by the incident proton as it passed through the target could then be dissipated through the entire target nucleus forming a single equilibrated source. The most reliable measurement of the source radius is extracted from the deuteron-deuteron correlation function. This particle pair does not contain peaks associated with resonant decays. These resonant decays distort the extracted radius. The radius of the thermalized source is measured to be approximately 9 Fm. This yields a freeze-out density of 0.17fgzégpo. If one takes a system of a given number of nucleons and varies the input excitation energy, the resulting initial temperature will also vary. The system has two ways of dissipating this excitation energy: expansion and fragment formation. If the system does not emit any fragments, then the temperature of the system, when it reaches the freeze-out density, will depend on the initial excitation energy. Therefore, the measured temperature should depend upon the energy of the incident beam as this is the major factor which determines the excitation energy generated during the initial phase of the collision. The slope parameter of the kinetic energy spectra generated a measure of the temperature that exhibited this expected feature. An alternate method of measuring nuclear temperature, through the excited state populations, however, yielded a constant and low value for the temperature. The study presented here was one of several studies that have attempted to analyze further the second method for determining nuclear temperatures. The populations of particle unstable excited states were measured through the detection of the daughter 151 fragments associated with the resonant decay. The individual states appear as peaks in the relative momentum spectra. An estimate of the populations can be made by integrating the peaks above a background for independent emission and correcting for the efficiency of the detection system. This analysis yields a temperature value 4 j: 1 MeV. These values are consistent with those established in heavy-ion induced reactions. A nuclear system should be able to reach a region of mechanical instability if the internal excitation energy exceeds a threshold value. At this point the disassembly will proceed simultaneously. A signature of this process can be found in the distribution of event shapes. If the reaction proceeds through a long-lived equilibrium process, the initial decays of the system will occur at the highest temperatures. These intial decays will define an axis along which the event shape will be elongated. If the reaction proceeds though a multi-fragmentation process, the event shape will be spherical in the limit of infinite multiplicity. The shape distributions observed in a set of Ar+V data at bombarding energies from 35 to 85 MeV/nucleon are compared to predictions from simulations. Both a sequential decay and a simultaneous multi-fragmentation are simulated. The experi- mental distributions are evaluated to determine which of the two models more accu- rately describes the reaction dynamics at a given energy. It is observed that the se- quential simulation reproduces the experimental distribution for the 35 MeV/ nucleon case. For the higher five energies, the centroids of the experimental distributions fall between the two predictions. The failure of the sequential model above 35 MeV / nucleon is seen as an indication that the multi-fragmentation process is con- tributing to the disassembly of the system. Thus, the onset of multi-fragmentation is conclusively observed. PPPPPPPPPP Appendix A Technical Descriptions of the Detectors Systems A The 4-by-4 Close Packing Array The 4-by-4 close packing array [Fox87] is a set of sixteen fast / slow plastic scintillator phoswichs [Wilk52]. These detectors were designed for a series of experiments which studied. charged-particle correlations at small relative momenta (one of which is de- scribed in chapter II). A goal of these experiments was to measure the correlation function down to a relative momentum of less than 1 MeV/c, therefore it was essential that the detectors close-pack. In addition, it was necessary to determine the relative angle between two detected particles to within 1 degree. This angular resolution was achieved using a multi-wire proportional chamber (MWPC) [HassS4, Tick80, Tick81, Char70]. This placed a limitation upon the dimensions of the array, because it was necessary for the array to fit behind the MWPC (which had been constructed for a previous experiment) which had an active area that was 15 by 15 cm. Each of the detectors in this array was a modified truncated square pyramid, refer to Figure A.1. Since perfect squares will not close-pack on the surface of a sphere, the detectors were skewed slightly. Ignoring the individual asymmetries in the lengths of each side, the front surfaces of the detectors were 1.5 inch squares. The front 152 153 Table A.1: Characteristics of the fast and slow scintillation material. Constants BC-412 (fast) BC—444 (Slow) Light Output % Anthracene 60 41 Rise Time (ns) 1.0 19.5 Decay Time (ns) 3.3 179.7 Pulse Width FWHM (ns) 4.2 171.9 Wavelength of Max Emission (nm) 434 428 No. of C Atoms per cm3 (x1022) 4.74 4.73 No. of H Atoms per cm3 (x1022) 5.23 5.25 Ratio HzC Atoms 1.104 1.109 No. of electrons per cm3 (x1023) 3.37 3.37 scintillator was a 1.588 mm thick wafer of BC412 plastic scintillator. The stopping scintillator is a 137 mm deep block of BC444 plastic scintillator. The characteristics of these scintillation materials are given in Table A.1. The back of the scintillators was approximately a 2.0 by 2.0 inch square. This gave the detectors a taper such that when placed 17 inches from the center of the target, each of the four edges pointed directly to the center. A two inch diameter Amperex photomultiplier tube (PMT) was directly coupled to the scintillator with an optical epoxy. The scintillators were painted with an opaque diffusive white paint to optimize light collection and minimize cross-talk between adjacent detectors. They were stacked in a four-by-four array, as illustrated in Figure A.2, when employed behind the MWPC. In this configuration, each detector covered a 5° range in polar angles and the entire array covered a total solid angle of 165 msr. B The Multi-Wire Proportional Chamber The multi—wire proportional counter (MWPC) was designed to provide better than 1" of angular resolution between multiple simultaneous hits [Ha5584 Tick80, Tick81, Char70]. It contained three wire planes, each of which contained 64 wires. The wires 154 MSU-81027 Figure A.1: A side view of four of the detectors. Figure A.2: Perspective view of the 4-by—4 array stacked for installation behind the MWPC. 155 on the X plane were arranged vertically, those on the Y plane were horizontal, and those on the Q plane were at 45°. Having three wire planes allowed correct position determination for multiple hits. Had only two wire planes been employed, there would have been ambiguities. The wires on each plane were Spaced 2.5 mm apart. To minimize the fluctuations, the wires were clustered in sets of two. Therefore, each wire plane had 32 signals. The signal from each pair of wires was connected to a discriminator channel, which was read by the PCOS III system. Hit information was recorded. No record of the amplitude or the time was made for the individual wires. The MWPC was pressurized to 450 Torr of 50/ 50 Argon/ Ethane. The X and Y planes were biased to 3127 V while the Q plane was biased to 2600 V. Figure A.3 displays 16‘scatter plots of the X and Y positions of the wires that were bit during a valid event for each of the telescopes. Each of the sixteen different plots requires a coincidence with a different one of the 16 phoswich detectors that were employed behind the array. For a valid event to be recorded, two phoswichs were required to fire in coincidence. Three wire planes were then needed to resolve the multiple hit ambiguities. The dark squares correspond to the wires in front of the given detector. Note that the 3rd and 14th wires in the X-plane were dead. The background counts that are plotted, but which are obviously not in front of the given detector, are generated because each valid event had two hits. Therefore, for each X-Y position in front of the selected detector there should also be a point elsewhere on the plot. The scales of the display have been adjusted to suppress many of the off-detector points. 156 Y Wire Figure A.3: Scatter plots of the X and Y positions determined by the MWPC when in coincidence with phoswichs 1-16 157 C The Multiplicity Array Detectors The multiplicity array detectors are a set of thirty fast / slow plastic scintillators. These detectors were elements of an array which was designed to detect the multiplicity of fragments emitted in the forwarddirection. At that time it was assumed that the charged-particle multiplicity was related to the impact parameter. Since the emitted fragments are forward focused, it seemed logical to design an array that covered angles forward of 45°. This would then allow the experimenter to tag an event with an observed multiplicity which could be used as a measure of the impact parameter. The major design goal of the project was to maximize the number of detector channels, therefore, the primary consideration of this array was cost per channel. It was decided to make these detectors as simple as possible. The front scintillator is a disk of fast plastic (BC412). This disk is two inches in diameter and 1.588 mm thick. The stopping scintillator is a six inch deep cylinder of slow plastic (B0444). An Amperex PMT is coupled directly to the scintillator. The scintillator is painted with an opaque diffusive white paint and then wrapped with black tape. D The MSU 47r Array The 47r array was designed to study violent high multiplicity reactions. The apparatus completely surrounds the target with detectors. The geometric shape of the device is a truncated icosohedron. This is a solid with 32 facets, 20 of which are hexagons and 12 of which are pentagons. Figure A.4 displays a schematic diagram of the external support structure for the 41r. Each facet comprises a single sub-array of the entire detection system. The subarrays will employ a logrithmic detection scheme [West85]. This enables the device to have a high dynamic range. This logrithmic detection system consists of three separate types of detector. These detector systems form 158 Figure A.4: A schematic diagram of the MSU 41r Array. 159 FAST/ SLOW PLASTIC SCINTILLATORS / BRAGG CURVE fl SPECTROMETER .J Figure A.5: A schematic diagram of a single module of the 47r Array. three shells, each with a higher stopping power than the one inside it. The innermost shell will be made up of low pressure gas counters. These will stop only the most highly ionizing fragments (mostly fission fragments). The second shell shall be made up of high pressure gas counters. These will stop the majority of the intermediate mass fragments. The outer shell is made up of phoswich scintillation counters. These detectors are optimized for light charged particles. A diagram of a single subarray is provided in Figure A.5. For the experiments described in this thesis, the 471' array was instrumented with only the outer shell of phoswich counters. For this shell only, each subarray is further divided into either five or six detectors (depending on whether it is a pentagonal or hexagonal module). This reduces the probability of multiple hits. The multiple hit problem is most severe for the light charged particles because they are produced in the largest abundance. The phoswich detectors in the 41r array are similar to those described earlier in this appendix. Their dimensions, however, are larger. The AE 160 scintillator is 3.175 mm thick, the stopping scintillator is 30 cm thick, and they are shaped like truncated triangular pyramids. One pentagonal facet had to be left open at both the entrance and exit of the array. This left a large uninstrumented region in the forward direction where the production cross-section is the highest. It was decided that the phoswich detectors described in sections A and 0 should be mounted in the empty forward wedge. These detectors now make up the forward array of the 471'. As they were not designed for this purpose they do not close pack, but they provide substantial coverage in a very important angular range. Appendix B The Light Response of Plastic Scintillator A Scintillation Theory Recently there has been an increase in the usage of organic scintillator as a detection material for nuclear physics. This has been brought about by an increase in the energy available from heavy ion accelerators and from the increased complexity of the experiments. As the energy of the projectile is increased, there is a comparable increase in the energy of the reaction products that must be detected. As the energy of the fragments is increased, the amount of detector material that is required to stop them must also be increased. It is a relatively simple and inexpensive task to make scintillation counters of greater and greater size. Solid state detectors, on the other hand, are limited to approximately 5mm in thickness. Though an experimenter can employ a multiple element stack of silicon detectors, each of the elements will require an extra channel of electronics making the cost per telescope rise steeply with total stopping power. The increased complexity of experiments makes it necessary to employ arrays with large numbers of detectors. For such arrays, the cost per channel is a major consideration. Additionally, plastic scintillator is an ideal detection material for close-packed arrays, because it can be machined to arbitrary shapes and 161 162 requires an extremely small amount of packaging between the sensitive areas of the detectors. The fact that a large number of close packed arrays have been built using plastic scintillator [Bade82, West85, Schi85, Boug86, Lide87, Sara88, Pou188, 0hap88, Lide88] demonstrates this point. Plastic scintillator has some disadvantages compared to other types of detection material. For example, the intrinsic energy resolution is inferior to that of solid- state detectors and the light response is non-linear. The intrinsic energy resolution is a weakness that must be accepted, however, with a little work, the light response function can be determined and its complexity will not compromise the energy mea— surement. The mechanism for light production in scintillation material is the de-excitation of electrons which have been perturbed from their ground state orbitals. As a charged- particle passes through matter, it interacts with the electrons throughout the lattice. These electrons are knocked out of their bound states into the conduction band of the material. As the electrons de-excite, they emit photons. If the band gap of the material is in the visible range, these photons will appear as scintillation light. The efficiency for conversion of the kinetic energy of the de-accelerating charged particle into photons with wavelengths in the visible range is reduced as the ionization density is increased, because the probability that multiple electrons will be excited at a single lattice site becomes non-negligible. This reduction in conversion efficiency occurs because only a single electron may occupy a given state at a given time, therefore, the additional electron must follow a different de-excitation route which may not involve emission of photons in the visible range. Additionally, an ion which is stopping quickly (high ionization density) may impart a greater kinetic energy to the electrons with which it interacts than an energetic light ion would impart. The additional energy of the electron will not be converted to light energy as the electron de-excites because 163 only the transition across the band gap will produce scintillation photons. These mechanisms account for the non-linearity observed in the light response of organic scintillation material. Early work on the light response function was done by Birks [Birk64]. He derived the following formula that relates the fluorescence per unit path length (dL/dr) to the specific energy loss for a charged particle (dE/dm): dL/dar: :: S(dE/d:v)/(1+ kB(dE/dz)) (B.1) where S is the scintillation efficiency and k3 is a quenching parameter. This the- ory has been refined and the number of adjustable parameters has been increased [Chou52]. However, the basic dL/da: formula, which contains a saturation as a func- tion of ionization density, is generally accepted. The total light produced by a particle stopping in a scintillator is then given by R L _—_ / S(dE(a:)/da:)/(1 + lemma/115))“ (B.2) 0 where R is the range of the particle. dE(m)/d:c is the function for the energy loss [Knol79]. For a particle that is stopped within the material, the energy loss function will be the Bragg curve. This theory has been supported by experimental results [Crau70]. One can not directly determine the scintillation efficiency or the quenching param- eter as a function of ionization density. The experimenter would prefer to determine the light response as a function of quantities that can be easily measured, such as the incident particle’s charge (Z), mass (A), and energy (E). Attempts to determine functional forms for the light response in terms of these quantities have been relatively successful [Becc76, McMa88, Pou188I. In order to get a reliable energy calibration for a set of scintillation detectors, one 164 typically would have to calibrate each detector during each experiment. These cali- bration functions would be specific to an individual detector and an individual particle species. However, this task becomes prohibitive for large numbers of detectors and for experiments for which a large number of different particle types are required. We have completed an extensive study of the light response for a set of fast / slow phoswich [Wilk52] detectors which are elements of the MSU 47r Array [West85]. Thirty detec- tors have been calibrated individually over the course of several experiments. The light response functions of all of the detectors were similar to one another and re- mained constant from experiment to experiment. We have generalized this function and applied it to the remaining 175 detectors in the array. B The Calibration Experiments 1 The Detectors The detectors that were used for these studies are elements of the MSU 47r Array. This array contains phoswich detectors of four different geometries. The relevant characteristics of each detector type are listed in Table B.1. For all of the detectors, the front scintillator is B0412 (3.3 nsec decay time) and the stopping scintillator is B0444 (180 nsec decay time). Refer to Table A.1 for details about these two types of scintillator. The detectors were biased from 1000 to 1600 Volts depending on the maximum fragment charge desired for a particular application. The signal from the detector is passively split three ways. One signal determines the timing and the trig- ger logic, the other two signals are integrated in LeCroy FERA QDCs. 165 Table B.1: A listing of the specifications of each of the four types of phoswich detector that were employed in the studies of the light response. Detector Geometry AE E PMT Type Thickness Thickness Type (mm) (mm) Truncated I Square 1.566 15.0 Amperex 2” Pyramid Right II Cylinder 1.566 15.0 Amperex 2” Truncated III Triangular 3.175 25.0 Amperex 3” Pyramid Truncated IV Triangular 3.175 25.0 Amperex 3” Pyramid 2 Elastic Scattering A calibration of the response of 16 type I detectors for protons and deuterons was performed using the SF U scattering chamber at the TRIUMF facility. The detec- tors were stacked in a four~by—four array behind a multi-wire proportional chamber (MWPC) which provided high resolution position information for the incident par- ticles. A scintillator paddle was positioned in the forward direction on a movable arm. This configuration is displayed in Figure B.1 Proton beams of 200, 300, and 500 MeV were scattered off CH2 and 0D; targets. A coincidence was required between the paddle and the array for a valid event. Using the position information from the MWPC and assuming elastic scattering, the energy of the recoil particle could be calculated. Given the recoil energy one could determine the amount of energy which was deposited in each of the two scintillation elements. Figure B.2 displays scatter plots of relative light versus calculated energy from the slow scintillator for protons 166 I Beam Axis Paddle Array of — l6 fast/slow telescopes M WPC Figure B.1: The experimental configuration for a calibration of the response to pro- tons and deuterons. Adjusted Channels ‘ 167 Figure B.2: A scatter plot of t against the calculated energy 0 60 O 60 Calibrated Energy 0 60 he magnitude of the signal from the slow scintillator f the recoil proton from the reaction p[p,p]p. 168 Table B.2: The thicknesses of the aluminum degraders that were used in the calibra- tion of the phoswich detectors and the energy deposited in each scintillator Beam Particle Degrader E( fa“) E(,1ow) Energy Type Thickness (mm) (MeV) (MeV) 212 MeV a 6.91 11.7 120 212 MeV 01 9.45 16.7 74 212 MeV 01 10.48 30.2 25 106 MeV d 6.91 2.3 85 106 MeV d 9.45 2.4 78 106 MeV d 10.48 2.5 73 106 MeV d 14.01 2.9 62 106 MeV d 19.22 4.0 40 106 MeV d 22.10 6.8 19 for each of the 16 detectors. Data from the three incident energies have been summed for these plots. All sixteen detectors were matched (refer to Appendix 0) and the spectra were added. Figure B.3 displays a scatter plot of this summed data. The centroids of the light versus energy curve for each energy channel were determined. Figures B.4a and B.4b display the results for the calibration of the slow scintillator for protons and for deuterons. 3 Degraded Calibration Beams Ten type I detectors were calibrated using degraded alpha and deuteron beams from the K500 cyclotron at MSU. The detectors were positioned on the rotating table in the 60 inch scattering chamber and were rotated to zero degrees one at a time. Degraders were positioned on the target ladder. The thicknesses of these degraders, the resulting energy of the calibration beams, and the energy deposited in each of the two scintillation elements are given in Table B.2. Figure B.5 displays the results of I this calibration. The solid lines correspond to the fits to the experimental calibration data. 169 Calibration of Phoswich Detectors 00 | "ca—uh“..- .4”. .. ””3..- 1010111111133 MOIS 10 indino 111511 « II: 5,}: A 1-1 V’ ('3 . I‘E 3' ‘. O I 3 1 3 2' 1 I r I r u u I r r I v v ‘3' 20 40 so so 100 120 Energy of Elostically Scattered Particle (MeV) Figure B.3: A scatter plot of the magnitude of the signal from the slow scintillator against the recoil proton energy. 170 CD 0 a) protons O) 0 4s 0 ll_zL LLIiJ_l ITTI IIPTII l l I'— T O .— CD O b) deuterons O) O vb (0 O O O IllTIlT—TIIIIIIIIITIIT—rll rrfiIIrrrIrrliITrrrTrir—r Calibrated Energy (MeV) N O LLLIIllLIlllllllllll llJl ILLllliilllLllLllLlllll—r Ali L11 LlilnlJlJLJ LLLIL 100 200 300 400 500 Light Output (Slow Scintillator). r. .- ,— 0 Figure 8.4: The centroids of the light distribution were determined for each recoil en- ergy. a) displays the proton response function, and b) displays the deuteron response function. The solid curves correspond to fits to the experimental calibration data. 171 150 .- Ifii—I W FT—W— T— I— I I I TT—I I I l I Ifi fli-I - d = deuterons . A E __ 4 I > 125 _01 — He _ 0) ~ - E I : >. 100 I— .1 an 1 i s - ./. L1] 75 I— “a '0 : I B — 1 (a 50 L. T T a E i 5 25 3— a a I. —l O b l l 4 LJ [ A l l L l T lmjc i AL m4 l_l 1 AL mu 1 O 100 200 300 400 500 Adjusted Channels Figure B.5: The light response as a function of energy for the slow scintillator for deuterons and 01 particles. 172 4 Fragmentation Beams Ten type I, three type II, and 12 type III detectors have been calibrated using frag- mentation beams at MSU. In our first application of fragmentation beams to pro- vide calibration points for phoswichs, the K500 cyclotron provided a beam of 50 MeV / nucleon 14N6+ which was stopped in a 35 mil tantalum production target which was positioned in Beam Box 2. The detectors were positioned on the table of the 60 inch scattering chamber and were rotated one at a time into the beam of frag- ments. The strengths of the downstream beam line magnets (quadrupoles Q10, Q11, Q12, and Q14, dipole BMZ, and quadrupoles Q14, Q15, Q16 and Q17) were varied to select the rigidity of the forward fragments produced in the target. The width of the accepted rigidity slice was determined by a one inch diameter collimator at the entrance to the scattering chamber. The beam line magnets, however, were not well calibrated (i.e. a 30% reduction in current did not correspond to a 30% reduction in field strength). Therefore, the exact energy of the outgoing fragments was measured with a silicon telescope and cross-checked with the estimated field strengths. In a later application of fragmentation beams, the six type III detectors were positioned in a test box behind the 20° exit port on the S320 spectrometer. The spectrometer was positioned at zero degrees with respect to the beam axis in order to receive the greatest flux of fragments. Once again, a beam of 50 MeV / nucleon 14N8+ was stopped in a tantalum target. The target was positioned in the target chamber of the S320. The field strengths of the magnets which make up the $320 (Quads Q20 and Q21, and dipole P M3) were set to select several different rigidities. An aperture at the entrance window limited the size of the accepted rigidity slice. This technique produced a beam which contained various isotopes of hydrogen, helium, lithium, beryllium, and boron. Given the charge, mass, and rigidity of each accepted particle type, the energy of the calibration points could be determined. The accepted particle types and energies are 173 listed in Table B.3. The energy in MeV is given for each particle type as a function of the rigidity settings of the S320 spectrograph. The rigidity values are presented in terms of the percentage of the original beam rigidity. The production beam was 50 MeV/n 14NM. The numbers presented in this table are the energy values that would have been accepted by the 8320; the cross section for the production of some of these particles types at the quoted energy was too small to be observed. This method produces a beam composed of many different species of fragments, and is only useful as a calibration technique if one is able to identify the different particle types in the detectors to be calibrated. The phoswichs that we were using were able to resolve all of the different calibration points. Figure B.6 displays a scatter plot of the fast signal against the slow signal for one detector at a single rigidity setting. The tails on the deuteron and alpha points are associated with particles which scattered out of the detectors. The detectors had not been collimated for these tests. These tails are noticeable only for the deuteron and alpha points because those points contain several orders of magnitude more counts than the others. The energy spread present in all of the points is a measure of the width of the accepted rigidity slice and not a measure of the resolution of the detectors. A similar fragmentation calibration was performed using the Nautilus scattering chamber at GANIL. C The Light Responses of the Scintillators 1 Response Function of the Slow Scintillator The spectra from the different detectors and from different calibration runs have been matched [Appendix C], and the results are displayed in Figures B7 to B9. The 174 Table B3: The energies of the observed fragment species during the fragmentation runs. I Particle Typel 100%J 90% L80%_I70% I 60%:I 50761 p 247 204 164 128 _ 96 67 d 135 109 87 67 49 34 t 92 74 59 45 33 23 31% 350 287 229 177 131 91 41% 270 220 175 134 99 69 6H6 183 149 118 90 66 46 8H8 138 112 89 68 50 35 °Li 404 330 262 202 149 103 7L1: 350 285 226 174 128 89 8L1; 308 250 198 152 112 78 9Li 275 223 177 136 100 69 7Be 610 498 397 306 226 158 9Be 483 393 312 240 198 123 10Be 437 355 282 216 159 111 11131: 399 324 257 197 145 101 9B 743 606 483 372 275 192 ”B 674 549 437 336 248 173 “B 617 502 399 307 226 157 1213 568 462 367 281 208 144 1°C 957 782 623 480 355 248 “C 877 716 569 438 324 226 ”C 809 660 524 403 298 208 13C 751 611 485 373 275 192 140 700 570 452 347 256 178 175 500 ‘ 400 L- ”' .44 ~ . x- i {y .;.‘ 300 +- = _ 153- wflP’-rgpm. 5,44 (UNCALIBRATED) (D (n ’5 .5 . 0 u ‘ . I 'I .. . . , . , no - , . o a I ' ' ' I l . . n ' . ' o 6 . ' .b. I. I . . .‘. o . ' .l 10". I. C ' ' . . . .u ' I i . 2 o C I I . . ..- o ‘ .. .." . o o . FAST SIGNAL 100 r-_i i611 ' . .ZH : '- '. i ‘ . 33.7? ‘Mtfl‘v‘fi if, ‘- .‘J-mo - ---1~.‘ ¢ o‘fl‘a .fi- . '. o i . U “ ' o i L L l L l o 100 200 300 400 500 SLOW S IGNAL (UNCALIBRATED) Figure B.6: A scatter plot displaying the raw data from a fragmentation calibration run. These data are from a single detector and a single rigidity setting. Several different isotopic species are simultaneously produced for each element. 176 400 T T T—r f—T 1—* —fi 7 d = deuterons - t. = tritons 1 A = 4 > ‘ it: all: j G) +- 6 = °Li E 300 e7 4 7L1 0 r 8 = °L1 >~. - 9 = °Li . E0 __ 0 =xoBe J (D _ 0 =33 . C: = B m 200 ~91 1:28 4 J L é =“B "O 0) 7 / 1 +" 7 CU , 1 S— F - "c6 ; - ‘ U P/ —/// .1 O ’1 !_I_L l 1 1 1 t l 1 I 14 l L 1. L I l u 1 J O 100 200 300 400 . 500 Adjusted Channels Figure B.7: This figure displays the light response of the slow scintillator as a function of energy for several different particle types. These data are from the fragmentation run in the 60 inch chamber using type I detectors. 400 a . r r , 4 4 p = protons I d = deuterons A [ t = tritons > 3 1 Q) r- a = E 300 r—H = —J _ 5 1 4 >\ ,_ 7 = :0 L. g: -+ Q) C T B = I :21 200 l—X = j "G i i 3 L . :3 t , : / g 100 r 2/ ‘J 76 A f", / U i/“/ i /‘.'. 0'11.11_n111n1. 111111 O 100 200 300 400 Adjusted Channels Figure B.8: This figure displays the light response of the slow scintillator as a function of energy for several different particle types. These data are from the fragmentation run using the S320 spectrometer and 6 type III detectors. 177 100C rent.7,nn,.flr, L J 7 Slow Cal. for A4 ' GANIL « 800 r- - L p X=Deuterons O +=Tritons 3 600 — _ 2 F 1% _ CD ” 4 L“ 1 (l) r LS 400 ~ 1 L -1 200 — _ l' .l l O I . 1 1 J , L L L 1. 1 l 1 0 500 1000 1500 2000 Channels (light output) Figure B.9: This figure displays the light response of the slow scintillator as a function of energy for several different particle types. These data are from a fragmentation run at GANIL [Gont90]. 178 solid curves are fits to the data using the formula Channel 2 C'.l471'4/Z0'8AO'4 (B3) where C' is a detector specific normalization constant, E is the energy of the fragment, Z is its charge, and A is its mass number. This formula for the light response agrees with the requirement that total light produced must decrease as a function of the average ionization density. The ionization density (dE/Dm) increases with Z and A, therefore, the light response should vary inversely with respect to these quantities. The ionization density is a non-uniform function of energy, however, the average ionization density decreases with total energy. Therefore, the light production must increase with energy. § 2 Response Function of the Fast Scintillator The response function of the fast scintillator is more complicated than that of the slow scintillator. The energy ranges of greatest interest are those for which the par- ticles stop in the slow scintillator. For these particles, the fast scintillator acts as a transmission detector. Particles which stop in the fast scintillator are of less interest because one can not identify the charge, mass or energy. The response function for transmission particles is different than for stopping particles because the region of highest ionization density, the Bragg Peak, does not occur within this scintillator material. Therefore, the ionization density is relatively uniform throughout the scin- tillator and is almost independent of Z and A. Figure B.10 illustrates this point. For a 1.5 mmAE scintillator, the 100 MeV deuteron passes through depositing 3.75 MeV uniformly with distance. For reference, the AE scintillators are 1.566 and 3.175 mm thick for the detectors used in these experiments. The generalization that the ionization density is independent of range, however, is not true for particles that deposit the majority of their total energy in the fast scintillator. The curve for the 179 deuteron Bragg Curves 80 'Tj—r 1V1? I 1 1 1 F[TTT TT—l l—l_F fi1—r—f] 1 I A L 4 E 60L n \ 1 1 i L 4 2 40— .4 L :Q d § 1 . \ L J l- _ 1' 1 1 1 l O 1h-1l14 n1inn n1 i1 1 1 1l1 1 11 LJ LIL 0 5 10 15 20 25 30 Range (mm) Figure B.10: An illustration of the ionization density as a function of depth in scin— tillation material for a 100 MeV deuteron (solid line) and a 25 MeV deuteron (dotted line). 180 25 MeV deuteron particle in Figure B.10 demonstrates the non-uniformities that are introduced when a major portion of the Bragg Peak is picked up in the fast scintilla- tor. Figure B.11 displays a compilation of the results from the calibration studies for the fast scintillator. The solid curves come from fits to the data using the following formula: Light(fast) = CEO‘5 (3.4) For transmission particles, the ionization density is uniform and, therefore, depends only on the total energy deposited in the scintillator and not on the particle’s charge or mass. Since light is a function of ionization density, the response of the AE _ Scintillator for transmission particles is a function only of the energy deposited. This is demonstrated in Figure B.11. Most of the calibration points fall along a single calibration curve. The points that miss the curve are the highest energy points for a given particle species. The highest energy point corresponds to a particle that just barely has sufficient energy to punch out of the AE scintillator. For those particles, a portion a the high ionization density region (the Bragg peak) will be picked up. These points will therefore produce less light than expected as is demonstrated in the figure. Figure B.12 displays a set of data close to the punch-in region where the AE response function is no longer independent of charge and mass. It is also possible for particle to stop within the AE scintillator. For low energy particles that are stopped, in entire Bragg peak will be contained within the fast scintillator. The solid curves are fits to the data which include responses for the fast scintillator in both transmission and stopping mode. There are no data for this region because the is no identification for particles that stop in the fast scintillator. However, if one assumes that the light response will be similar in form to that determined for the slow scintillator, one can generate these complete response functions that span both the stopping and the 181 AB Calibration 150 Li I IT I l I71l17 T Ijj I I? T lfT' I737? _ .: sz : g 125 L0=d ;; J g : a... 1 V : 4:01 69 >6 3 :6 100 TfizaHe ___j L, — _. g : *1"le 1E1 :1 j m 75 _{-—X=Be X 4, re 5 =8 - g t: 1 c6 50 —— 4‘ ~— ‘7 C I g _ .1 <6 L .1 o 25 1 D _ r _ OE L, l Ll l i ll 1 1i 1 IA l i l LJ 1 i l l Lid 0 50 100 150 200 250 300 Adjusted Channels Figure B.11: A compilation of the data from the various calibration runs. This figure displays the light response of the fast scintillator as a function of energy for several different particle types. 182 Forward Array Deha.E Cahbratkni 200 _ I l V I I T j I I l I T T I r1 1 I I T'r I T J _ )1 4 150 e- __ s _ . , Q) h- —-I 5 100 _ .4 L11 f 7 <1 " _ 50 L. . __‘ " 1 7‘ .1 O .— L L l_ l i J_J l L i L l l l J l I 1 l_ i l J J l .1 0 25 50. 75 100 125 LlghtF‘ast Figure B.12: A calibration of the AE scintillator for energy near the punch-in region. 183 transmission regions. D Conclusions This work presents functional forms for the light response of plastic scintillator. This is the first study to determine reliably the mass dependence of the functions, because it is the first to present data containing calibration points from multiple isotOpes of more than one element. This work also examines the response function for organic scintillator operating as a transmission detector. The calibration functions presented here are general enough to be applied to other work using organic scintillator provided one has at least one calibration point in order to determine the calibration constant. (Appendix C describes how an experimenter using phoswich detectors can normalize to the forms using the shapes of the particle identification bands on a histogram of the fast versus slow signals. Using that technique one does not even need a single calibration point.) / Appendix C Phoswich Analysis Routines A phoswich detector is a device that utilizes two or more coupled scintillation ele- ments (a phosphor sandwich) [Wilk52] for the detection and identification of charged particles. The light produced as a particle traverses these scintillators can be collected by a single photodetector, commonly a photomultiplier tube (PMT) or a photodiode. Typically the front scintillator is much thinner then the second. The light produced from this first scintillator which is related to the energy loss (AE) can be used as a measure of the rate of energy loss (dE/dx) since the distance traversed (Ax) by the particle is known to be the thickness of the scintillator. The light produced in the second scintillator (E) is then a measure of the total energy of the particle. Since dE/dx is a function of the particle’s energy, charge, and mass, a histogram of the signal from one scintillator versus the signal from the other will yield bands for the different elements and isotopes and can be used to identify the particle’s charge (Z) and mass (A). Figure (3.1 displays the electronics setup for processing the signals from the phoswich detectors used in the MSU 47r Array [West85]. The setup for any other experiment employing phoswichs would be similar. The important feature of the elec- tronics diagram is the passive three-way signal splitter. One branch (Logic/ Timing on the left) goes to a discriminator and then through a logic circuit which determines 184 _; 185 I VACUUM I I PMT l Ribbon Cable Voltage I Divider j BNCSHV—_! Box T— HV in from LeCroy Splitter Box Phillips Disc. .,_____l L________. tr 1 11 Timing AB . E Delay Delay Delay TFC t Timing AB E F ERA FERA F ERA I t l C CAMAC Dataways D Figure 0.1: The electronics diagram for the analysis of signals from the phoswich detectors used in the MSU 47r Array. 186 if the event should be recorded. If the trigger requirements are satisfied, then gates are generated for the charge-to-digital converters (QDCs). The arrival time and the duration of these gates are important. The other two branches (Fast and Slow) are delayed, to allow the Logic/ Timing circuit time to satisfy the trigger requirements and to generate the gates for the QDCs, and then integrated. The gating of the signals on the Fast and Slow branches is important for optimal detector operation. The gates must arrive at the correct time, must be an appropriate length, and must not wander more than a few nanoseconds from event to event. The solid curve in Figure C.2 shows an idealized phoswich signal for a detector with a fast plastic (BC 412, 3.3 nsec decay time) front scintillator and a slow plastic (BC 444, 180 nsec decay time) stopping scintillator (refer to table A.1 for more details about the scintillators). The positions of the integration gates are indicated at the top of the figure. The fast gate should arrive before the signal starts and should end approximately at the inflection point where the contribution from the slow signal starts to dominate. The slow gate should start about 50 nsec after the the end of the fast gate, and it should be about 200 nsec in duration. There is a certain degree of latitude in the position and duration of the gates. Both the timing and the length of the gates affect the amplitudes of the integrated signals, however, they also affect the degree to which the light from one scintillator is picked up in the gate intended for the other. A decomposition of the signal into its fast (dotted) and slow (dot-dash) components is displayed beneath the curve for the total signal. This illustration demonstrates that there is some pick-up of slow component in the fast gate and vice-versa. Figure 0.3 displays a scatter plot of the signals from the fast versus slow gate. The ridge of high intensity which slopes away from the axis for the fast signal is populated by events from particles which were stopped in the fast scintillator. These 187 Idealized Phoswich Signal 20 rr T r I If I l I I f T —I I 1 —I r I Fri I T j + t- FflSt -I l t b Gate S ow Ga 3 4 A o ____\\ . ..................................................... .— 1n - . ,- fl 0 F I ' 5 ‘ :> - , ° « E b 4 a --20 - ’ .1 £1 7 4 .99 - « o _ . :1: _ - 1 g -40 _. __ Total Signal .1 ‘3 . , n‘ : ----- Fast Component ~60 -- - - - Slow Component -7 U 1 J J J #L J J l__L L L_ln L 1 l l J 1 L l 1_ O 100 200 300 400 Time (nanoseconds) Figure 0.2: An idealized phoswich signal for a charged particle incident on a detector with a fast plastic front scintillator and a slow plastic stopping scintillator. 188 120 — TOO — (n C) l C) 0) O I 4; O l . n I. ~ . . v f... . . . . :‘J‘A‘ " ‘ . QDC Signal - Fast Gate (Channels) ' q I. .‘la' ’5'. .‘:I":‘ ' . ' ' ' ,ul. '. "0 ‘Ir; . "v’ ‘ '1' . . . . . . " . «1152”.- . "- - #9" - ..~.‘.- "1 - ' 31:29 £4" , .' .1 ,4. . .3. . ,'.._‘. .. ‘.. .. . -' ‘ .‘a’.¢~"-- :‘I A '- . '1'.“ - . ' «op , _ u ' -‘ ‘ f. . u" is. 9?. 1. ' . ' a ’. ‘ .~ :. ' . -. . o a ., \ _ ' ’ ‘.-.‘ u‘ .I .. _- -'_‘ . ”-a.‘ _ N O l l l l 60 80 100 120 QDC Signal - Slow Gate (Channels) Figure 0.3: A scatter plot of the signal in the fast gate versus the signal in the slow gate. The data are from a 477 phoswich at 23°, the reaction is 35 MeV/ nucleon Ar+V. 189 events produce signals with no slow component. The fact that the slope of the line (.Mp) is not infinite demonstrates the degree to which the fast signal is collected in the slow gate. The ridge of high intensity that rises above the axis for the slow signal is made up of events for which the triggering particle was either a gamma or a neutron. These particles interact with matter very differently than charged particles. Since slow scintillator makes up the bulk of the detector material, these events are most likely to produce signals with no fast component. The slope of this line (Mn) is a measure of the fraction of the slow component recorded in the fast gate. The intersection of the Punch-In and the Neutral lines corresponds to a signal with no fast or slow components. The location (X0, ’0) of this intersection is a measure of the offsets in the QDCs. Adjustments in the pedestals of the fast encoding readout ADCs (FERAs) effect the position of this intersection. Having determined the crosstalk between the two components of the signal and the offsets in the individual QDC channels, the fast versus slow scatter plot can be corrected to produce a plot of AL versus L, where AL is the light produced in the fast scintillator and L is the light from the slow scintillator. Each point in the raw plot is mapped using the following algorithum: AL : (Fast —— 1’5) — (Slow — X0) x Mn (0.1) L 2 (Slow ~Xo)——(Fast—1’B)/Mp (C.2) Figure 0.4 displays the corrected AL versus L plots for 16 Fast / Slow telescopes. The final stage of the analysis is the gain matching of the AL-L plots from in- dividual detectors. This conformal mapping is done with two operations, a stretch in the AL direction (GAL) and another in the L direction (GL). The final mapping algorithms are: AL : GAL X [(Fast -— 13) — (Slow - Xo)lVIn] (C3) 190 Figure 0.4: AL—L plots for 16 phoswichs. The gains are from the reaction 500 MeV p+Ag. set for as. The data are L = GL x [(Slow — X0) - (Fast — Y3)/M,,] (0.4) These stretch operations represent the gain of an individual photomultiplier tube relative to the average response of the other PMTs and the relative lengths of the fast and slow gates. Figure 0.5 displays the summation of the same 16 spectra that have been presented in Figure C.4. The light response of organic scintillator is non-linear with respect to a particle’s charge, mass, and energy (details of the response function are given in Appendix B). We have made use of this response function to calculate the expected trajectories of the various particle bands in the AL-L space. 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