. a" 'k‘ LIBRARY Michigan State University This is to certify that the dissertation entitled The Emission of Complex Fragments From Highly Excited Nuclear Systems presented by David James Fields has been accepted towards fulfillment of the requirements for Ph.D. degree“. Phy51cs 6 Astronomy 1 /7 , - / a/ { 1/ ’- ' Lwfiz 9116((M Major pfofé’ssor I) ’ , ~ 'bt Dateix ‘ L S‘s MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES n RETURNING MATERIALS: Place in book dr55_to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. THE EMISSION OF COMPLEX FRAGMENTS FROM HIGHLY EXCITED NUCLEAR SYSTEMS By David James Fields 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 1986 ABSTRACT THE EMISSION OF COMPLEX FRAGMENTS FROM HIGHLY EXCITED NUCLEAR SYSTEMS by David James Fields The emission of complex fragments is investigated for the following reactions: 12C + 197Au at E/A-lS and 30 MeV, 120 + Ag at E/A=15 and 30 MeV, 14N + Ag at E/A-35 MeV, and 328 + Ag at E/A-22.5 MeV. In all reactions, intermediate mass fragments, Af>4, are produced with single particle inclusive cross sections which generally decrease with increasing element number. The energy spectra and angular distributions indicate substantial non-equilibrium contributions, particularly for lighter fragments. Measurements of light particles and target-like residues coincident with intermediate mass fragments provide information about the dynamical configuration of the fragmenting system. From these measurements it appears that intermediate mass fragments are emitted in highly damped collisions. The average fragment multiplicities are low, of the order of one. Intermediate mass fragments are typically accompanied by a significant number of nucleons, approximately 10 in the form of light particles alone, emitted prior to the attainment of full statistical equilibrium. As a consequence, only about 0.7 of the beam momentum is transferred to the fragment-residue system. Angular correlations between light particles and intermediate mass fragments indicate enhanced emission in the entrance channel reaction plane and, therefore, that dynamical effects are important in the fragment emission mechanism. Furthermore, the angular correlations indicate that fragment emission is not restricted to central collisions. The target residue angular distributions and the mass, isotopic, and excited state distributions indicate substantial emission of intermediate mass fragments in excited states. The effects of the sequential decay of particle-unstable states are studied in schematic statistical calculations in which the states of nuclei are assumed to be populated according to a thermal equilibrium distribution. These calculations indicate that the characteristic structures in the fragment mass spectra and the apparent suppression of excited state emission are the result of the decays of particle-unstable states. A model is discussed in which particles are emitted at statistical rates from a source which is in the process of equilibration with the target system. The resulting calculations are discussed and compared to measured fragment multiplicities, momentum transfers, and energy spectra. The comparison indicates that descriptions of dynamical evolution and the fragment emission mechanism must be integrated in a realistic model of nuclear reactions. I had intended to dedicate this work to my wife, Anh Phuong, and our son, Andrew David. On her suggestion, I dedicate it to her family in Viet Nam, as representatives of those against whom viscious governments have conspired and for whom there is no easy escape. ii ACKNOWLEDGMENTS Graduate study has been both collaborative and tutorial, and at its conclusion I take some satisfaction in acknowledging the efforts of those with whom I have worked and from whom I have learned. Dr. C. Konrad Gelbke has been the consummate thesis advisor. His advice has been excellent; his assistance, reliable; his patience, remarkable; and his attitude, thoroughly professional. I am particularly grateful to Drs. William G. Lynch and M. Betty Tsang. Their efforts were essential to the work presented in this dissertation. I have profited from knowing them as teachers and co- workers, and take pleasure from knowing them as friends to both myself and my family. I have benefited from the experience and efforts of a great number of other individuals. Josef Pochodzalla deserves special mention. T.C. Awes, G.R. Young, R.L. Ferguson, F. Plasil, F.E. Obenshain, R.L. Robinson, M.L. Halbert, D.C. Hensley, D.C. Sarantites, and L.G. Sobotka were instrumental in experiments at ORNL. R. Bougault and D. Horn analyzed the recoil distributions in the 14 N + Ag experiment. I also acknowledge pleasant collaborations with V.E. Viola, Jr. and K. Kwiatkowski. iii My graduate committee consisted of C.K. Gelbke, W.G. Lynch, W.W. Repko, G.F. Bertsch, and J. A. Nolen. I wish to acknowledge the early assistance H. Toki. I can also acknowledge the graduate students with whom I have worked. Charles Chitwood, a fellow graduate of Tennessee Technological University, David Klesch, Tapan Nayak, and Hongming Xu were fellow students at MSU. Mirek Fatyga and Douglas Fields, from Indiana University, I also acknowledge as co-workers. It is appropriate that I acknowledge the contributions of my undergraduate professors : R.L. Kozub, F.L. Culp, K. Kumar, J.F. Mateja, and J.C. Wells. I owe a great deal to my parents, for obvious reasons. I am grateful to my family in Michigan for their help and hospitality. Finally, I am most grateful to my wife, Anh Phuong, for her continuing efforts on my behalf, and on behalf of our son. iv TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES Chapter I. Introduction A. Motivation . B. Scope and organization . II. Experimental Methods and Analysis A. 12C+ Ag,Au at E/A-15,30 MeV/A 1. Experimental objectives 2. Detectors and geometry 3. Analysis and calibration B. 14N + Ag at E/A-3S MeV . 1. Experimental objectives 2. Detectors and geometry 3. Analysis and calibration C. 32$ + Ag, E/A-22.5 MeV, particle-particle coincidence measurements . . . . . . 1. Experimental objectives 2. Detectors and geometry 3. Analysis and calibration D. 32 S + Ag, E/A-22. 5 MeV, particle— gamma coincidence measurements . . . 1. Experimental objectives ix xi Page 12 12 12 l3 l3 l6 l6 l7 19 22 22 23 27 29 29 Chapter 2. Detectors and geometry 3. Analysis and calibration III. Single Particle Inclusive Data A. Features of the inclusive data . 1. Light particle spectra 2. Intermediate mass fragment spectra B. Two source parametrization of the cross sections 1. Moving source parametrizations 2. Description of present parametrization 3. Results of parametrization 4. Limits of the parametrization . C. Integrated cross sections 1. Elementally resolved yields 2. Isotopically resolved yields IV. Light Particle-Intermediate Mass Fragment Correlations. . A. Spectra of light particles coincident with .intermediate mass fragments B. Angular correlations l. The correlation functions 2. Azimuthal correlations 3. In-plane correlations C. Discussion . V. Associated Multiplicities A. Associated multiplicity defined B. Multiplicities of light particles associated with intermediate mass fragments C. Intermediate mass fragment multiplicities vi Page 32 36 43 43 43 SO 52 52 53 55 6O 61 61 68 71 71 77 77 79 87 87 93 93 95 97 Chapter VI. VII. 1. light particles 2. Fragment multiplicities associated with Fragment multiplicities associated with other intermediate mass fragments . D. Summary Velocity Distributions of Target-Like Residues A. Velocity distributions . 1. 2. 3. General characteristics . Dependences of the peak position Integrated probabilities B. Kinematic analysis . l. 2. 3. 4. 5. Method of missing momentum analysis Quantities of interest Results of missing momentum analysis Errors and uncertainties Sequential decay C. Summary Statistical Aspects of Fragment emission A.Introduction B. Statistical Emission and the Population of Excited States 1. A schematic model . 2. Mass distributions 3. 4. Isotopic yields Excited state populations 32 C. Comparison with S + Ag data 1. Isotopic ratios vii Page 97 101 102 104 104 104 106 112 113 113 114 115 122 122 123 126 126 127 127 130 136 143 150 151 Chapter 2. Population ratios D. Summary of results VIII. Equilibration and decay in nuclear collisions A. Model for statistical decay during equilibration . 1. Motivation 2. Formulation of the model B. General features of the calculation. 1. Instantaneous emission rate 2. Time evolution C. Results of the model . D. Summary IX. Summary and conclusion A. Summary of present results B. Concluding remarks APPENDIX A : Kinematic bias in target velocities APPENDIX B ': Gamma ray efficiency calibration APPENDIX C : Jacobian for relativistic transformation APPENDIX D : Sequential decay . APPENDIX E : Quantum statistical calculations LIST OF REFERENCES viii Page 155 161 164 164 164 165 171 171 176 182 194 196 196 200 202 204 208 210 222 228 TABLE II-l II-2 II-3 III-l III-2 III-3 LIST OF TABLES The positions and solid angles of the light particle detectors (LP) and the intermediate mass fragment detectors (IMF) used to study the 325 + Ag system, as discussed in Section II.C. The positions, solid angles, and construction of the solid state fragment detectors used in the 328 + Ag particle - gamma coincidence experiment, as discussed in Section II.D. The positions and typical efficiencies of the Compton supressed germanium detectors, as discussed in Section II.D. The number in parenthesis next to the detector number is the parameter number for the Compton shield. The position of each detector in the ORNL Spin Spectrometer is given in the second column. The remaining columns give the absolute position with respect to the beam axis and the absolute efficiency of each detector at 300 keV. The best fit parameters from the parametrization discussed in the text for fragments produced in 12C induced reactions on Au at E/A-30 MeV. The estimated cross sections for the fast and slow sources are also given. The best fit parameters from the parametrization discussed in the text for fragments produced in 14N induced reactions on Ag at E/A-35 MeV. The estimated cross sections for the fast and slow sources are also given. The best fit parameters from the parametrization discussed in the text for fragments produced in 328 induced reactions on Ag at E/A-22.5 MeV. The estimated cross sections for the fast and slow sources are also given . ix PAGE 25 33 34 57 58 59 VI-l 3-1 The estimated average total multiplicities, M, of light particles, p,d,t,a, associated with intermediate mass fragments with momenta, ’ emitted in 328 induced reactions on Ag at E/A—22.5 MeV. MT is the total nucleon multiplicity;

and are the average total longitudinal momentum and average total energy carried away by non-equilibrium light particles. Momenta and energies are given in units of MeV/c and MeV, respectively. The multiplicities are inferred from cross sections measured at 0y-40°, 6-27.5°, averaged between A¢=90° and 180° . . . . . . . . . . . . . . . . . . . . . . . . . 96 The observed and calculated kinematic properties of systems for which the peak in the distribution of heavy residue velocities, 32, was within the experimental acceptance when detected in coincidence with an intermediate mass fragment with a momentum at an angle 91. . . . . . . . . . . . . . . . . . . . 119 The properties of the sources used to provide absolute efficiency calibrations. The gamma ray source is given and the nucleus which emits the gamma ray is given in parenthesis, E is the transition energy, J: is the initial and J; is the final spin and parity of the transition, L is the multipole, 6 is the multipole mixing ratio, B is the number of partner gamma rays emitted in coincidence with each transition, and A2 and A4 are the angular correlation coefficients for the transition pairs. . . . . . . . . . . 207 FIGURE I-l II-l II-2 II-3 II-4 II-5 LIST OF FIGURES The cross sections for fragments of mass Af produced in the p + Xe reaction [HIR84] are shown as solid points. A power law dependence on fragment mass is indicated by the solid curve A scatter plot of AE vs. E-AE for the reaction 12C + 197Au at E/A-30 MeV measured at 0-70°. The solid curve represents the locus of peak kinetic energies predicted by a generalization of the systematics of fission Coulomb kinetic energies [V1085] A schematic diagram of the experimental layout for the 14N + Ag coincidence studies. The position sensitive detector is labelled as "HR" and the ionization chamber-solid state telescopes are labelled as "IMF". A schematic diagram of the triple Frisch-grid ion chamber system used in two of the experiments discussed in this thesis. The solid state detectors are contained in the gas volume. A schematic diagram of the multiwire avalanche counter used in the 14N + Ag experiment. The ionization avalanches occur about the wires in the anode planes labelled a and c. The signals induced on the wires undergo a position dependent delay before amplification. A signal induced on the foil cathode, b, is used as a time reference A schematic diagram of the 32$ + Ag particle-particle coincidence experiment. The intermediate mass fragments are labelled as "IMF", the light particle detectors, as "LP", and the position sensitive detector, as "HR". xi PAGE 15 18 20 21 24 FIGURE II-6 II-7 II-8 II-9 II-lO II-ll III-l III-2 A schematic diagram of the Breskin design position sensitive detector. The ionization avalanches occur at the central wire anode. The signals induced on the evaporated stripes on the foil cathode undergo a position dependent delay before amplification. Raw position spectra from the Breskin detector placed at the three laboratory angles from target-like fragments detected in coincidence with lithium fragments detected at 0-27°. Detector originated modulations of the spectra are easily recognized . Final angular distribution from the raw spectra in Figure II-7. The distribution is smooth and relatively free of spurious structures Schematic diagram of one of the Compton suppressed germanium detectors used in the particle-gamma coincidence experiment . The relative efficiency curve (a) for 7 detector 1 from a 152Eu source (circles) and 182Ta source (squares). The solid curve indicates the analytic interpolation discussed in the text. The ratio of the source data to the interpolation is shown in the bottom part of the figure (b). The spectrum of gamma rays coincident with 10B ions, without Doppler shift, is shown in the upper part of the spectrum, (a). The bottom part of the figure, (b), shows the corrected spectrum, indicated by the histogram, and the estimated background, indicated by the solid points. The background is produced by Doppler shifting gamma rays coincident with other nuclei, as discussed in the text . Differential cross sections, dzo/dEdfl, measured at 0-40°, 70°, and 130°, for light particles, p, d, t, and a, from 328 induced reactions on Ag at E/A=22.5 MeV. Also, cross sections measured at 6-27.5° and 52.5° are shown for a particles. The solid curves are the result of the parametrization discussed in the text . Differential cross sections, dza/dEdQ, measured at 0-30°, 50°, 70°, and 120°, for fragments, 552510, from 12C induced reactions on Au at E/A-30 MeV. The solid curves are the result of the parameterization discussed in the text. xii PAGE 26 30 31 35 38 39 44 45 FIGURE PAGE III-3 Differential cross sections, dza/dEdfl, measured at 0-32.5°, 45°, and 57.5°, for fragments, 45259, from 14 N induced reactions on Ag at E/A-35 MeV. The solid curves are the result of the parameterization discussed in the text. . . . . . . . . . . . . . . . . . . 46 III-4 Differential cross sections, dza/dEdO, measured at 9-32.5°, 45°, and 57.5°, for fragments, 1052515, from 14N induced reactions on Ag at E/A-35 MeV. The solid curves are the result of the parameterization discussed in the text. . . . . . . . . . . . . . . . . . . 47 III-5 Differential cross sections, dza/dEdfl, measured at 0-27.5°, 40°, and 52.5°, for fragments, 35258, from 32 S induced reactions on Ag at E/A-22.5 MeV. The solid curves are the result of the parameterization discussed in the text. . . . . . . . . . . . . . . . . . . 48 III-6 Differential cross sections, dza/dEdfl, measured at 0-27.5°, 40°, and 52.5°, for fragments, 952524, from 32 S induced reactions on Ag at E/A-22.5 MeV. The solid curves are the result of the parameterization discussed in the text. . . . . . . . . . . . . . . . . . . 49 III-7 Differential cross sections, da/docm, for various fragments, 552518, from 12C induced reactions on Au at E/AP3O MeV. The solid curves are drawn to guide the eye. Emission from fission would produce a flat curve. . . . . . . . . . . . . . . . . . . . . . . . . . . 51 .III-8 Average differential cross sections, , over the region 50°505120° for fragments of charge, 2, from 12C induced reactions on Au at E/A-lS and 30 MeV (solid and open points, respectively). . . . . . . . . . . 62 III-9 Average differential cross sections, , for fragments of charge, 2, from 12C induced reactions on Ag at E/A-lS ( 50°50570°, solid points ) and 30 MeV ( 40°50570°, open points ). The solid curve depicts a -2.6 strict power law dependence, Ya Z 63 xiii FIGURE III-10 III-11 III-12 III-13 IV-l The extrapolated total cross sections, ax, for fragments produced in 12C induced reactions on Au at E/A—30 MeV are shown as open circles. The solid circles and open squares show the average differential cross sections, , measured over the angular ranges 30°505120° and 50°585120°, respectively . . . . . . . . . . . . The extrapolated total cross sections, ax, for fragments produced in 14N induced reactions on Ag at E/A-35 MeV are shown as open circles. The solid circles show the average differential cross sections, , measured over the angular range 32.5°50557.5°. The solid curves depict a power law dependence de'z'g The extrapolated total cross sections, ax, for fragments produced in 325 induced reactions on Ag at E/A-22.5 MeV are shown as open circles. The solid circles show the average differential cross sections, , measured over the angular range 27.5°50552.5°. The solid curves depict a power law dependence YOCZ.1'6 Relative isotopic yields for isotopes of elements, 352x58, are shown as solid points. The solid curves represent the predictions of a simple statistical model discussed in the text. Energy spectra of protons (top of figure) and alpha particles (bottom of figure) in coincidence with Li fragments (left side of figure) and C fragments (right side of figure). The circles and diamonds correspond to the light particles being detected at 0-40° and 70°, respectively. The solid and open points correspond to relative azimuthal angles between the light particle and intermediate mass fragment of A¢-180° and 90°, respectively. The inclusive spectra are shown as solid curves. xiv PAGE 65 66 67 69 73 FIGURE IV-2 IV-3 IV-4 IV-5 IV-6 The ratio of the differential coincidence cross sections to the single particle inclusive cross sections for protons as a function of proton energy. The ratio is shown for protons detected at 9p-40° (left hand side) and 0p-70° (right hand side) in coincidence with Li nuclei (top of figure) and C nuclei (bottom of figure) detected at 6x-27.5° and at relative azimuthal angles of A¢-180° (solid points) and A¢-90° (open points) . The ratio of the differential coincidence cross sections to the single particle inclusive cross sections for alpha particles as a function of alpha particle energy. The ratio is shown for alpha particles detected at 6p-40° (left hand side) and 0p-70° (right hand side) in coincidence with Li nuclei (top of figure) and C nuclei (bottom of figure) detected at 0 x-27.5° and at relative azimuthal angles of A¢-180° (solid points) and A¢=90° (open points). . . . . . . . . . . . . . . . The correlation function for alpha particles detected at 0a-40° in coincidence with fragments, Zx- 2-7, as a function of their relative azimuthal angle, Ad. The coincidence fragments, X, are detected at laboratory angles of 0x-27.5° (open points) and 0x-52.5° (solid points). The correlation function integrated with low and high low particle energy thresholds (open and solid points, respectively) for alpha particles detected at 0a-40° in coincidence with fragments, 2x. 2-7. It is shown as a function of their relative azimuthal angle, Ad. The coincidence fragments, X, are detected at 0x-27.5°. The in-to-out of plane ratio, A¢, defined in the text, for fragments of charge, Zx-2-7, detected at 9x-27.5° and light particles, Y-p,d,t,a, detected at 0y-40°. The solid and open points correspond to the ratio for correlations integrated for particle energies above a higher and lower low energy threshold. XV PAGE 75 76 80 81 83 FIGURE IV-7 IV-8 IV-9 IV-lO V-l PAGE The in-to-out of plane ratio, A¢, defined in the text, for fragments of charge, Zx-2-7, detected at 9x-40° and light particles, Y-p,d,t,a, detected at 0y-40°. The solid and open points correspond to the ratio for correlations integrated for particle energies above a higher and lower low energy threshold. . . . . . . . . . . . . . . . . . . . . . . . . 84 The in-to-out of plane ratio, A¢, defined in the text, for fragments of charge, 2x-2-7, detected at 9x-27.5° (open points) and 52.5° (solid points) and light particles, Y-p,d,t,a, detected at 9y-40° . . . . . . 85 The in-to-out of plane ratio, A¢, defined in the text, for fragments of charge, Zx-2-7, detected at 9x-27.5° (open points) and 52.5° (solid points) and light particles, Y-p,d,t,a, detected at 0y-70° . . . . . . 86 The correlation function for alpha particles in coincidence with fragments, Zx- 2-7, detected in the same plane (Ad-0° or 180°) as a function of the polar angle, Ga, of the alpha particle. The coincidence fragments, X, are detected at laboratory angles of 0x-27.5° (open points) and 0x-52.5° (solid points) . . . . 88 The estimated average total multiplicities, Mx’ of particles, Zx-2-7, associated with alpha particles in 328 induced reactions on Ag at E/A-22.5 MeV. The results were obtained from the correlations measured at 0a-40° and 0x-27.5° at A¢-90° and 180°. . . . . . . . . 98 The correlation function, ny, between light particles (y) and intermediate mass fragments (x), measured at 0 -40° and 9x-27.5°, and averaged between Ad- 180° and 90°, is shown as a function of Zx’ the charge of the coincident fragment. . . . . . . . . . . . . 100 xvi FIGURE VI-l VI-2 VI-3 VI-4 VI-S PAGE Velocity distributions for target-like residues detected in coincidence with intermediate mass fragments detected at 01-27.5° with momenta, P1. The left hand side shows the distribution as a function of the polar angle, 02, of the projections of the recoil velocity vector onto the reaction plane. The right hand side shows the measured distributions of .p |v2|. The arrows show the values expected for binary reactions. . . . . . . . . . . . . . . . . . . . . . . . . 105 Probability distributions for heavy residues as a function of recoil angle, 92, detected in coincidence with boron fragments of different momenta, Pl’ detected at 01-40°, (upper part) and coincidence with different intermediate mass fragments gated by the same momentum bin (lower part). The arrows indicate the maxima of the distributions. . . . . . . . . . . . . . 107 The differential cross sections for heavy residues detected in coincidence with Be fragments (lefthand side) and 0 fragments (right hand side) of various momenta detected at 01- 45°. . . . . . . . . . . . . . . . 109 The value of the average recoil angle <92> for angular distributions of heavy residues detected in coincidence with 0 fragments detected at 01- 45° as a function of fragment momentum. The solid curves are expected values for 100% and 73% momentum transfer. . . . 110 The results of kinematics calculations for coincidences between heavy residues and carbon nuclei detected at 01-27.5° with an average momentum of 1448 MeV/c are shown as solid curves. The hatched area represents the estimated uncertainties. The dashed curves are obtained when the carbon nucleus is assumed to be a secondary fragment produced in the decay of a particle stable 0 nucleus. A detailed discussion is to be found in Section VI.B. . . . . . . . . 116 xvii FIGURE PAGE VII-l Correlations between protons and 7Li nuclei (bottom) and between alpha particles and 7Li nuclei (top) measured in 40Ar + Au reactions at E/A-60 MeV as functions of the relative momenta between the two particles. Peaks in the correlations which correspond to states in 11B and 8Be are indicated in the top and bottom figures, respectively . . . . . . . . . . . . . . . 129 VII-2 Mass distributions calculated from Eq. VII-l for emission from a Xe nucleus at T-5 MeV. Histogram : primary distribution; solid points : final distribution; dark and light shaded regions show contributions from bound ground and excited states, respectively . . . . . . . . . . . . . . . . . . . . . . . 131 VII-3 Mass distribution from proton induced reactions on Xe (solid points) and final distribution predicted from Eq. VII-1 (histogram). . . . . . . . . . . . . . . . . . . 133 VII-4 Mass distributions as a function of the emission temperature for emission from a nucleus, A-13l and Z-54. The primary (histogram), final (solid points), and stable ground state (shaded region) distributions are shown for emission temperatures of T-3,5,7, and 10 MeV . . . . . . . . . . . . . . . . . . . . . . . . . . 135 VII-5 The calculated isotopic distributions for N nuclei emitted from a nucleus, A-13l and 2-54, at T-S MeV. The distribution of stable ground states is shown as squares. The primary and final distributions from calculations which include excited states are shown as the histogram and solid points, respectively. . . . . . 137 VII-6 The calculated isotopic distributions of Zx-3-8 from a nucleus, A-131, 2-54 and T-S MeV are shown as histograms. The solid points represent the measured isotopic ratios for p+Xe reactions [HIR84] . . . . . . . . 139 VII-7 The calculated primary isotopic distributions for N nuclei emitted from a nucleus, A-l31 and Z-54, at temperatures, T(MeV)- 1 (solid diamonds), 3 (open squares), 5 (star) and 10 (solid circle) . . . . . . . . . 140 VII-8 The calculated final isotopic distributions for N nuclei emitted from a nucleus, A-13l and Z-54, at temperatures, T(MeV)- 1 (solid diamonds), 3 (open squares), 5 (star) and 10 (solid circle) . . . . . . . . . 142 xviii FIGURE VII-9 VII-10 VII-11 VII-l2 VII-13 VII-14 VII-15 PAGE The calculated primary isotopic distributions for the stable ground states of nitrogen nuclei emitted from a nucleus of charge, Z-54, and mass, A - 120 (open diamonds), 130 (solid squares), and 140 (stars), at a temperature of T- 5 MeV . . . . . . . . . . . . . 144 The calculated final isotopic distributions of nitrogen nuclei emitted from a nucleus of charge, Z-54, and mass, A - 120 (open diamonds), 130 (solid squares), and 140 (stars), at a temperature of Ta 5 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Temperature dependence of population ratios, Rp/Rm, for specific states in ‘He, 6Li, 6Li, 8Be nuclei. Ratios measured [POC86p] for the ‘°Ar+197Au reaction at E/A-60 MeV are shown by hatched regions. Dotted curves : temperature dependence of primary population ratios; solid curves : final ratios predicted from Eq. VII-1; dashed and dashed-dotted curves : final ratios predicted by quantum statistical calculations for densities of p/po- .05 and .9, respectively. . . . . . 148 Isotopic distributions from 32$ + Ag at E/A-22.5 MeV (solid points) are compared to calculations (histograms) for emission from a nucleus, A-l30, 2-58, at a temperature of T-2 MeV. . . . . . . . . . . . . 152 Isotopic distributions from 3ZS + Ag at E/A-22.5 MeV (solid points) are compared to calculations (histograms) for emission from a nucleus, A-130, 2-58, at a temperature of T-4 MeV. The solid curves correspond to Eq. III-5 with T-3.8 MeV . . . . . . . . . . 153 Isotopic distributions from 32$ + Ag reactions at E/A-22.5 MeV (solid points) are compared to calculations (histograms) for emission from a nucleus, A-130, Z-58, at a temperature of T-lO MeV . . . . 154 Energy spectra (shown as histograms) of gamma rays in the regions corresponding to transitions in coincident intermediate mass fragments produced in 328 + Ag reactions at E/A-22.5 MeV. The background is indicated by the solid points. . . . . . . . . . . . . . . 156 xix FIGURE VII-l6 VII-17 VII-18 VIII-l VIII-2 VIII-3 VIII-4 VIII-5 VIII-6 VIII-7 The fractions, F7, of intermediate mass fragments undergoing specified transitions are indicated as a functions of temperature for calculations from Eq. VII-l both with (solid curves) and without (dashed curves) the contributions from sequential decay. The observed fractions are indicated by the hatched regions. The apparent emission temperatures of intermediate mass fragments emitted in 32$ + Ag reactions at E/A-22.5 MeV, as discussed in the text . The fractions, F1, of intermediate mass fragments undergoing specified transitions are indicated as a functions of temperature for quantum statistical calculations at p/po-.l (solid curve) and .9 (dashed curve). The observed fractions are indicated by the hatched regions. The dependence on source and fragment mass introduced by the preexponential terms in Eq. VIII-9. The mass spectrum of the instantaneous emission rate for particle stable nuclei as a function of source size in a composite system of a-220 and z-80 . The mass spectrum of the instantaneous emission rate of nuclei emitted in their particle stable ground states as a function of source temperature for a compound nucleus of a-112 and z-56 The mass spectrum of the instantaneous emission rate of particle stable nuclei calculated as a function of the Coulomb parameter, A, as in Eq. VIII-12. The time evolution of compound nuclear systems with initial a-112 and z-52 starting from four temperatures The time evolution of the 12C + 197Au system at E/A-30 MeV, with four different initial source sizes. The accretion rate used in these calculations was [das/dt]a- 3 nucleons/(fm/c) The time evolution of the 12C + 197Au system at four different incident energies, calculated with as(t-0)-24 and [das/dt]a- 3 nucleons/(fm/c). XX PAGE 157 158 160 173 174 175 177 178 180 181 FIGURE VIII-8 VIII-9 VIII-10 VIII-ll VIII-12 VIII-l3 VIII-l4 The time evolution of the 12C + 197Au system at four different incident energies, calculated with as(t-0)-24 and [daS/dt]a- l nucleons/(fm/c). The momentum transfer systematics [FAT85] for l2C,14N induced reactions on 197Au are indicated by solid points. The solid curves represent calculations for the 12C + 197Au system using three different accretion rates. ' The experimental elemental cross sections for 12C carbon induced reactions on 197Au at E/A-lS and 30 MeV, shown as solid and open points, respectively, are compared to calculations with accretion rates of [daS/dt]a- 1 (solid curve) and 5 (dashed curve) nucleons/(fm/c). The experimental elemental cross sections from 12C + 197Au reactions at E/A-30 MeV, shown as open points, are compared to the yield curves for the preequilibrium (dashed curve) and equilibrium (solid curve) stages of the reaction. . . . . . . The energy spectra measured in 120 + 197Au reactions at E/AF30 MeV at 9-30°, 50°, 70°, and 120°, shown as solid points, are compared to the calculation described in the text. The calculated energy spectra for 120 + 197Au reactions at E/A-30 MeV at 0-30° (left) and 90° (right), shown as solid curves, are decomposed into a preequilibrium component (dotted curve) and an equilibrium component (dashed curve) The calculated energy spectrum for 12C + 197Au reactions at E/A-30 MeV at 9-30°, corresponding to the upper solid curve, is decomposed into the contributions from stable primaries (lower solid curve), neutron-unstable primary fragments (dotted curve), proton-unstable primary fragments (dot-dashed curve), and alpha-unstable primary fragments (dashed curve) xxi PAGE 183 184 186 188 190 191 193 FIGURE PAGE D-l Transmission coefficients calculated as a function of energy with the form Tz-x-exp(-2w) for three decay channels of 12C : n(bottom), p(midd1e), and a(top). The calculations are shown as solid curves for partial waves 2-0 to 8 (from left to right). . . . . . . . 213 D-2 Transmission coefficients calculated as a function of energy with the form Tfi-exp(-2w) for three decay channels of 12C : n(bottom), p(midd1e), and a(top). The calculations are shown as solid curves for partial waves £-0 to 8 (from left to right). . . . . . . . 215 D-3 Final mass distributions calculated for the system A-131, 2-54, and T-5 MeV using three forms of the transmission coefficients : T2- exp(-2w) (histogram), T£-x-exp(-2w) (open circles), and a sharp cutoff approximation (solid squares). . . . . . . . . . . . . . . 216 D-4 The ratios of populations of levels in 4He, 5Li, 6Li, and 8Be are shown as functions of emission temperature for calculations with different transmission coefficients : T1- exp(-2w) (solid curves), Tfi-x-exp(-2w) (dotted curves), and a sharp cutoff approximation (dashed curves) . . . . . . . . . . . 218 D-5 A comparison of the mass distributions calculated according to Eq. VII-l using branching ratios calculated from transmission coefficients (histogram) and using branching ratios that are fixed to be equal for all allowed decay channels (solid points) for a system A-l31 and 2-54 at T-S MeV . . . . . . . . . . . . . 219 D-6 A comparison of the fractions, F7, of nuclei in their excited states as calculated according to Eq. VII-l with branching ratios calculated according to statistical rules (solid curves) and branching ratios fixed to be equal for allowed channels for a system of A-130 and 2-58. The fractions observed in 328 induced reactions on Ag at E/A-22.5 MeV are indicated by the hashed regions. . . . . . . . . . . . . . . . . . . 220 xxii FIGURE E-l E-2 PAGE The primary and final mass distributions for the fragmentation of a A-131, Z-54 system at a density of p/po-.4 and at temperatures, T- 1,3,5, and 10 MeV, are shown as histograms and solid points, respectively . . . . . . . . . . . . . . . . . . . . . . . 224 The primary and final mass distributions for the fragmentation of a A-l3l, 2-54 system at a temperature of T-S MeV and at densities, P/Po' .05, .3, .6, and .9, are shown as histograms and solid points, respectively . . . . . . . . . . . . . . . . . . . 225 xxiii Chapter I Introduction I.A, Motivation : To understand the properties of highly excited nuclear systems, it is necessary to understand the processes by which such systems decay. Nuclei excited to sufficiently high energies are known to decay by division into two smaller nuclei. Decay by symmetric division is termed fission, while extremely asymmetric decay into a H or He nucleus and a larger residual nucleus is termed light particle evaporation. Great progress has been made in understanding theSe modes of decay, particularly for systems in which all internal degrees of freedom can be considered fully equilibrated. Between these two extremes in mass asymmetry lies the emission of fragments of intermediate mass. These fragments are heavier than alpha particles but lighter than typical fission fragments. In many cases, the angular distributions of these fragments indicate that they are emitted before the composite systems reach equilibrium. The emission of intermediate mass fragments was first observed in reactions induced 2 by relativistic protons [STE67] [P0871] [HYD71] [WES78]. It was thereafter observed in proton induced reactions at lower energies [GRE80] and in heavy ion collisions both at relativistic [00877] [WAR83] and non-relativistic energies [MOR75b] [FRA81] [JAK82] [CHI83] [SOB83]. In general, the emission of intermediate mass fragments has been associated with the most violent nuclear collisions. The recent observation [GALS84] of the radioactive emission of 14C indicates, however, that intermediate mass fragment emission is likely a "normal" nuclear decay mode [POE85], the relative importance of which can be understood in terms of penetrabilities and binding energies. Existing single particle inclusive data exhibit a few simple characteristics. The energy spectra are peaked at energies somewhat below the Coulomb barrier energy between the fragment and daughter nuclei. The cross sections decrease approximately exponentially with increasing fragment energy. The angular distributions of many fragments contain at least two components. One is approximately isotropic in the center-of—mass of a fusion-like (target-like) residue. The other is strongly forward peaked in the laboratory and nearly isotropic in a frame moving with a velocity between the center of mass and the projectile velocities. (Unless otherwise stated, normal kinematics are assumed, i.e. the projectile is lighter than the target.) The integrated cross sections decrease smoothly with increasing fragment mass. The total fragmentation cross section increases with increasing energy [CHI83] until it reaches a limiting value at relativistic energies [GRE80] [KAU80]. Theoretical investigations of intermediate mass fragment emission have evolved from two directions. Relativistic reactions were expected 3 to produce energy densities in excess of the nuclear binding energy. Nuclear systems with such excitation energies would be intrinsically unstable and would "explode" [FAI82] [BON85b]. As a result, models applied to these data are generally based on the assumption of high particle multiplicities. The relative particle yields are calculated by applying statistical partition rules. The thermodynamical models are based on the assumption that equilibrium is effectively reached ( or, as a weaker statement, that the ensemble average of all reactions approaches the equilibrium limit ) in a uniform region of nuclear matter. This hot region is assumed to expand until it reaches a temperature and density where the interactions between fragments cease and the distribution "freezes out" [MEK78]. The fragment yields are calculated either from the microcanonical ensemble [RAN81] [FA182] [FA185] or from the grand canonical ensemble corresponding to a system in chemical equilibrium [STOC83] [JAC83] [HAH86]. In general, these models neglect the Coulomb interaction between particles, though it has been shown that the Coulomb interaction is important even at high energies [CR082]. Models of non-thermodynamical partitioning attribute the emission of fragments to mechanical instabilities which develop during the collision process. In some models the projectile nucleons "cleave" or "shatter" the target nucleus. In another, the interacting system develops "cracks" and "bubbles" which result in the fragmentation of the system. In the first of these (cleavage) the partition is determined solely by geometrical considerations [BOH83]. In the shattering and cracking models it is determined by the average partition of any finite set of objects (nucleons) [AIC84a] [AIC84b] [FIE84] [BON85b] [50885] or A by percolation calculations in a "nuclear lattice" [BAU85] [BAU86]. A more fundamental understanding can only be provided by microscopic calculations. Monte Carlo integrations of transport equations indicate that in small systems density fluctuations can persist and result in fragmentation [JEN86]. Molecular dynamics calculations [VIC85] suggest that these density fluctuations result in fragmentation only if the system expands to low densities. The nucleon-nucleon potential is attractive at long range and repulsive at short range, analogous to the Van der Waals force. Nuclear matter may, then, exhibit liquid and gas phases [JAQ83] [60084] [JAQ84], much as a Van der Waals gas. Droplets formed in a classical gas near the critical point are produced with probabilities [FIS67] [SIE83]: PAC: A-Texp(-b(T)A2/3) , (1-1) where A is the fragment size, b(T) is a temperature dependent parameter measuring the surface tension of the drops, and r is the critical exponent. At the critical point the argument of the exponential becomes zero and a pure power law dependence remains. In measurements of macroscopic systems, the critical exponent is approximately rz2.5 [KIA70]. Mass distributions in proton induced reactions on Xe and Kr [FIN82] exhibit a similar power law dependence (see Figure I-l) with rz2.6, and from this it was inferred [HIR84] that the emitting systems had passed through the critical point for nuclear matter. The results of a systematic study [PAN84] of mass distributions in energetic collisions were interpreted to indicate a critical temperature of T=12 MeV and T-l.8. MSU-86-32I |()8 L_ I T I l i If ‘ F: 5 ' '] '07? E a, _ .1 E s 3'0 :.'-' ‘z o ; : 0 - . '05":- ': r 1 I04 L I l I J l l. o 5 IO IS 2025 30 Af Figure I-l The cross sections for fragments of mass Af produced in the p + Xe reaction [HIR84] are shown as solid points. A power law dependence on fragment mass is indicated by the solid curve. 6 The validity of the phase transition model is vigorously debated. The analogy between small, electrically charged, quantum mechanical clusters of nucleons and macroscopic droplets is somewhat strained. A number of more specific objections have been raised. Molecular dynamics calculations indicate that fragmentation in small systems does not occur in the region of the phase diagram suggested by the phase transition theories [VIC85]. The signatures of any critical phenomenon would be seriously affected by the small number of nucleons in the experimentally accessible nuclear systems and by the properties of the nuclear surface [BOA84a] [60084]. In addition, the range of fragmentation data is not consistent with the single mechanism provided by the model [CUM85a]. Intermediate mass fragments may also be emitted from nuclei at much lower excitation energies. At these lower energies emission may proceed by a mechanism analogous to light particle evaporation [FRI83a] or the fission-like decay [MOR72] [MOR75a] of equilibrated nuclei. In fact, it was suggested that the study of intermediate mass fragment emission may assist in the unification of evaporation and fission into a single description [MOR75a]. These low energy mechanisms explicitly include the Coulomb interaction between the fragment and the parent system. A statistical evaporation model [FRI83a] [FRI83c] of fragment emission based on the Weisskopf formula [WEI37] successfully described relative fragment cross sections and energy spectra from proton induced reactions [HIR84]. It also explained the relative fragment yields in 12C induced reactions on Ag and 197Au at E/A- 15 and 30 MeV, as well as the measured dramatic rise in the cross sections between these two incident energies [CHI83]. 7 Aspects of complex fragment emission have also been explained by transition state fission theories [MOR72] [MOR75a] for equilibrated systems. The energy spectra and angular distributions of complex fragments emitted at backward angles in 3He + Ag reactions at E-90 MeV [SOB83] were consistent with equilibrium emission of fragments from compound nuclei. The energy dependence of the fragment cross sections was used to test models for fission barriers at different mass asymmetries [McM85] and to explore the target mass dependence of the shape of the fission saddle point along the mass asymmetry coordinate [SOB84]. At lower energies, up to E/A210 MeV, the target and projectile may form a dinuclear system and separate without losing their identity [BAB76], a process termed deeply inelastic scattering. At energies above E/A-lS MeV, the memory of the initial configuration, the defining characteristic of deeply inelastic scattering, is lost [GRASS] and the fragment mass distributions become broad. Intermediate mass fragment emission becomes increasingly important at beam energies above E/A-lS MeV [CH183]. Cross sections for complete fusion, the other principal low energy mechanism, decline to zero with increasing bombarding energy [GAL82] [FAT85], signalling the rapid rise in pre-equilibrium processes [AWE81b] [AUB82], i.e. emission prior to the equilibration of the target-projectile system. Intermediate mass fragment cross sections clearly indicate contributions from such non- compound emission mechanisms [CHI83] [SOB83]. At the lowest energies fragments are emitted from equilibrated compound nuclei. At the highest energies, the reactions can result in a distinct division of the system into a participant region and the 8 projectile- and target-like spectators. At intermediate bombarding energies, there is no distinct separation between the spectator and participant components. A wide variety of emission mechanisms may contribute, including projectile fragmentation [MUR84] [MUR86], pre- equilibrium emission, and some remnant of deeply inelastic processes. It is difficult to distinguish clearly, either experimentally or conceptually, between these mechanisms. The greater part of the experimental data has been in the form of single particle inclusive cross sections. The information content of these data is limited, since the inclusive distributions are averaged over all reaction channels, impact parameters, and orientations. They contain no direct information on particle multiplicities, collective effects, localization, or energy and momentum balance. Although several models are based on thermodynamic concepts, the inclusive data support this assumption only weakly : the energy spectra of fragments and light particles appear to be random and phase space dominated, and the relative fragment yields can be reproduced by statistical models. Yet there are non-thermal models which can reproduce these cross sections as well [AIC84d] [FIE84]. Additional information is necessary to adequately characterize intermediate mass fragment emission. I.B, Scope and organization : The object of this dissertation is the characterization of fragmentation processes in intermediate energy heavy ion collisions through single and two-particle inclusive measurements. Of particular interest is the identification of non-statistical, dynamical aspects of 9 these reactions, as well as those aspects which can be treated with statistical theories. Useful experimental observables which can provide such information include the particle multiplicities, momentum transfer to the target, energy dissipation, and angular correlations. Of particular relevance for statistical treatments are the relative probabilities of emitting nuclei in excited states and the relative time scales for particle emission and equilibration. This thesis addresses three issues: 1) single particle inclusive cross sections, 2) two particle coincidence cross sections, and 3) the statistical aspects of intermediate mass fragment emission at intermediate energies. The data presented here are drawn principally from experiments with the following projectile-target systems : 12C + 197Au at E/A- 15 and 30 MeV [cures] [FIE84], 12c + Ag also at E/A- 15 and 30 MeV [CH183], 1“N + Ag at E/A- 35 MeV [BOU86s], and 323 + Ag at E/A-22.5 MeV [FIE86] [XU87]. The experimental procedures employed to study these systems are discussed in Chapter II. Single particle inclusive cross sections from all of these systems are presented in Chapter III. The object of this chapter is to present and characterize the single particle inclusive data and to provide a possible link between complex fragment emission from collisions at intermediate and high energies. Through the use of moving source parametrizations, the non-compound aspect of the differential cross sections is highlighted. The inclusive charge and isotopic distributions are also discussed. The dynamics of fragment emission, as evident in the two particle coincidence data, will be the object of study in Chapters IV-VI. The data presented were obtained from the 32$ + Ag system and , to a lesser 10 extent, the 14N + Ag system. Non-equilibrium aspects of reactions leading to intermediate mass fragment emission are established. Chapter IV addresses the cross sections for coincident light particles (p,d,t,a) and intermediate mass fragments. Angle and energy dependent correlations between these particles provide clear evidence for the importance of the reaction dynamics. Associated multiplicities of light particle and intermediate mass fragments are evaluated in Chapter V. These measurements help to determine the validity of high multiplicity fragmentation models in intermediate energy reactions. In Chapter VI the velocity distributions of the target-like residues from fragment emission are studied. These provide information on the global energy and momentum balance in the projectile-target system. In Chapters VII and VIII statistical aspects of fragment emission are addressed. Chapter VII presents calculations using a schematic model in which excited states of nuclei are populated with thermal probabilities, and compares these calculations with data to demonstrate the plausibility of such a statistical mechanism. The impact of the sequential decay of particle unbound states on the mass and isotopic distributions is investigated. Consequences of thermal populations of the excited states on measurements of excited state populations are discussed. Chapter VIII describes statistical model calculations which treat particle emission during and after the equilibration of the composite system. These calculations use a simplifying assumption, that a localized region of excitation from which particles are emitted at statistical rates is formed and evolves towards the fully equilibrated residual nucleus. They are compared to experimental charge distributions, momentum transfers, and energy spectra. 11 Chapter IX contains a summary of the present results and a few comments on the outlook for progress. Chapter II Experimental Methods and Analysis In the following discussion of the experiments and their analysis, a polar coordinate system is employed. The polar axis is defined by the A beam unit vector, b, and the origin by the target. The polar angle, 9, of a unit vector, v, is defined as 0-cos'l(v-b). The azimuthal angle, ¢, is defined to be positive clockwise when facing in the beam direction. II.A. 12C+ Ag.Au at E/A=15L30 MeV/A : II.A.l. Experimental objectives : At the time that the K500 cyclotron at Michigan State University became operational (Spring,1983), little was known of intermediate mass fragment production in heavy ion collisions. The objective of this first experiment was the measurement of single particle inclusive cross sections of intermediate mass fragments in order to establish the existence of fragment production in intermediate energy heavy ion collisions, and the comparison of the results with data from low energy 12 l3 heavy ion collisions and high energy proton-nucleus collisions. II.A,2. Detectors and geometry : Beams of 12C ions accelerated to energies of 180 and 360 MeV were used to bombard self supporting Ag and Au targets with areal densities of 0.6 mg/cmz. Fragments of 2-4-20 were detected by a single telescope positioned at polar angles greater than the grazing angles of Coulomb trajectories ( flgrz 11° and 25° for 12C+Au reactions and 6 rz7° and 16° for 12C+Ag reactions at E/A-30 and 15 MeV, respectively). The telescope consisted of an ion chamber and two 0.4 mm thick silicon surface barrier detectors with active areas of 450 mmz. The ion chamber was a 10 cm deep Frisch grid ion chamber [BAR75] and was operated with an Ar(90%)- CH4(10%) gas mixture at a pressure of 80 Torr. The solid angle subtended by the telescope was 5 msr. II.A.3. Analysis and calibration : The normalization of the absolute cross sections is calculated from the target thickness, detector solid angle, and integrated beam current according to the formula 4 99 A N Z - 2.66-10' , (II-1) do ‘ o p I where da/dfl is the differential cross section in mb/sr, A is the atomic weight of the target, N is the number of counts, 0 is the solid angle of . . . . 2 . the detector in sr, p IS the target thickness in mg/cm , I is the l4 integrated beam current in nanocoulombs, and 2 is the charge state of the beam particles collected in the Faraday cup. The estimated uncertainty is less than 20%. The energy calibrations for the solid state detectors were assumed to be linear over the measured energy range, i.e. of the form E=m(x-o), where E is the particle energy, x is the ADC channel, and m and o are calibration parameters. These parameters were obtained in a two step process. Firstly, a series of known charges was injected into the input of the preamplifier to produce a linear curve of pulse height vs. energy, which determines the offset, 0, of the calibration curve. Secondly, a thin, high resolution source of alpha particles was used to provide an absolute calibration point which determines the slope, m, of the pulse height vs. energy curve. The resulting energy calibration of the solid state detectors is accurate to within 3%. The energy calibration for the ionization chamber was obtained by comparing the pulse heights of signals from fragments of known element and energy with calculated energy losses for those heavy ions. The energy calibration of the gas counter is accurate to within 10%. Particle identification was obtained from the elemental localization in a AE-E scatter plot, such as is shown in Figure II-l. The elemental cross sections were obtained for both Ag and Au targets at E/A- 15 MeV and 30 MeV. For the 12C + Au reaction at E/A-30 MeV, the statistics were sufficient to produce differential cross sections for fragments up to Z-lO. Energetic lighter fragments, 254, were not stopped in the detector; therefore, differential energy spectra are not shown for these fragments. Integrated and average cross sections are estimated from the cross sections at larger angles, where the cross sections for 15 MSUX-83476 i . l2C +Au, T E l T /A = 30 MeV, Blob = 70° E-AE (MeV) AE (MeV) 197 Figure II-l : A scatter plot of AE vs. E-AE for the reaction 12C + Au at E/A-30 MeV measured at 0-70°. peak kinetic energies predicted by a generalization of the systematics of fission Coulomb kinetic energies [V1085]. The solid curve represents the locus of 16 the emission of penetrating particles are small. When more than one particle arrives in the telescope within its resolving time (zl ps), the event will produce a AE-E signal which does not fall on a locus corresponding to one of the incident particles. Such events generate a background which is most evident at low energies and can be seen in Figure II-l in the region between AE-lO and 40 MeV and around E-AE-20 MeV. The low energy thresholds of energy spectra presented in Chapter III are set at values where this background becomes small. A method for minimizing this contamination is discussed in Section II.C.3. II.B, 14N + Ag at E/Aa35 MeV : II.B.1. Experimental objectives : This experiment was designed to measure both the single particle inclusive spectra of intermediate mass fragments and the angular distributions of target-like residues detected in coincidence with these fragments. The inclusive spectra provide information on the relative importance of equilibrium and non-equilibrium fragment emission processes. The angular distributions of coincident target residues provide information on two aspects of fragment producing reactions. Firstly, they determine whether fragments are typically produced by the complete disintegration of the target nuclei. Secondly, if heavy target residues remain, their angular distributions are related to the momentum transferred to the target system by the projectile, the relative 17 sequence of particle emission, and other dynamical aspects of the reaction. The 14N beam at E/A-35 MeV was chosen because it was the beam with the highest energy per nucleon available at the NSCL at the time. This high energy assured reasonably large fragment cross sections, as well as a sizable cross section for incomplete fusion reactions. The Ag target was chosen because it is relatively heavy and yet has a low fission cross section, somewhat simplifying the interpretation of the data. In addition, other measurements with Ag targets were available for comparison [CUR85t]. II.B.2. Detectors and geometry : The reactions were produced in a 1.1 mg/cm2 thick self-supporting Ag target by 490 MeV 14N ions. The arrangement of the detectors is depicted in Figure II-2. Intermediate mass fragments were detected in three detectors at laboratory polar angles of 6-32.5°, 45°, and 57.5°, angles significantly larger than the grazing angle of a Coulomb trajectory, Ogrz6°. These detectors define the azimuthal angle ¢=0°. Coincident target-like residual nuclei were detected with a position sensitive multiwire detector with a 20 cm x 16 cm active area. The detector was located 32 cm from the target and, in separate measurements, at polar angles of 6=30° and 9=45° on the opposite side of the beam from the intermediate mass fragment detectors (¢=180°). Thus, coincidence measurements were performed for the range of polar angles in the reaction plane, 12°50560°. The three intermediate mass fragment telescopes were contained in a l8 MSU-86-286 Figure II-2 : A schematic diagram of the experimental layout for the 14N + Ag coincidence studies. The position sensitive detector is labelled as "HR" and the ionization chamber-solid state telescopes are labelled as "IMF". 19 single three detector system, as shown in Figure II-3. Each telescope consisted of a 10.5 cm long Frisch grid ion chamber (AE1), a .44 mm thick silicon surface barrier detector (AE2), and a 5 mm thick lithium drifted silicon solid state detector (E). The ion chamber was operated with a Ar(90%)-CH4(10%) gas mixture at a pressure of 90 Torr. Each telescope subtended a solid angle of 9.1 msr. A schematic diagram of the position sensitive detector is shown in Figure II-4. This multiwire avalanche counter consists of two planes of parallel wires which act as anodes separated by a conducting foil cathode plane. The wires in each plane are arranged in parallel and are read from a tapped delay line. The wires of each plane are perpendicular to the wires of the other. The avalanche at the anode induces a signal at the cathode foil which serves to provide a reference time signal. The relative time between the cathode time signal and the anode signal, which has a position dependent delay, is directly related to the position of the incident ion. II.B.3. Analysis and calibration : The normalization for absolute cross sections and the energy calibrations were obtained as described in Section II.A.3. The normalization is accurate to 10%. The energy calibrations for the intermediate mass fragments were also obtained with uncertainties similar to the experiment in Section II.A. The position calibration for the target recoil detector was obtained with a metal mask placed over the front face of the detector. The position of this mask with respect to the laboratory coordinate MSU-86-285 L_ E CATHODE A T" FRISCH/ Anooe’ , ': : GR") 5 ENTRANCE , 3:.._"—A '95 cm ————". WINDOW Figure II-3 : A schematic diagram of the triple Frisch-grid ion chamber system used in two of the experiments discussed in this thesis. The solid state detectors are contained in the gas volume. MSU-86-253 I PARTICLE I47) LEFT LIN RIGHT ” E 4H7 W' D (b) a. "° “V Figure II-4 : A schematic diagram of the multiwire avalanche counter used in the 14N + Ag experiment. The ionization avalanches occur about the wires in the anode planes labelled a and c. The signals induced on the wires undergo a position dependent delay before amplification. A signal induced on the foil cathode, b, is used as a time reference. 22 system-was determined by optical alignment. The uncertainties are less than 1°. Single particle inclusive cross sections for intermediate mass fragments were obtained in addition to angular distributions of coincident target-like nuclei. These angular distributions were analyzed as functions of fragment charge and momentum. II.C. 328 + Ag. E/A=22.5 MeV, particle-particle coincidence measurements : II.C.l. Experimental objectives : This experiment had two principal objectives. The first was to measure the distributions and multiplicities of non-equilibrium light particles emitted in collisions producing intermediate mass fragments. These provide information on the effects of the nuclear mean field on particle emission and on the momentum and energy transfer from the projectile to the target. The second objective was the determination of the velocity distributions of target-like residues from reactions which produce fragments. The projectile, 325 at E/A-22.5 MeV, was chosen because of its large total energy. The Ag target was chosen because it is heavy and non-fissile, and because other measurements with Ag targets were available for comparison. This experiment was performed at the Holifield Heavy Ion Research Facility at Oak Ridge National Laboratory. 23 11.6.2. Detectors and geometry : Two self-supporting Ag targets were used. For the light particle- intermediate mass fragment measurements, a target with an areal density of 3.2 mg/cm2 was used in order to maximize coincidence count rates. For the measurements of coincident intermediate mass fragments and heavy residues, a .75 mg/cm2 thick target was used in order to reduce the energy loss of the heavy residues in the target. This thinner target was also used to obtain the single particle inclusive cross sections. The experimental geometry is depicted in Figure II—S and summarized in Table II-l. Light particles were detected at the azimuthal angle ¢-180°, and the polar angles 6- 40° and 70°, and at the azimuthal angle of ¢--90°, and the polar angles 9=40°, 70°, and 130° ( see Figure II-S for an illustration ). Light particle telescopes consisted of 400 pm thick silicon surface barrier detectors as AE elements and 10 cm thick NaI(Tl) scintillation detectors as E elements. Intermediate mass fragments were detected at laboratory polar angles, 0-27.5°, 40.°, and 52.5° with respect to the beam axis and at an azimuthal angle of ¢-0°. They were detected with the three telescope system described in Section II.B.2. The ion chamber was operated with an Ar(90%)-CH4(10%) gas mixture at a pressure of 150 Torr. The position sensitive detector was located 50 cm from the target and had an active area of 11 cm x 11 cm. It was placed at the azimuthal angle, ¢-180°, and at polar angles, 0=1l°, 16°, and 22°. This detector [BRE79] [BRE84] is shown schematically in Figure II-6. It consisted of two 50pg polypropylene cathode foils with parallel conducting strips evaporated onto their surfaces, separated by a grid of thin wires which 24 MSU-85-566 Figure II-S : A schematic diagram of the 32$ + Ag particle-particle coincidence experiment. The intermediate mass fragments are labelled as "IMF", the light particle detectors, as "LP", and the position sensitive detector, as "HR". 25 Table II-l : The positions and solid angles of the light particle detectors (LP) and the intermediate mass fragment detectors (IMF) used to study the 32$ + Ag system, as discussed in Section II.C. | detector | 9 (deg) | ¢ (deg) | A0 (msr) | L l l l L I | I | | | LP-l | 40 | 180 | 27. | | LP-2 | 40 l -90 | 21.8 | | LP-3 | 70 | 180 | 64.6 | ] LP-4 | 70 | -90 | 34. | | LP-5 | 130 | -90 | 64.6 | ] IMF-1 | 27.5 | 0 | 7.9 | | IMF-2 | 40 | 0 | 7.9 | ] IMF-3 ] 52.5 | 0 | 7.9 | 26 stop evo oroted stri s D D \\ induced charge _'_x' ggthgdg porficm Figure II-6 : A schematic diagram of the Breskin design position sensitive detector. The ionization avalanches occur at the central wire anode. The signals induced on the evaporated stripes on the foil cathode undergo a position dependent delay before amplification. 27 acts as the anode plane. The parallel strips in each foil are connected in parallel to a delay line. The two foils are oriented so that the evaporated strips will be mutually perpendicular. The position is calculated from the relative time of arrival of the anode signal and the signal induced on the cathode delay line. II.C.3. Analysis and calibration : The normalization of the absolute cross sections, obtained as discussed previously, is accurate to 10%. The energy calibrations of the solid state detectors in the intermediate mass fragment telescopes and AE elements of the light particle telescopes were obtained with a pulser and an alpha source and are accurate to 5%. The ionization chamber calibrations are accurate to 10%. The NaI(Tl) scintillation detectors were calibrated with peaks from protons emitted in the elastic scattering of 328 ions from a polyethylene target. The NaI(T1) detector energy calibrations are accurate to about 5%. Intermediate mass fragments were identified elementally using the localization of the elements in AE-E scatter plots. For detectors of large solid angle, the spectra are contaminated by events in which two particles are detected simultaneously in the same telescope. Light particles may arrive with heavier fragments which deposit most or all of their energy in the gas AEl element. Such events produce a low energy contamination of the energy spectra of intermediate mass fragments consisting of misidentified particles. With a two element telescope, this cOntamination is impossible to eliminate. With three element telescopes, as were used here, the events which produce signals in three 28 elements can be rejected by double indentification: particles which are identified differently in the AEl-(AE2+E) and AEz-E scatter plots are rejected. The position response of the parallel plate multiwire detector was calibrated with respect to the edges of the active area. The calibrations are accurate to one degree. The known flight times of elastically scattered beam particles provided the absolute time calibration. The period of the cyclotron beam was used to calibrate the conversion constants of the TDC. The resulting velocity determinations are accurate to within 5%. In order to measure the velocities of target like recoil nuclei, it is necessary to measure their times-of-flight. The quantity measured was a time interval between a signal which begins the clock (supplied by the overlap coincidence between an intermediate mass fragment telescope logic signal and a logic signal generated from the anode pulse) and the signal from the anode of the position sensitive detector, which is delayed so that it stops the clock. This time interval differs from the actual time of flight by a constant, which is provided in the time calibration. Because of energy losses in the target foil, the measured velocity spectra of the target-like residues are not the original velocities of the emitted particles. A correction for this energy degradation would increase the velocities by 5%-10%, depending on the mass and charge of the residue. These quantities were not measured. However, an additional kinematic effect connected with the finite acceptance of our detectors biases the velocity spectra towards higher velocities. This produces as much as a 10% increase in the measured velocities. A more complete 29 discussion of this effect is included in Appendix A. Because of the uncertainties involved in estimating the effects of either mechanism, they are assumed to cancel and no corrections to the velocity distributions have been applied. The raw position spectra, as shown in Figure II-7, are modulated by spurious structures resulting from non-linearities in the detector response. These have been eliminated by correcting the position histogram by a factor which represents the deviation of the raw, ungated position spectra from a smooth linear dependence on angle. The position spectra from the three detector positions were corrected, normalized, and combined to produce a distribution as shown in Figure II-8. The distributions are normalized to represent the probability of detection, P(0), per degree; here 0 is the polar angle of the projection of the recoil velocity onto the reaction plane. 32 II D S + A E A- 2.5 MeV article- amma coincidence measurements: II.D.l. Experimental objectives : In this experiment the fractions of intermediate mass fragments emitted in their particle stable excited states were measured. This set of measurements provides information about the partition of excitation energy among the available intrinsic degrees of freedom which are represented by the excited states of fragments. Statistical models of fragment production make certain predictions concerning the relative population of these excited states. Comparisons between such calculations and measured values may indicate the applicability of 30 MSU—86-400 I . . a . I . . r Ag(3zs,Li), au=27.5° 640 MeV/c 5 PL, s 960 MeV/c Cacf=:].].o l 1 l l l l 1 N rP 5 10 15 £9cf=::l63° Yield l LL11 O (Icf==23230 6 (degrees) Figure II-7 : Raw position spectra from the Breskin detector placed at the three laboratory angles from target-like fragments detected in coincidence with lithium fragments detected at €-27°. Detector originated modulations of the spectra are easily recognized. 31 MSU-86—401 1.50 "lTr'ifirlliTrrI[l1|T['IIII'IIIJ I AsCZSLi), 9u=27.5° I 1 25 "—640 MeV/c é Pu s 960 MeV/0.: ? : O...¢¢ : I ' ’4» . 33 1.00 :— ¢ _‘ i—u _ ¢ - Q0 _ + - CD _ ¢ - 3 0.75 :— ¢ __ A - ( x 100 ) a. _ CD : . : E: 0.50 :— .. ‘5. 0.25 -— _. 0.00 L.'l'l1"'lllllilllLLl I ll 111L- 6 (degrees) Figure II-8 : Final angular distribution from the raw spectra in Figure II-7. The distribution is smooth and relatively free of spurious structures. 32 statistical theories to fragment emission and the state of the systems giving rise to fragments. The 328 + Ag projectile-target system was chosen because of the availability of other measurements to which these results might be complementary. II.B.2. Detectors and geometry : The fragments are produced in reactions induced in a 1.1 mg/cm2 thick natural Ag target by a beam of 328 ions accelerated to an energy of 720 MeV at the Holifield Heavy Ion Facility. The intermediate mass fragments were detected by solid state detector telescopes at polar angles, 9- 20°, 25°, 30°, 45°, and 50°. The telescopes consisted of two transmission solid state detectors, AEl and AE2’ and a solid state detector which measured the total remaining energy, E. The detector dimensions are summarized in Table II-2. Discrete gamma rays were detected with the Oak Ridge Spin Spectrometer. The results discussed here were obtained from six Compton- shielded germanium detectors which were placed in the Spin Spectrometer. A recent version of the Compton shielded germanium detectors is shown in Figure II—9. The resolution of the detectors is of the order of 2 keV for 1 MeV gamma rays. The maximum opening angle for each detector is approximately 12°. Typical efficiencies for gamma rays emitted from the target position are given in Table II-3. The geometrical arrangement of the gamma ray and particle detectors is summarized in Tables II-2 and II-3. Table II-2 33 The positions, solid angles, and construction of state fragment detectors used in the coincidence experiment, as discussed in Section II D. 32 S + Ag particle - gamma I det I 6(deg) I ¢(deg) I AO(msr) Idetector depth the solid (#m) I 1 l 1 1 1 A31 1 AEZ 1 E 1 I I I I I I I I I I 20 I 90 I 9.75 I 75 I 100 I 1500 I I I 25 I -90 I 10.1 I 75 I 100 I 1500 I I I 30 I 0 I 15.35 I 75 I 100 I 1500 I I I 45 I 90 I 36.0 I 50 I 75 I 1500 I I I 50 I -90 I 28.5 I 44 I 75 I 1500 I 34 Table II-3 : The positions and typical efficiencies of the Compton supressed germanium detectors, as discussed in Section II.D. The number in parenthesis next to the detector number is the parameter number for the Compton shield. The position of each detector in the ORNL Spin Spectrometer is given in the second column. The remaining columns give the absolute position with respect to the beam axis and the absolute efficiency of each detector at 300 keV. | y-detector | SS-pos | 9 (deg) | d (deg) | 6 (@300 keV)| I I I I I I I 1 (18) I 00 I 63.4 I 36.0 I .00105 I I 2 (66) I L0 I 116.6 I -72.0 I .00068 I I 3 (48) I 10 I 116.6 I 72.0 I L00079 I I 4 (30) I F0 I 63.4 I -108.0 I .00092 I I 5 (24) I E0 I 63.4 I -36.0 I .00095 I I 6 (12) I 00 I 63.4 I 108.0 I .00101 I 35 O'~L-M 03d 76"! 75 HIGH VOLTAGE PM TU“: «07 . j ' § \I SIGNAL , ,/ 7 ,/ // , I '9" W. Iii?”*"5:5=§s§s§s§a§s§s Uc‘ 9“ ME! I! §_;I .‘I . N 2 5° “‘5 0A I 13:15:}: 5:323:23 , :: ‘5':- 3“ , 'z' :-:-:-:-.':1 V, _ ”I , W“ " I- 0552 ' {7", M .11 //flagq///// . on runs 9 . _ I t” "//_’// / '7 I) \ 0212' / /// /////% _\.\\\ 2501894ou I I /%\\N\\ i 7' i 50 Oman ‘~—ucuf smcw _.I_. ALUMINUM sure —/ I “WWW I-c-j 6.25 ALL DIMENSIONS m INCHES Figure II-9 : Schematic diagram of one of the Compton suppressed germanium detectors used in the particle-gamma coincidence experiment. 36 II.D.3. Analysis and calibration : Cross section normalizations and particle energy calibrations were obtained as discussed previously; they are accurate to 10% and 3%, respectively. The energy calibrations of the germanium gamma ray detectors were obtained from peaks provided by a number of standard sources, and are accurate to 1 keV. This experiment requires an accurate isotopic identification of fragments, even at the expense of detection efficiency. Instead of the particle identification method discussed previously, two different particle identification functions, K K Fl- (AE1+AE2+ E) -(AE2+ E) and (II-2) F2- (AE2+ E)K- EK , (II-3) were generated, where K is a constant value, Kal.74. An accepted particle event was required to satisfy consistent constraints on both identification functions. Less than 4% of accepted particles are estimated to be misidentified. The resulting detection efficiency is in the range of 80%-90%. The efficiency of the germanium detectors was determined in a two step process. In the first step, the relative efficiency of the detectors as a function of photon energy is determined by comparison of . . . . . 152 the integrated photopeak inten31t1es from tran51tions from Eu and 182Ta sources to the known relative transition intensities. This 37 results in relative efficiency curves as shown as solid points in Figure II-10a. These relative efficiencies were fit to a function, C E - (m-E+b)-exp(aIE-E0|T)-(1- 2 (E-d) + f (II-4) for purposes of interpolation, where E is the gamma ray energy, E is the parametrized relative detector efficiency, and the other quantities represent fitting parameters. This fit produces the solid curve shown in Figure II-lOa. The ratio of the data to the fit is shown in Figure II- lOb. The maximum deviation of the measured response from the fit is less than 10%. The normalization for absolute efficiency was obtained from sources which emit two coincident photons, specifically 758e, 88Y, and 60Co. The absolute efficiency of a detector is the probability, after correcting for the angular correlations between the two photons, that one photon is detected in that detector given that the other photon in the decay is detected in another detector (see Appendix B). These measurements were then used to normalize the relative efficiencies curves to absolute efficiencies. The errors in the resulting efficiencies are of the order of 5%. A spectrum of gamma rays measured in coincidence with 108 is shown in Figure II-lla. The gamma rays are Doppler shifted in the laboratory, so that E*- E QM). (II-5) 38 2 MSU—86-402 10 I I I I I I I I I .. I I .. .. (0) w " + 7 r 7 Detector 1 7 . 152Eu ‘ 18 1 II ZTYi II 101 -_I I l I I J I I I LI — 1.2 _ I I T I I I r I I r I _' (b) g 1.1 _. . -l : LIN I I 3 U) 1 0 :T I . «I I: E ++* ‘il 5 . 0A9 :-' .1 08 1 I 1 1 1 1 1 1 1 1 I " 100 1000 E7 (keV) Figure II-lO: The relative efficiency curve (a) for 7 detector 1 from a 152Eu source (circles) and 182Ta source (squares). The solid curve indicates the analytic interpolation discussed in the text. The ratio of the source data to the interpolation is shown in the bottom part of the figure (b). 39 MSU-86—403 1.0 _ I I I I I I f I I E 10B (2.154 - 1.740) 3 0'8 _ a) uncorrected spectrum ‘3 0.6 :— -3 '3 E : CD 0.4 g: . 0.2 :- _- 12 : : :3 0,0 : I : : .5 : 4 I 1 £0 E b) Doppler shifted spectrum I F-I " .. CD 0-9 .T' and background (0) 7 Di :| 3 0.8 I‘ 1 ! - I I 3 0 7 I ’I I " I I ' —- I '1 ". ‘ 0.6 I I . i 0.5 , 1 . . . i L . . -100 O 100 E—E7 (keV) Figure II-ll: The spectrum of gamma rays coincident with 10B ions, without Doppler shift, is shown in the upper part of the spectrum, (a). The bottom part of the figure, (b), shows the corrected spectrum, indicated by the histogram, and the estimated background, indicated by the solid points. The background is produced by Doppler shifting gamma rays coincident with other nuclei, as discussed in the text. 40 where cosfi-E-Ly, E: is the transition energy, E7 is the photon energy in the laboratory rest frame, fl is the velocity in the laboratory of the nucleus from which the gamma rays are emitted, g and z? are unit vectors in the direction of the detected particle and photon, respectively, and 0 is the relative angle between these two vectors. Due to summations over different particle energies and detection angles, lines from discrete transitions in intermediate mass fragments are Doppler shifted and broadened. Gamma rays from transitions in the relatively slow moving target residues are not greatly Doppler shifted in the laboratory. Therefore, gamma ray energies from each event were corrected for the Doppler shift on the assumption that the photon was emitted from the detected intermediate mass fragment. The corrected spectrum is shown as the histogram in Figure II-llb. The transition from the intermediate mass fragment is well resolved, while the lines from target-like residues are broadened. There is a change in the effective solid angle and efficiency of the photon detectors due to the relativistic Jacobian of transformation, |J|, between the solid angles in the laboratory and fragment rest frames: 38_ (l-Bcosfi)2 * IJI‘ " an 1-52 3 (II-6) * where 0 and 0 are the solid angles of the gamma ray detectors in the laboratory and fragment rest frames, respectively. (See Appendix C for a derivation.) The spectra were corrected on an event by event basis for 41 this change in efficiency and for the difference in the detector efficiencies at the energies 87 and E:. The gamma rays of interest lay on top of a background which contains noticeable contributions from discrete transitions in target- like nuclei. As already mentioned, when the gamma ray spectrum is Doppler corrected for the motion of the fragment in the laboratory, the fragment gamma peak is narrowed while the target related gamma peaks are broadened into complicated structures. This background must be subtracted from the area under the fragment gamma peak, so it is necessary to identify any structures under that peak. This target related background is only weakly dependent on the intermediate mass fragment; the laboratory background spectra can be assumed to be the same for spectra of neighboring fragments. This feature is utilized to estimate the background underneath the peak of interest. To calculate the background for gamma rays coincident with fragment A, the spectrum of gamma rays coincident with a fragment B with a similar mass and charge is utilized. The calculated background, * O BA(Ey)’ is given by -3? * d0 €(E1) B(E)-N2 2B(E)0(E) A7 EBA37 AAdn*2(E) 7 7 * gl-Qcosfiz x6(E , E . ) , (II-7) 7 (1432)”2 where BB(E7) is the gamma yield at energy E7 coincident with fragment B, aA(EA) is the cross section of fragment A at energy EA’ N is a 42 normalization constant, B is the velocity of the fragment A at the energy EA’ and 6(x,y) is 1 when x-y and 0 otherwise. The background for the 1OB gamma ray spectrum, generated from gamma rays in coincidence with neighboring nuclei, is indicated by the solid points in Figure II- llb. Five gamma ray transitions of interest were analyzed. These are the transitions from the following states : 8Li (E*=O.981 MeV, Jfl- 2+), 7Be(E*-O.429 MeV, J"-1/2'), lONE-2.154 MeV, J"- 1*), 123(E*-o.953 MeV, J”- 2+, E*-2.621 MeV, J"- 2'), and 13C(E*-3.854 MeV, J”=5/2'). The transition from the first excited state in 7Li (E*=O.477 MeV, Jfl=l/2-) was not analyzed because of contamination of the particle spectra by 83e nuclei which, after decay into two alpha particles, are misidentified as 7Li nuclei [WOZ72]. The transition from the first excited state in 1 OB (E*-O.718 MeV, Jfl- 1+) was also not analyzed. -Because the lifetime of this state is rather long, r-l.02 ns, nuclei excited in this state could have travelled a significant distance from the target (23cm), introducing a considerable uncertainty into the effective photon detection efficiency of the Compton shielded Ge detectors. This experiment yielded relative isotopic fragment cross sections in addition to coincidence gamma ray spectra. Chapter III Single Particle Inclusive Data Single particle inclusive energy spectra, dza/dEdfl, for light particles and intermediate mass fragments are shown as solid and open points in Figures III-1 through III—6. III.A. Features of the inclusive data : III.Aml. Light particle spectra : The energy spectra for p,d,t, and a particles produced in 32S induced reactions on Ag at E/A-22.5 MeV are shown in Figure 111-1. The cross sections in the forward hemisphere are dominated by products of non-equilibrium processes [AWE82a], i.e. the angular and energy distributions are inconsistent with emission from equilibrated compound nuclei. In fact, the energy thresholds in the present experiment exclude most of the contributions from equilibrium evaporation from the target. Previous studies of non-equilibrium light particle emission show that it is associated with violent nuclear collisions [AWE8lb] and that it is 43 44 MSU— 85— 504 lvffilTII Ag<°zs X) E /A=22.5 MeV—g r rvv ”C 03 :>' Q) E \ JD 3 c: “U [3:] 'U \ b N 'U 1 i i? 1(]-'1 .1. .‘ ”I!“ r.,.. l. L: _0....50..1.00 150 o 50 100 150 ENERGY (MeV) Figure III-l : Differential cross sections, 70°, and 130°, for light particles, p, d, t, and a, from 32S induced reactions on Ag at E/A-22.5 MeV. Also, cross sections measured at €-27.5° and 52.5° are shown for a particles. The solid curves are the result of the parametrization discussed in the text. dza/dEdO, measured at 6-40°, 45 MSU-86—404 I l I fi—vr I l’ I If r I I I .Lu “Aural 111ml].unuJ.-JJou.uum1_unuJ- IIIIIIII um dza/dEdQ (,ub/MeV-sr) l I um] .1 1.1an -uLfllLLJLfll-I‘LLIJMIJJlUHJ- .LJLUII‘» -L JAM-Ll “LII-J41 LI LAMA—J ILMIJ 10—2 h.. ..1... .1. ...| .... O 100 200 100 200 ENERGY (MeV) iv I 3- i 3- i I '2— I 0 Figure III-2 : Differential cross sections, dza/dEdfl, measured at 0-30°, 50°, 70°, and 120°, for fragments, 552510, from 120 induced reactions on Au at E/A-30 MeV. The solid curves are the result of the parameterization discussed in the text. f 46 MSU—86—405 TIT’V‘I’IIITIrTYIrVUllf'vrl’lITII’ 104 % Ag(14N,X) E/A = 35 MeV 141M unL..L ule__J_lJuflll_-l_Lu.Lfll_L_L uni—um maul—mum] 4.1441111- dzo/dEdQ (,ub/MeVsr) LLJ IJMLII -4L LIAN-J ._J_L.lflLfl|-_JJ unnl-.4 L1 1 l l AAAAAAA LIILIJJLLJIAI O 100 200 300 O 100 200 300 ENERGY (MeV) Figure III-3 : Differential cross sections, dZa/dEdfl, measured at 0-32.5°, 45°, and 57.5°, for fragments, 45259, from 14N induced reactions on Ag at E/A-35 MeV. The solid curves are the result of the parameterization discussed in the text. 47 MSU— 86- 406 ,. ,....,....,,...,...,,...-, 103 — E/A 35 MeV j 102 E_ X=Na_% 101 :— Q‘ ”I: 100 - é 10‘1 5 n 1cfi3 ;_ )(== )(= I) 1 E 321? -'!~‘%5h .3 10 ? (x10) ' ‘3 I 45°(x3) \ I 515° l....l ........ 1L .Lla11 O 50 100 150 0 50 1010 150 ENERGY (MeV) _lruwl.nunmL t—I‘ O 0 m1 dzo/dEdQ (,ub/MeV sr) 5.; ad i Mg =Al Si 4 H O O I I I 111'}! 1 p y- )- .— y- b P u- ‘3 3 I J J Figure III-4 : Differential cross sections, dza/dEdfl, measured at 0-32.5°, 45°, and 57.5°, for fragments, 1052515, from 14N induced reactions on Ag at E/A—3S MeV. The solid curves are the result of the parameterization discussed in the text. 48 MSU—BS—SOS I....,.---,....,-..,..-.,.-..,.. ,. 104 if Ag(3zs,X) E/A = 22.5 MeV 1: 10‘3 : ' ‘ 1 102 «2:150 X BeE 101 40° 1: 100 52.5° .5 10‘1 I dzo/dEdQ (pb/MeV-sr) . 1 f: — 1 1 1 ....... 1....1....1..‘ 101 0 100 200 300 0 100 200 300 ENERGY (MeV) Figure III-S : Differential cross sections, dza/dEdfl, measured at 0-27.5°, 40°, and 52.5°, for fragments, 35258, from 328 induced reactions on Ag at E/A-22.S MeV. The solid curves are the result of the parameterization discussed in the text. 49 MSU—85—567 'I'Il""I""l""I"'jIIH[H'II'H'I'I'II'I'II"II[H'IIII'II'II'II"Il'If' 104 r 32 1 ; Ag( 310 E/A = 22.5 MeV : N >< II CD Zx=10 r—b H H O O O O H N Y Y'VTIV1 I I 'VI'VI1 Y V‘I'IV‘I V V 5% ..1 . .......1 . ......| V 7' "1 I f7‘m1 V 'V'IVVII AAAA 101 r dga/dEdQ (,ub/MeVosr) I I IIIWII V 7 VIII AAAAAJ A AAAAAAA‘ A AAA-AA V VI’V'V'1 I VVY'V'V' I A AAALAAJ A AAAA‘AJ A AAAAA I VII VIII! H O H D p > h h D D D L .— . D D . .— D D . p , . D D p— p p D . .— D p » D .— . . D D H O N H O O """| Y 'V'V'I'l Y V'V'W'1 ml .1......l A .1 “Al V 'V'V'V1 T 'V'w'1 V V A AAAA 101 A AAAAAJ A A AAAA‘J A AAA‘ VIII'I' 1 'VVI'VV1 I I III / 11......1 . .......l . .. '1 V I VY'TII' V I A AA AAA] All ' p 10—1 -...11.-.1-...1.... ....1..-.1..-.J...- A.1..-.1--..1.1.1 -...1....1....|.-1- O 200 0 200 O 200 O 200 ENERGY (MeV) Figure III-6 : Differential cross sections, dza/dEdfl, measured at 0-27.5°, 40°, and 52.5°, for fragments, 952524, from 328 induced reactions on Ag at E/A-22.S MeV. The solid curves are the result of the parameterization discussed in the text. 50 principally sensitive to the laboratory bombarding energy per nucleon of the reaction [WESBZ]. The spectra slope exponentially towards higher energies and show no particular structure. III.AaZ. Intermediate mass fragment spectra : The energy spectra of intermediate mass fragments display several of the characteristic features observed previously in intermediate energy heavy ion and high energy proton induced reactions [$0383] [P0871] [HYD71] [WES78] [HIR84]. They exhibit broad maxima at energies slightly below the Coulomb energies for two touching charged spheres. At higher fragment energies the cross sections decrease exponentially with increasing energy. The slopes of the energy spectra become steeper with increasing fragment angle, changing the character of the energy spectra from Maxwellian to roughly Gaussian at very backward angles [SOB83]. The fragment spectra become increasingly Gaussian with increasing mass. The angular distributions in the center-of-mass system, do/d6C m for intermediate mass fragments produced in 12C induced reactions on Au at E/A-3O MeV are shown in Figure III-7. The cross sections are forward peaked for the case of light fragments but are approximately isotropic for the heaviest fragments. The energy spectra of all fragments are generally smooth and structureless. In this respect, they are reminiscent of evaporation spectra of neutrons and light particles from low energy fusion reactions. These have been interpreted as the result of a fully statistical, phase-space dominated emission process [WEI37] [HAUSZ]. While other features of the cross sections, such as the angular 51 MSU—83-580 F l I I I I I _ 8 '2C + Au -> X _ C Elob/A=3O MeV .. 0 - lq; , Ea __ _ B F 3‘ 3 b ’ 2 U - _ _ Mg(x0.5L\o\¢7 _ P Si(x0.4) .—+——'/O - S(X.3) H—_.\./‘ 102 :- ANX-I) DAN/o : — 1 I 1 1 l 1 1 ‘ IO 30 50 7O 90 HO I30 9cm.(deg) Figure III-7 : Differential cross sections, da/d9c for various m! fragments, 552518, from 126 induced reactions on Au at E/A-3O MeV. The solid curves are drawn to guide the eye. Emission from fission would produce a flat curve. 52 distributions, rule out equilibrium emission from the compound nucleus as a possible production mechanism, statistical emission as a general mechanism still provides an attractive framework for understanding fragmentation. III.B. Two source parametrization of the cross sections III.B.1. Moving source parametrizations : It was an early observation in high energy heavy ion reaction studies that particle spectra could be parametrized as isotropic thermal distributions corresponding to emission from one or more sources, moving in the laboratory with velocities different from the center-of—mass velocity [P0871]. Early work in relativistic collisions pictured three sources [WES76], clearly separated in momentum and coordinate space, corresponding to a target-like source, a projectile-like source, and an overlap, participant region with an intermediate velocity. Parametrizations of this form are also applicable at lower energies and to heavier particles. Close examination of intermediate mass fragment differential cross sections reveals that they can be associated with either a component which is approximately isotropic in the rest frame of the composite system [SOB83] or a component which is isOtropic in a frame moving with a velocity between those of the projectile and the center of mass. The forward peaked component is strongest for the lightest fragments but persists for fragments which are heavier than the projectile. This is also seen in 3He induced reactions [KWI86] where projectile fragmentation and transfer reactions would not be expected to 53 play a role in intermediate mass fragment production. At very forward angles, the cross sections (in 40Ar + Ag reactions at E/A=27 MeV [BOR84], for example) contain a strong component which is approximately isotropic in the rest frame of the projectile. However, this component contributes only weakly at the intermediate angles which are the subject of discussion here. In general, the angular distributions cannot be explained by isotropic emission from the compound nucleus and/or an excited projectile remnant [BOR84]. III.B.2. Description of present parametrization : In order to organize the present set of data and to provide reasonable extrapolations to unmeasured scattering angles, the differential cross sections of intermediate mass fragments are fit with a two source parametrization which models emission from both a fusion- like and a non-equilibrium (intermediate rapidity) source, each of which emits isotropically with a Maxwellian energy distribution in its respective rest frame. A projectile velocity source is not included because the measurements were taken at relatively large angles, where contributions from such a source should be small. The emission of intermediate mass fragments is evidently more complicated, involving a continuum of "sources" from the quasi-elastic regime to the compound nucleus. However, the present single particle data do not justify a larger parameter space than the restricted two source space utilized here. Although the specific choice of parametrization is not unique, it may facilitate the formation of a qualitative picture of the relative time scales for particle emission and equilibration. Similar points of 54 view have been adopted elsewhere [TRO85p], facilitating the comparison of different data sets. This parametrization is formulated in a rest frame, Oeq’ which moves with a velocity, v with respect to the laboratory frame; Ve is eq’ chosen to be close to the average velocity of fusion-like residues. In this frame the cross sections are parametrized as ~ d2}; E'-vC ax(6 ,E ) - EE7EE_ = I [ Neq-(E -Vc) exp(- T . eq Ef + Nf /(E’—Vc)- Ef exp(- Tf ) ] 2 -1/2 (Vc’ va)2 x (2nwx) exp[- 2 ] dVC, (111-1) 2 w X with Ef- E'-vc + Ed- 2-/(E'-VC)-Ed c056' , and (III-2) 1 2 Ed- 2 Mx(vf-veq) . (III-3) Here, the factor R-(Mp+ MT- Mx)/(Mp+ MT) is due to momentum conservation, where Mp’ MT’ and Mx denote the masses of the projectile, target, and fragment, respectively; E' is the energy of the fragment in the frame, Oeq; vf is the velocity of the fast, non-equilibrium source with respect to the laboratory; and N and T (N and T ) are the f f eq eq normalization and "temperature" parameters which characterize the fast (slow) source. To avoid sharp cut-offs at low energies, Eq. III-l contains a weighted average over a Gaussian distribution of Coulomb SS barriers [P0871], Vc. The parameters, VX and wx, are the mean and the standard deviation of this distribution. Comparison to experimental data is made after transforming Eq. III-1 to the laboratory rest frame to yield the laboratory cross sections 3X(0,E). The values of veq- 0.79-v0, 0.75-v0, and 0.86ovo are used for 12C, 14N, and 328 induced reactions, respectively, where v0 is the beam velocity. These values are consistent with the systematics for linear momentum transfer observed in measurements on fissile targets [FAT85]. We have assumed in the formulation of Eq. III-l [GOL78] that particles are emitted from the surface of both the equilibrium and non- equilibrium sources. Temperature parameters obtained under this assumption are approximately 10% lower than those extracted from parametrizations of volume emission. In order to reduce the number of variable parameters, the Coulomb widths of the distributions are set to fixed values. For the data from the 328 induced reactions on Ag, the values wx- 2, 4, 11 and 17 MeV for Zx- l, 2, 3 and 452x524, respectively, are used. For the other fragments, the value is fixed at wx- 8 MeV. The temperature parameters of the fusion-like source were fixed at Teq- 4.7 MeV (12C + Au), 6.9 MeV 14 3 eq f f f x free parameters. III.B.3. Results of parametrization : The resulting fits are displayed as solid lines in the Figures III- 1 - III-6,. The parametrization provides a reasonably good fit to all fragments. The fits are best for the heavier fragments (Zx>10), where 56 the angular distributions are nearly isotropic in the rest frame Oe , and for the products of the 328 induced reaction. There are some problems for the lighter fragments (Zx - 2 sin 91 , (III-4) i for fragments from 12C induced reactions on Au and Ag, respectively at 62 MSU-86—407 TlllrllTIITlllrlilllrlr111f lllll Ill] 0 0 E/A = 15 MeV ° 0 E/A = 30 MeV O H 0 C23 llllllll 11111111 9 o o 9 ¢ ¢ 0 +¢ ¢¢¢¢¢¢¢¢ . C l I ‘0' I lllllll llllllll + + + +"’ +[++++++ l l (,ub/sr) S N H O H I l [Illll 11111111 I l 100 [III [III llllllllllllllllll 0 5 10 15 20 25 Zx Figure III-8 : Average differential cross sections, , over the region 50°505120° for fragments of charge, 2, from 12C induced reactions on Au at E/A-lS and 30 MeV (solid and open points, respectively). 63 MSU—86—408 4 I] 111111 I IT IT] I I I I l l TrT I I II 10 L— 12 ‘3 : Ag( C.X) _ : 00 o E/A = 15 MeV - Z 0 E/A = :30 MeV _ ‘1: - 02-26 - 00 E :3~103 :— _5 /\ C I C -— _ Po \ - _ b .5 ~ _ v 102 .— 7 L11] I I 1] I [III lllllLl II I 111 L1 0 5 10 15 20 25 ZX Figure III-9 : Average differential cross sections, , for fragments of charge, Z, from 126 induced reactions on Ag at E/A-lS ( 50°50570°, solid points ) and 30 MeV ( 40°50570°, open points ). The solid curve depicts a strict power law dependence, Ya 2-2'6. 64 E/A- 15 and 30 MeV. The cross sections exhibit a strong energy dependence, with the fragment cross sections in the 12C + Au reaction increasing by a factor of 10 over the energy range from 180 MeV to 360 MeV. For both targets, the cross sections decrease rapidly with increasing fragment charge up to Z=lO. The cross sections for the 12C + Au reaction begin to increase for 2x2 14. This increase is most likely due to the tails of the fission distribution, which becomes broader with increasing excitation energy and angular momentum [TSA83]. Total elemental cross sections and average differential cross sections for intermediate mass fragments from 12C + Au, 14N + Ag, and 32$ + Ag reactions are shown in Figures III-10,11, and 12, respectively. The total cross sections, ax, are extrapolations from the parametrizations discussed previously and are shown as open points. Because of the limited range of angles fit by this parametrization, the estimates do not include possible contributions from projectile-like sources. For comparison, the angular average of the experimental cross sections, , as defined in Eq. III-4, is also shown in these figures as solid points and open squares. ( In Figure III-10, the averages 1 and 2 correspond to the average over all measured angles and over the three most backward angles, respectively.) The elemental yield curves decrease smoothly with fragment charge. In contrast to elemental distributions measured at lower energies, where deeply inelastic processes dominate, the yields do not display an enhancement in the region of the projectile [MOR75]. The elemental yields can be approximated by a power law, oxa 2", as have the mass distributions in high energy proton-induced fragmentation reactions [FIN82] [HIR84]. Such a power law dependence, with an exponent of 65 MSU—86—409 102 E— 112 I l I ""3102 5 Au( C,X), E/AzBO MeV 3 I O 0,, I /\ I o o 1 - >3 (3 A 101 “:— o D 2—§ 101 E ,o E E S E : o : A >< — o o - B t) I. C3 C7, 100 :- 0 ° 0 1100 \ C [3 I 00 : D : E, : a . I C] ., I. 10—1 :— u a 110—1 -I I I I I I LI I I I I I I I I I LII I I II 2 4 6 8 10 12 Figure III-10: The extrapolated total cross sections, ax, for fragments produced in 120 induced reactions on Au at E/A-3O MeV are shown as open circles. The solid circles and open squares show the average differential cross sections, (dax/dfl>, measured over the angular ranges 30°$05120° and 50°505120°, respectively. 1o2 ”’23 v 101 >< b 100 10‘1 66 MSU-86—41 I I l I I I I I I I I I I I I Ag(14N,X), E /A=35 MeV I E- 0 0x I 0 : sz-zs II IIIIII I I IIITIII ‘ _ IIIIIIII I I I IIIIIII lIIlIlII l 102 /\ q >< \ Q... :3 V 1()1 ./-\ B .O" \ (1) C3 100 10 1 Figure III-ll: The extrapolated total cross sections, ox, for fragments produced in 14 N induced reactions on Ag at E/A-BS MeV are shown as open circles. The solid circles show the average differential cross sections, , measured over the angular range 32.5’56557.5°. curves depict a power law dependence YaZ -2.8 The solid 67 MSU-BS-SOB IIIlIIIIlIIIIIIIIIIIIIIIIIII jxég(3£%3’)() E/A=22.5 MeV III] I I I o 0 0x 105 . (dax/dfb _ 105 o __ Y cc Zx—La I 11111 0x (Mb) H o 4:. (.Is/q'fl) I IIITII I l 103 —- ‘2- 103 llllllllllllllILlllIllllIlll o 5 10 15 20 25 30 Zx Figure III-12: The extrapolated total cross sections, ax, for fragments produced in 328 induced reactions on Ag at E/A-22.5 MeV are shown as open circles. The solid circles show the average differential cross sections, , measured over the angular range 27.5°50552.S°. The solid curves depict a power law dependence YaZ-l'6. 68 r-2.6, was interpreted as a signature of cluster formation near the critical point in the liquid-gas phase diagram of nuclear matter. If this were true then all measured fragment distributions should be steeper, i.e. r>2.6. A similar dependence (r-2.6) was observed in the fragment cross sections resulting from 12C + Ag collisions at E/A=30 MeV as shown in Figure III-9. However, a very different value, 7:1.6, describes the data from 328 induced reactions. A similar value (121.7) was observed for 40Ar induced reactions on Au at E/A=29.7 MeV [TR086p]. The liquid-gas models do not explain this variation. The general shape of the elemental yield curves, as well as their energy dependence, can also be understood in the framework of a statistical evaporation calculation [FRI83a]. In that model, the calculated yield curve results from a dependence on the binding energies and Coulomb barriers of the emitted fragments. Nonetheless, the dependence of the fragment cross sections on projectile mass suggests that parameters other than binding energy and Coulomb barriers, e.g. angular momentum and shape degrees of freedom, affect the fragment distributions. These dependences have, however, not yet been established quantitatively. III.C.2. Isotopically resolved yields : The relative isotopic yields for intermediate mass fragments, 352x58, emitted in the 32S + Ag reaction at E/Aa22.5 MeV, are shown as solid points in Figure III-l3. These were obtained fitting the double- gated energy spectra (in the gamma ray-particle coincidence experiment described in Section II.D.) with the parameterization discussed in MSU- 86- -411 10.00 :- l [Bé r r I I ‘g 5A ( s, ), Iii/1131:1255 MeV; 1.00 E— _ 1:“ __ '3 5 Zx—B . 35 Zx—6 i I 9 I: O 2 I3.l() F‘ 'E?‘ . -5 E ‘P as a I 1 i: 2 PC I I " I ‘ "63' (3131 I I °"" 1.00 -— 1:— _ 1- D>q E 212::14 EE 22x-7’ a L O “t . <0 I °/\o 1’ 1 .2 0.10 =— i— 1 +9 E 55 . : 2 F “g ‘ I j 3:) 001 I I ' ' ' I ' I I I 1-00 E'z ‘2'? Zx=8 '3 ; :: - o 1 b dI- . .. (3.1C) E' 'E?‘ -E 001 h " I I I I | I d '-3210 -2—101234 Figure III-13: Relative isotopic yields for isotopes of elements, 352x58, are shown as solid points. The solid curves represent the predictions of a simple statistical model discussed in the text. 70 Section III.B.2. All parameters except the normalizations were fixed at the values which characterize the elemental distributions (see Table III-3). The ratios of the two normalization constants, Nf and Neq’ were also constrained to be the same as in the elemental distributions. Thus, there is a single normalization parameter which results from this procedure. The error bars depict estimated systematic uncertainties. Additional uncertainties, of less than 20%, may also result from inefficiencies in the particle identification process. The 7Li yield may be over-estimated by 5-25% because of the inclusion of 8Be fragments in the spectra. The solid lines are relative yields estimated with a simple statistical calculation. For each element, the relative isotopic yields, Y(X), were assumed to be simply a function of the binding energy of the daughter system of the reaction A»B+X [ART77]: Y(X)a exp( Bsep/ T ). (III-5) where Bsep- Bx+ BB are the binding energies of the fragments. The parent,A, was assumed to be 107Ag; the binding energy for the daughter, BB, was calculated with a liquid-drop formula; the emission temperature is Tz3.8 MeV. The relative normalization for each element is chosen for appearance. This simple estimation supports the assumption of statistical emission, which in turn may provide a tractable framework for understanding fragment emission. Chapter IV Light Particle-Intermediate Mass Fragment Correlations The single particle inclusive cross sections discussed in Chapter III do not discriminate between different possible classes of reactions. More information is contained in two particle coincidence data. In this chapter, the spectra and angular distributions of non-equilibrium light particles coincident with intermediate mass fragments emitted in 325 induced reactions on Ag at the incident energy of E/Aa22.5 MeV [FIE86] will be explored. The distributions of coincident light particles provide information on the dynamical aspects of the heavy ion collisions which produce intermediate mass fragments. This type of study has been applied to relativistic nuclear collisions [MEY80] [WAR83]. IV.A, Spectra of light particles coincident with intermediate mass fragments : The single particle inclusive spectra of light particles, discussed in Chapter III, exhibited rather structureless exponential slopes. It is 71 72 conceivable that such structureless spectra result from the superposition of several different mechanisms, each contributing differently to the inclusive distributions. Reactions in which intermediate mass fragments are emitted might constitute such a distinct class of reactions for which the light particle spectra differ significantly from the single particle inclusive distributions; coincidence studies may reveal such differences. Energy spectra of non-equilibrium light particles detected in coincidence with intermediate mass fragments are compared to single particle inclusive cross sections in Figure IV-l. The single particle inclusive spectra are indicated by the solid curves. The spectra were measured at oy- 40° (circles) and 70° (diamonds) at relative azimuthal angles from the intermediate mass fragment, detected at 9x-27.5°, of A¢-180° (solid points) and A¢-90° (open points). The proton spectra are shown in the upper part of the figure, with the spectra coincident with Li and C on the left and right hand sides of the figure, respectively. The singles and coincidence spectra are remarkably similar. The coincidence spectra fall exponentially with increasing energy, becoming steeper at larger angles, and thus retain the essentially random, statistical character of the inclusive spectra. The slopes of the spectra are nearly unchanged by the imposition of the coincidence requirement. The corresponding inclusive and coincidence spectra of alpha particles are displayed in the lower part of the figure. The coincidence spectra measured at A¢-180° are virtually identical to the inclusive spectra, as was true for the protons. At A¢-90°, however, the spectra possess slightly different slopes, and the coincidence cross section is 73 MSU—86-412 I ' I '7 ' I ' . _....f....,.,.-.. ’ 32 108 _ Ag( 8. XY) , E/A=22.5 MeV _ o A¢=90° , 6Y=40° o A¢=90° . ay=7oo E 0 A¢=180°. 6Y=40° o A¢=180°, 0Y=70° 3 X=Li,Y=p X=C,Y=p U I rUUUIIl I rV'UIUl Relative Yield (a.u.) 3 CI 1 I U'UFIII I TUUUUU' L l L L 1 l l l L L L l l l l L 1 l 50 100 0 50 100 Energy (MeV) Figure IV-l : Energy spectra of protons (top of figure) and alpha particles (bottom of figure) in coincidence with Li fragments (left side of figure) and C fragments (right side of figure). The circles and diamonds correspond to the light particles being detected at 6-40° and 70°, respectively. The solid and open points correspond to relative azimuthal angles between the light particle and intermediate mass fragment of A¢-180° and 90°, respectively. The inclusive spectra are shown as solid curves. O 7A supressed. The ratio of the coincidence and singles spectra can be used to display small differences between them. This ratio is defined as f dEx a ‘(Ey,EX,0y,9X,A¢) R(E ,0 ,6 ,A¢)- * (IV-l) x a E ,6 Y Y y( y y) where, for brevity, dhovx d20 ayx dE d0 dE do and ”y' dE d0 ' (IV'Z) Y Y X x Y Y The ratio is calculated separately for each detector, and is less sensitive to errors in calibration than the spectra themselves. The spectrum ratios for protons are shown in Figures IV-2. Ratios for lithium and carbon coincidences are shown in the upper and lower parts of the figure. The spectrum ratios measured at 0p-40° (left side) and 70° (right side) are shown as solid and open points to indicate the relative azimuthal angles of the coincident particles, A¢=l80° and 90°, respectively. The ratio, R(Ey), for protons is relatively independent of energy for all angles, demonstrating the insensitivity of the proton spectra to coincidence requirements. The proton cross section is mildly suppressed in non-coplanar geometries. The corresponding spectrum ratios for alpha particles are shown in Figure IV-3. For this heavier particle, R(Ey) measured at A¢=180° is also independent of energy. However, at A¢-90° R(Ey) decreases with increasing energy, confirming that the slopes of the coincidence spectra 7S MSU-86—413 :.-..T....,....I....,....:....l....Te...l... : Ag(323,Xp) j: 0: A¢=180° : ’ E/A=22.5 MeV 1' 0: A95: 90° 0 ' 9,527.50 ' X=Li , 0p=40° X=Li , 6p=70° A 0.1 3‘ ¢ ":7 g - v4 _ I .. .IIOII .II. + '0 : ”0008332234? I E 0°0000 0 ; m I + .. ‘I . v .. . "a. . .. [3:] r If . I . :}:r0 {0%. M.LI ..1%H%§%§PL V :- «I- d 0: X=C , 0p=40° X=C , ap=70° 0.10 a“ ‘5 ‘: ’ ¢ 2: . I . g°°'°°... * $+ ooooo.$$¢+ ‘ .. Ooooooo¢¢¢$ "- Oooo¢ 0-01 _. 1 . L....l 1 7i 1 1 1 ‘0 O 80 O 20 4O 60 Ep (MeV) 0 20 40 Figure IV-2 : The ratio of the differential coincidence cross sections to the single particle inclusive cross sections for protons as a function of proton energy. The ratio is shown for protons detected at 0p-40° (left hand side) and 9p-70° (right hand side) in coincidence with Li nuclei (top of figure) and C nuclei (bottom of figure) detected at 0x-27.5° and at relative azimuthal angles of A¢-l80° (solid points) and A¢-90° (open points). 76 MSU—86—414 :..]....lfi.. WIWII :.I...-l.1..l....l ; Ag(SZS, Xa) 0; o: A¢=180° ' E/A=22.5 MeV " 0: A96: 90° ‘ 9,527.50 ’ X=Li , 901:400 X=Li , 6a=70° ':- 0.1 F 0000............¢¢?? goo-o.ot+ ? l . .. . E I- 0000 :I: O O O V : O OOOO¢¢ ¢¢ : O 0¢¢¢ Q I I . - I m I I W: 0% I I I I v 1- 11- d m X=C , 9a=400 X=C , 9a=70° 0-10 r 1? ‘; E 00......0...‘¢’¢§+3E +......¢+ . 0000 I . 0 ¢ .. ‘I o OO¢¢¢ 00¢ III __ I I _ 0-01 T. L. .1.. ..1 1. .1 .1. 0 1. .1. .10 20 4O 60 80 lOOIZO 20 4O 60 80 Ea (MeV) Figure IV-3 : The ratio of the differential coincidence cross sections to the single particle inclusive cross sections for alpha particles as a function of alpha particle energy. The ratio is shown for alpha particles detected at 0p-40° (left hand side) and €p-70° (right hand side) in coincidence with Li nuclei (top of figure) and C nuclei (bottom of figure) detected at 9x-27.5° and at relative azimuthal angles of A¢-l80° (solid points) and A¢-90° (open points). 77 are steeper than those of the single particle spectra. The difference in the temperature parameters which characterize the in- and out-of-plane coincidence spectra is of the order of 10%. The ratios also indicate a definite suppression of the coincidence cross sections at A¢=90°. IV.B, Angular correlations: IV.B.1. The correlation functions : The angular correlations between intermediate mass fragments and coincident light particles are explored further with the correlation function, which is defined as c<0y.ax,A¢)- fvfx dEvdEx avx(Ev,Ex,9v,0 ,A¢) I v ( (IV-3) x dEydEx by(E§,0y) ax x x where the limits of the integrations can be chosen. If the coincidence cross sections are the result of independent emission of particles with their respective single particle distributions, then the correlation function is constant. The correlation function allows meaningfull comparisons between angles and energies with very different coincidence and singles cross sections. The phase space acceptance of the detection system can influence coincidence observables, e.g. the coincidence cross section as a function of relative energy possesses structures which are solely functions of detector positions; such effects are not present in the correlation function. Experimental uncertainties in detector placement, solid angle, and energy calibration, also, do not influence 78 the correlation function since the single particle and two particle cross sections were measured simultaneously and, therefore, errors cancel in the ratio. The subtle correlations which may arise from collision dynamics [TSA84b], phase space constraints [TSASAC] [LYN82] [HASSS] [FOX86] [CH186b], and single nucleon scattering processes [TAN80] [TAN81] can be accurately determined with this technique. Final state interactions between particles and the decay of particle unbound states produce distinct correlations between the particles. The effects of these processes are seen in the correlation function principally at small relative momenta [BER84] [BER85] [CH186a] [GUS85] [LYN83] [POC85a] [POC85b] [POC85c] [ZAR81]. Since dynamical effects which are independent of these processes are of interest here, the energy integrations in the correlation functions have been restricted to regions where final state interactions and sequential decay are of diminishing importance, i.e. where the relative energy between the coincident fragment and alpha particle is greater than a value, E0, where EO-6, 7, 6, 9, 11, and 10 MeV for fragment charges Z-2- 7, respectively. Above these thresholds the correlation functions do not exhibit sharp structures as functions of relative energy, indicating that final state interactions and sequential decays are less important. The restriction to large relative energies influences the correlation function only at small relative angles. Because the correlation functions between intermediate mass fragments and light particles other than alpha particles were only measured at large relative angles, no similar restriction on relative energy was applied to these correlations. 79 IV.B.2. Azimuthal correlations : Figure IV-h displays the correlations between alpha particles with energies, Ea>40 MeV, and fragments of charge Zx-2-7, with energies Ex/Ax> 5 MeV, except for Ea> 40 MeV, as a function of relative azimuthal angle, A¢. The alpha particles were detected at a polar angle of 0y-40°. The solid and open points represent correlations for fragments detected at 6x-52.5° and 27.5°, respectively. For all fragments, the correlations are enhanced for coplanar particle emission, i.e. the coincidence cross sections are largest at A¢-O° and 180° and smallest out of the plane, at A¢-90°. An interesting feature of this azimuthal anisotropy is its symmetry about A¢=90°. While there is a modest preference for particle emission to opposite sides of the beam axis, the correlation between intermediate mass fragments and non-equilibrium light particles is only slightly sensitive to whether they are emitted to the same or to opposite sides of the beam axis. The energy dependence of these azimuthal anisotropies is examined in Figure IV-S. The measurements were made at 9a-40° and 0x-27.5° at relative angles, A¢-O°,90°, and 180°. Here, the open points correspond to correlations integrated with all energies, Ex/Ax> 5 MeV and Ey> Et where Et- 15, 20, 20, and 40 MeV for p, d, t, and a particles, respectively. The solid points correspond to correlations integrated over higher energy thresholds: Ex/Ax> 10 MeV and Ey> Et+ 20 MeV. This comparison demonstrates that the same side - Opposite side symmetry largely persists for high energy particles. The dependences of the in-plane enhancements of the correlation on the particle masses, angles, and energies are further demonstrated in f‘ 0.0 3 1.0 ’9‘. <1 0.5 >~ ‘5. a“ 0.0 0 1.0 0.5 0.0 Figure IV-A L5 L0 05 80 MSU-86—415 ' I I I I I I f I I I r I- u- ” ' g(3zs Xa) . P I- ’ .. D I - _ .. E/A=22.5 MeV _. P ' q in .. J n p q I- I- u — — — I- I- u: I- p a I- n q ' " I l ‘ 41 l l l l l P l L J l I l I U I I I U I l T I I I- d. d D C. d _ —— q . up 1 :- db .1 b all- q +- 1. cl — —_ _ b 1. d u- q- . I d. C ' I I | "’ I I ' 1 1 1 1 1 1 1 1 1 1 1 Q1 1 1 1 1 1 1 1 I T I l U I I r 1 r I I I I T r I I f I l I I I D D d p I- .. ru— — _ i- l- I: D D d I- I- d D I _ L _ i- n d h D d h I I b n d 1 I l l l l l l l l l L l 1 L 1 _L L A¢ (deg) 90 A95 (deg) : The correlation function for alpha particles detected at 0a-40° in coincidence with fragments, 2x. 2-7, as a function of their relative azimuthal angle, A¢. The coincidence fragments, X, are detected at laboratory angles of 0x-27.5° (open points) and 6x-52.5° (solid points). 81 MSU- 86- 416 - I If r . . I .47.~Te. I .‘~ I Ag(3ZS,Xa) ax=27.5° .. 0.1:: >51». MeV E912. 1.0 — E/A=225 MeV ‘_' 0 Ex) 10A MeV ,E,>Et+20 MeV ‘ . .. XE=LJ . H0 00 .0 01 .0 o C (6.6..Aqb) (b") H C) .0 01 0.0 A¢ (deg) Asb (deg) Figure IV-S : The correlation function integrated with low and high low particle energy thresholds (open and solid points, respectively) for alpha particles detected at aa-40° in coincidence with fragments, Z - 2- 7. It is shown as a function of their relative azimuthal angle, Ad. The coincidence fragments, X, are detected at 6x-27.5°. 82 Figures IV-6-9, which show the azimuthal anisotropy, _ 0 (9y, 9x, A¢-180°) ¢ 0 (9y, ax, A¢ -90 ) (IV-4) as a function of fragment charge. In Figure IV-6, this ratio is shown for light particles, y-p,d,t,a, detected at 9y-AO° and intermediate mass fragments detected at 0x-27.5°. The solid and open points correspond to correlations integrated over particle energies above the lower and higher energy thresholds discussed with reference to Figure IV-S. A number of features are evident. The anisotropy increases with increasing mass of the intermediate mass fragment and with increasing light particle mass. The anisotropy also becomes stronger as the energy of either particle increases. This same energy and mass dependence is displayed in Figure IV-7, which shows the anisotropy measured, instead, at 0x- 40“. In Figure IV-8, the anisotropies measured at 0x=27.5° and at 0x-52.5° are compared for a fixed value of 0y-40°; in Figure IV-9, of 0y-70°. The anisotropy becomes larger as the polar angle of detection for either particle increases. The increase in the anisotropy with detection angle is stronger for alpha particles than for protons, though there is some suggestion that this angular dependence is less pronounced for the heavier fragments. 83 MSU-86—417 I I 7 T l T’ I I 10 _— OzEx/A>5 MeV :- Ag(azS,XY) 7 ' EY>Et 2: E/A=22.5 MeV ‘ j OzEx/A>10 MeV j: ax=27.5°, 9Y=40° j EY>E,+20 MeV 'Y=p ‘-Y=d .’0¢ '. ‘. '| " C) C) O O . 8 O o O O O O _ O O O __ __ <3 1 : : % + : : % % 10 _Y=t _— Y=a + - L . o + ; ' ¢ ‘ O o . o c ’ ° 0 O 00 00 o0 o O O 1 — 1 l I l "’ 1 1 1 I ‘ Figure IV-6 : The in-to-out of plane ratio, A¢, defined in the text, for fragments of charge, Zx-2-7, detected at 9x-27.5° and light particles, Y-p,d,t,a, detected at 0y-hO°. The solid and open points correspond to the ratio for correlations integrated for particle energies above a higher and lower low energy threshold. 84 MSU-86-418 I I I l I I I I 10 POZEx/A>5 MeV j:- Ag(SZS,XY) i ' EY>E, :: E/A=22.5 MeV ‘ j OzEx/A>10 MeV j: 9x=40°, 0Y=40° . EY>Et+20 MeV : Y=p “ Y=d. + ++ ‘ ‘. w o o O... 000 O 1 00000 s1'%%%}"%%%%‘ <1: 10- -- - :Y=t Y=a . +0 O 0 o 1' 00 + o ‘ o? + 00 00 0000 l Figure IV-7 : The in-to-out of plane ratio, A¢, defined in the text, for fragments of charge, Zx-2-7, detected at 0x-40° and light particles, Y-p,d,t,a, detected at 0y-40°. The solid and open points correspond to the ratio for correlations integrated for particle energies above a higher and lower low energy threshold. 85 MSU—86-419 F I I I I I I I 10 .- 0 9x=27-5° :‘Ag(SZS.XY) ‘ .l O 6x=52.5° I E/A=22.5 MeV j; a, = 40° ”Y=p “-Y=d "|II" 0.0000 8 8 Q 8 O O O I I Y=t EEY=a .¢ 0 .° 0 ° 0 ,.8 00 .0. 0 OO Figure IV-8 : The in-to-out of plane ratio, A¢, defined in the text, for fragments of charge, Zx-2-7, detected at 6x-27.5° (open points) and 52.5° (solid points) and light particles, Y-p,d,t,a, detected at 0y-40°. 86 MSU—86—420 I I I I I I I l 10 r O 6x=27'5° ‘J" Ag(323.XY) i i o a,=52.5° :I E/A=22.5 MeV I ' I 9y=70° : ‘{::I) Y::d 9i, o 09 o 0 Oe O o O O 0 00080 .3 1 ' I I I I "" I I I I ‘ Figure IV-9 : The in-to-out of plane ratio, A¢, defined in the text, for fragments of charge, Zx-2-7, detected at 6x-27.5° (open points) and 52.5° (solid points) and light particles, Y-p,d,t,a, detected at 6y-70°. 87 IV.B.3. In-plane correlations : Figure IV-lO shows the correlations between alpha particles and fragments, Zx-2-7, detected in a coplanar geometry (A¢-O° or 180°), as functions of the polar angle of the alpha particle, 0a. Positive values of 0a correspond to emission to the same side of the beam axis as the fragment. When the fragments are detected at 9x827.5° (open points), the measured correlations are insensitive to the polar angle of the alpha particle. When the fragment is detected at a larger angle, 6x-52.5°, the correlations show a suppression of the correlation function at forward angles. This feature becomes clearer as the fragment mass increases. IV C Discussion : The comparison between the coincident and single particle inclusive spectra of light particles provides no evidence for the existence of a peculiar class of fragmentation events. On the contrary, it suggests that intermediate mass fragments are associated with the rather large class of events characterized by non-equilibrium light particle emission. Such reactions have been determined to be non-peripheral, as light particle spectra gated on projectile-like fragments do not account for the inclusive spectra [HASBS]. It has been determined through the analysis of spectra gated on momentum transfer that the principal component of the non-equilibrium light particle cross sections at large angles originates in the early stages of very damped, central collisions [AWE79] [AWE81b]. Energy and angular distributions of light particles have been 88 MSU-86-421 . T I I r I . I I I I I I 1.5 ? 0 9, 275° . Ag(328.Xa) : . 9:: 52.50 E/A=22.5 MeV LO 0.5 9‘9 L190 AllllllLllll ‘I'UU'I'U IAIAJLJILALI l 1 O O l l I l l I l l l l l . I I I I I I. I I I I I A 7 . . . ,o I 1 I v '- . . . \ -I 1.0 — \ 1’ —I A : ‘x ‘: \.~ \ 1 ~e_ _ G\c\ \ \ —O‘O .. W ‘ <1 L T ‘ x .1; \ o _‘ 5‘ 0.5 _ . ‘L . .1 <5. I zx=4 1 zx=5 3 N 0 0 ' L I I I I " I I l I I S - . I I I I I .. . I I I I I . U ; .~ I \ 1 O ' \\ J” \ ‘ ’ : M \ $12) :: G\g>’ detected at 9x-27.5° are shown in Table V-l. The estimates are deduced from the coincidence cross section at 6y=40° averaged between A¢-l80° and A¢-90°. These results indicate that intermediate mass fragments are accompanied by an average of about 10 nucleons in the form of non-equilibrium light particles, Ays 4. In light of the coincidence spectra of Section IV.A., it is reasonable to assume that these nucleons are emitted with energy distributions which are similar to the corresponding single particle inclusive distributions. Non-equilibrium light particles are estimated to remove a total longitudinal momentum of zllOO MeV/c from the system. This corresponds to more than 15% of the projectile momentum. The average kinetic energy carried away by these 96 Table V-l : The estimated average total multiplicities, M, of light particles, p,d,t,a, associated with intermediate mass fragments with momenta, , emitted in 32S induced reactions on Ag at E/A-22.5 MeV. MT is the total nucleon multiplicity;

and are the average total longitudinal momentum and average total energy carried away by non- equilibrium light particles. Momenta and energies are given in units of MeV/c and MeV, respectively. The multiplicities are inferred from cross sections measured at 9y-40°, 0-27.S°, averaged between A¢-90° and 180°. Li | l | 1 I 1 I l I l 820. 1082. 1348. i 832. 1144. 1429. 1719. i 1164. 1448. 1745.I Mp’n I 2.0 1.8 1.7 I 1.7 2.1 2.0 1.8 I 1.9 2.0 1.9 I Md I s 0.4 0.4 I 0.5 0.5 0.5 0.4 I 0.5 0.5 0.5 I Mt I .3 0.2 0.2 I 0.2 0.3 0.2 0.2 I 0.2 0.3 0.2 I Ma I 1.2 1.1 0.9 I 1.0 1.2 1.2 1.1 I 1.2 1.3 1.2 MT I10 4 9.5 8.3 I 9.1 10.8 10.5 9.4 I 10.4 11.1 10.1

I1170. 1073. 942. I 1026. 1218. 1182. 1060. I 1170. 1253. 1143.I I 167. 152. 136. I 145. 174. 168. 150. I 165. 177. 161 I 97 fast light particles is about 160 MeV, or more than 20% of the projectile laboratory energy. An energy of approximately 60 MeV is contained in the center-of-mass motion of the light particles. The remaining 100 MeV is contained in the random velocities of the light particles about their center-of—mass. There is a perceptible dependence of the correlation function on fragment momentum. non-equilibrium light particle emission is strongest for fragments emitted near the peak in energy distributions. As the momentum of the fragment increases, the multiplicity of the light particles decreases slightly. This qualitative dependence is expected from energy conservation. The decrease in light particle multiplicity for fragment momenta lower than the maxima of the distributions is not so easily understood, and may be related to the sequential decay of excited primary fragments (see Section VI.A.2.). While this might also be related to contributions from the target-like partner of peripheral reactions, the velocity distributions of target-like residues discussed in Section VI do not indicate that low energy fragments are particularly associated with peripheral reactions. V C ntermediate mas ra ment multi li ities: V.C.1. Fragment multiplicities associated with light particles : The associated multiplicities of intermediate mass fragments are estimated in the same manner. Multiplicities of fragments, ZX-2-7, associated with non-equilibrium alpha particles are shown in Figure V-l. 98 MSU—86—428 I I I l l | I I A8(BZS,X) _ E /A=22.5 MeV 1. 1130 l I l I II llllll 0.10 rlllll . O llllll 0135 I l F Figure V-l : The estimated average total multiplicities, Mx’ of particles, Zx-2-7, associated with alpha particles in 328 induced reactions on Ag at E/A-22.5 MeV. The results were obtained from the correlations measured at 00-40° and 9x-27.S° at A¢-909 and 180°. 99 These are deduced from the correlations function measured at 6y-40°, 9x-27.5° averaged between the two relative azimuthal angles, A¢=l80° and 90°. The mean associated multiplicity of any one of these fragments is low, of the order of 0.1. The azimuthally averaged correlations between light particles detected at Oy- 40° and fragments detected at 0x=27.5° are shown in Figure V-2. They do not exhibit any significant dependence on fragment charge. The proportionality factor between the inclusive cross sections and the associated multiplicity is the two—particle correlation function. Thus, the coincidence cross sections scale approximately with the inclusive yields. This is very similar to result of relativistic heavy ion reactions [WAR83]. If an average value of Cz0.45 is assumed for heavier fragments and if the particles are further assumed to be emitted independently, the value for the total associated multiplicity of fragments with charge 352524 is estimated to be 0.75 . This value corresponds to the emission of about 12 nucleons in the form of complex fragments. The parametrizations of the singles cross sections indicate that there is a large non-equilibrium component in the fragment cross sections, so that energy and linear momentum are carried away prior to the attainment of full statistical equilibrium by reaction products other than light particles. This increases the momentum lost to non-equilibrium emission by about 40%. Thus, a total of about 25% of the beam momentum is carried away by pre-equilibrium emission. These estimates have considerable uncertainties, because the correlations functions depend on particle masses and the polar angles at 100 MSU-86—424 _ I I I I l I I I . :323 . 0.8 _ Ag( S,X) _. I E/A=22.5 MeV l I - ° Y=p ' y=a ‘ A 0.6 - '— T .0 I I I v ' ID I. o . o ‘ g 0.4 — I. - >¢ . . Q _ . 0.2 - ,. ‘ O O " l L l l l L l L 0 2 4 6 8 ZX Figure V-2 : The correlation function, ny, between light particles (y) and intermediate mass fragments (x), measured at 9y-40° and 0x-27.S°, and averaged between A¢- 180° and 90°, is shown as a function of Zx, the charge of the coincident fragment. 101 which the fragments are detected. They vary by 10-20% between intermediate mass fragments Zx-2 and Zx-7 and between protons and alpha particles. They increase with the angle at which either particle is detected, by about 10% between 6x-27.5° and 0x-52.5° and by about 20% between 0y-40° and 0y-70°. Nevertheless, these estimates of the associated multiplicities are sufficiently accurate to establish that complex fragments are emitted with low average multiplicities and in the same class of reactions for which pre-equilibrium emission is important. V.C.2. Fragment multiplicities associated with other intermediate mass fragments : While the multiplicity of intermediate mass fragments associated with light particles is lower than expected for complete fragmentation of the composite nucleus, it does not necessarily preclude the dominance of high multiplicity events in the fragment production mechanism. If the emission of intermediate mass fragments occurs in a small fraction of the events producing non-equilibrium light particles, then the fragment- fragment multiplicities might still be high. The fragment-fragment associated multiplicities provide a better test. The correlation functions between all combinations of Li and C 1 fragment detected at 0x1=27.5° and 0x2-52.5° at A¢=0° are C z.55b- 12 The corresponding values for light particle-intermediate mass fragment correlations are typically Clzz.45 b-l. The multiplicity of intermediate mass fragments is, therefore, of the same order of magnitude (=1) as those indicated by fragment-light particle coincidence data. 102 One must exercise caution in interpreting this result since the estimates are deduced from measurements taken over a very restricted angular range. They are, therefore, subject to the effects of momentum conservation and final state interactions between coincident particles. In addition, the azimuthal and mass dependence of the two particle correlations suggests that there will be large in-plane enhancements in the coincidence cross sections. Nonetheless, these associated multiplicities suggest that intermediate mass fragments do not originate from a peculiar class of reactions with high fragment multiplicities, such as the complete shattering of the target nucleus. Instead, intermediate mass fragments produced in this reaction are emitted with low probability. The fragment multiplicities exhibit little memory of prior emission. V D ummar ° The multiplicity data indicate that intermediate mass fragments are emitted with low probability from a class of reactions with a total cross section on the order of C'1z2b. The geometric cross section for nuclear radii, r-l.2»A1/3 fm, is 2.8 b. If fragment emission is associated with central collisions, then the fragmentation cross section corresponds to all impact parameters of 8 fm or less. This is consistent with the conclusion of Chapter IV, made on the basis of the azimuthal correlations, that fragment production was not restricted to central collisions covering only a small range of impact parameters. This class of reactions is characterized by substantial pre-equilibrium light particle emission, with over 10 nucleons being 103 emitted in that manner. These particles carry a significant part of the beam momentum and energy. In addition, about the same number of nucleons are emitted in the form of heavier fragments, with the average multiplicity of these fragments being slightly less than one. Apart from the enhanced angular correlations, the two particle correlations are approximately constant. This indicates that intermediate mass fragments are emitted statistically (with a Poisson-like distribution) with probabilities which are independent of prior particle emission. There is some evidence that higher multiplicity events may be selected by heavier fragments emitted at larger angles. However, the increase in multiplicity is marginal. Violent multifragmentation, with fragment multiplicities much larger than 1, can be excluded as a dominant process. Chapter VI- Velocity Distributions of Target-Like Residues In this section, the velocity distributions of target-like residues coincident with intermediate mass fragments are investigated. These distributions provide information on the overall dynamics of the reactions producing fragments by determining the momentum and energy balance. One may infer from these distributions the characteristics of the particle emission in the forward direction, in the absence of direct measurements. Unless otherwise stated the data presented are taken from 328 induced reactions on Ag at E/A=22.5 MeV [FIE86]. These data are supplemented with data from 14N induced reactions on Ag at E/A-35 MeV [BOU86s]. VI.A. Velocity distributions: VI.Awl. General Characteristics : 0n the left hand side of Figure VI-l, are shown angular distributions of target-like residues coincident with lithium and carbon 104 0.01 0.00 0.01 13092) (deg—1) 0.00 Figure VI-l 105 USU— 85-503 I I I V V r ' I—r r b I A. A A 1 L; L l I U . .1 -. . - , Ag(328 X) E/A=22.5 MeV 8,: 27. 5° 9 (a) X=Li , P,=960—1280 MeV/c I I U I r “It. (b)X= I C, P1=1260-1600 MeV/c db ‘ d A I l..I'ogl¥I 00...... h- .. 0.. .00' O. O p A A A L LTA L A A I A A : AIL l A A A A l A A A I (c)X=C,P1=1920—2240 MeV/c .. 4 '4 l T) J I T. .. ‘ o I U V : V I“ d C 4 o 9 ‘ . 3'. I 4,0: ., L L .‘20...§....-J 0 10 20 92 (deg) 0 1 2 v2 (cm/n5) 1 20 H 00 (81mm 919) plan 00 DO : Velocity distributions for target-like residues detected in coincidence with intermediate mass fragments detected at 0 -27.5° with momenta, P1. The left hand side shows the distribution as a function of the polar angle, velocity vector onto the reaction plane. 4 measured distributions of |v2|. binary reactions. 105 02, of the projections of the recoil The right hand side shows the The arrows show the values expected for 106 nuclei detected at 61-27.5° in 328 induced reactions on Ag at E/A-22.S MeV. The residues are detected in coincidence with fragments with momenta, P1, in the ranges (a) PLi-960-l280 MeV, (b) PC-1280-l600 MeV/c, and (c) PC-l920-2240 MeV/c. The distributions are shown as functions of 92, the polar angle of the projection of the residue velocity vector, 32, onto the reaction plane defined by the fragment and the beam axis. They are normalized to represent the probability per degree that a heavy residue will be detected with the coincident intermediate mass fragment. The distributions exhibit broad maxima, which have widths of about 25°. The peak positions expected for complete fusion followed by a binary decay into a fragment and heavy residue are indicated with arrows. They are located at angles smaller than the those of the measured maxima, indicating that a substantial part of the projectile momentum is carried away by other particles emitted into the forward direction. Velocity distributions of heavy residues coincident with intermediate mass fragments are shown in the right hand side of Figure VI-l. They, too, are inconsistent with a fusion-fission process where 100% of the projectile momentum is transferred to the composite system. (The velocities expected for purely binary reactions are indicated by arrows.) VI.A.2. Dependences of the peak position : The location, ogax , of the maximum in the angular distribution of the coincident target-like residues depends both on the momentum, P1, and the mass, M1, of the coincident fragment. This is illustrated in Figure VI-2. The upper part of the figure shows angular distributions of 106 107 MSU-85-577 .“T'I'm' ..... 5,4" - . .1.I 1.1I. I ........ I....I.... Ag( 3X1) . E/A= 22.5 MeV 0'02 T X1=B , aI=4O° I T . 21 I» I; I 3+ q - t 0 ' : : < ,0 9.3“, .. .953. ~35.- +4” ”1+ - ~ : i f. 2 ‘5 o ’ ~ 0” 3 _ 0.01 "' .” , "" . . .0. 2 0 ¢ . I I 0 I : $5 * = .. + 845 MeV/c I 1130 MeV/c 1415 MeV/c 1892 MeV/c I in..l...nl.it A1..l.1..l ........ 1141 ........ 1....1... ........I . ...I....I-..-I-...I....I..-.I.-..I....I. 960 MeV/c

1000 MeV/c ), 02 increases with increasing values of PI’ qualitatively consistent with momentum conservation. However, as P1 decreases below the Coulomb barrier, 093x . max increases again; 02 is smallest for fragment energies near the maximum in the energy spectra. Similar observations were made for 14N induced reactions on Ag at E/A-35 MeV. Figure VI-3 shows the angular distributions of target-like residues measured in coincidence with beryllium and oxygen fragments from this reaction. They share many characteristics with reactions induced by 328. The angular distributions have broad maxima at angles somewhat larger than expected for binary reactions. They also show a similar dependence on fragment momentum, with the positions of the centroids, <02), increasing with increasing fragment momenta above the barrier, reaching a minimum around the barrier, and increasing again for momenta below the barrier. Figure VI-h demonstrates this dependence over a broad range of oxygen fragment momenta. The minimum of <02) is seen at the momentum corresponding to the peak of the inclusive fragment spectra, which is indicated by the arrow. Fragment energies less than the Coulomb energy can be generated by the sequential particle decay of heavier particle unstable nuclei which were originally emitted at or above the Coulomb barrier energy [BER84]. For the example presented in Figure VI-h, it may be assumed that the primary fragment, a 20Ne nucleus, is emitted at 0-45° with the barrier energy (E-SS MeV, P-1440 MeV/c) and an excitation of about 8 MeV. The 108 30 20 10 56 36 18 90 SIGMA Ipb/MeVsr’I 60 30 120- 80 60 Figure VI-3 109 21:1: L 12F = 8 . P, 3 1586 14th P, a 1584 mV/c _ ‘2 5 III I +1111 9" WW ”1+ II HIM 3“ Ii. . E11911 **‘1#*....1‘. : +++hI ! . ‘ ':72 ”HI” + ++I I +#II+H++I+ - “ “+1. 9"" “g :2“ l + I; . 11916 HLcV/Lc l P; 8.17961HIVL/E :0 ' bi”: HIHIII I I IIII++++II+ 1 o :1 “1 II“ iIII ‘ _ *6 +9 fi‘ -«20 . . . ’1- 1 ...... 1.‘ I _I+I1IIIIII+ II I T” 3' +1 I I ++III 1" .. 6 I M HI d 14 ’ .1.??? ”’31 1111‘ 12 28 96; 60 76 12 28 5’; 60 76 5101111 Iph/Hev .sr’I : The differential cross sections for heavy residues detected in coincidence with Be fragments (lefthand side) and 0 fragments (right hand side) of various momenta detected at 91- 65°. llO MSU—86-425 —lFf|TTTI|IlFl|Ill-1 6O — —1 ' Ag(“N,0), E/A=35 MeV ‘ P 61 = 45° 01 o I <62> (deg) .1; _ O I o 30— 311 ./. i000 1500 2000 P1 (MeV/c) L l Figure VI-a : The value of the average recoil angle <02> for angular distributions of heavy residues detected in coincidence with 0 fragments detected at 01- 45° as a function of fragment momentum. The solid curves are expected values for 100% and 73% momentum transfer. lll alpha decay energy to 16O is about 5 MeV. After the decay, the remaining oxygen fragment will have a momentum between 950 and 1330 MeV/c, corresponding to energies between 30 and 60 MeV, all at or below the peak of the oxygen spectrum. Such a mechanism has been discussed with respect to the spectra of light particles emitted from unstable resonances [BERBA]. Two-particle correlations at small relative angles [POC85c] suggest that the sequential decay of excited primary fragments does occur (see Chapter VII). This mechanism may be a major factor determining the energy spectra about the Coulomb barrier (see Section III.B.4). Model calculations which include this effect are discussed in Chapter VIII. Low energy fragments might originate from the target-like partners of peripheral reactions. The residues move with low velocities, and fragments emitted from them will have small velocity boosts compared to fragments emitted in full momentum transfer events (see Section VI.B.3.). The bottom section of Figure VI-2 shows the angular distributions for residues coincident with different nuclei of similar momentab The values of ogax depend not only on the momenta of the outgoing fragments but also on their mass: ogax increases with the mass of the coincident fragment. This behavior is also displayed in the residue angular distributions in 14N induced reactions as shown in Figure VI-3. Part of this behavior may be related to the energy dependence of the angular distributions. A fixed momentum will be at positioned differently relative to the peak of the distribution for different fragments, so the behavior about the barrier may influence the mass dependence. The . . . . 2 . . angular distributions for heav1er fragments in the 3 S induced reaction, 112 Zf29, do not exhibit maxima within the angular acceptance of the detector, presumably because the distributions are peaked at angles beyond the detector acceptance. VI.A.B. Integrated prdbabilities : We can extrapolate the measured angular distributions to estimate the probability that a heavy residue accompanies an intermediate mass fragment. For this purpose, the distribution is assumed to be a Gaussian function of angle, and the widths in- and out-of-plane are assumed to be the same. For the case of carbon nuclei detected at 0x-27.5° with momenta between 1280 and 1600 MeV/c, the width of the distribution in the reaction plane is approximately 25°. This narrow distribution argues against any high multiplicity fragmentation process. The emission of an additional carbon nucleus from the decaying residue of 100 nucleons would result in a final angular width of z50°, considerably larger than the observed distributions. Additional emission would widen this further. The integrated probability for detecting a residue over the measured angular range 6-9.5°-21.5° is about 0.13. Extrapolation over the full angular range gives approximately unit probability for the detection of a coincident heavy residue; there is a single target-like residue remaining after the emission of, at most, a few fragments. 113 V1.3, Kinematic analysis: VI.B.l. Method of missing momentun.analysis : In order to provide a more quantitative analysis of the data, a kinematic analysis has been performed using the most probable values of the heavy residue velocity distributions. For this analysis the mass number, M1, of an intermediate mass fragment of charge, 21, is taken to be Ml-ZZI. The momentum of the intermediate mass fragment is given by the expression: fil- Pl- J 2M1E1 , (VI-l) A where P1 is the unit vector of the fragment momentum and E is the 1 measured energy of the fragment. The momentum of the heavy residue is given by ?2(M2)- v2. M2 , (VI-2) where 32 is the measured velocity of the residue and M2 is the mass of the residue, which is not measured. The mass is treated as a parameter in the kinematics calculations. For each assumed value of M2, the "missing mass", M3- M0- M1- M2, (VI-3) 114 and the "missing momentum", —> —) —> -v P - P - P - P , (VI-4) carried away by undetected particles in the reaction are calculated. Here, M0 and PO denote the total mass and momentum in the reaction. VI.B.Z. Quantities of interest : We define the "sum kinetic energy", Ek- 2». (v1.51 as the sum of the kinetic energies of the two detected fragments and the kinetic energy corresponding to the motion of the center-of—mass of the missing mass. The difference between the projectile energy and the sum kinetic energy may be associated with the energy dissipated into other, "internal", degrees of freedom, i.e. degrees of freedom different from 1’ P2, and P3. Small values of ER correspond to violent collisions in which a large the nine translational degrees of freedom represented by P amount of energy is converted to excitations of these "internal" degrees of freedom. For example, complete fusion followed by symmetric binary fission would result in Ek- E1+ E2” 250 MeV. Another quantity of interest is the magnitude of the center-of-mass velocity of the missing mass, v3-|P3/M3|. For example, values of v3 close to the projectile velocity would indicate that the missing 115 momentum is carried away by projectile fragments, as might be expected for a breakup-fusion process. Information about the linear momentum transfer is provided by (P3)z’ the component of the missing momentum along the beam axis. Finally, the polar angle of the missing momentum, 93 - cos-1[(P3)z/P3], is evaluated. In this sign convention positive values of 03 indicate that the missing momentum is directed to the same side of the beam axis as P1. VI.B.3. Results of missing momentum analysis : As a specific example, the kinematics of coincidences between heavy recoil nuclei and carbon nuclei detected at 01-27.5° with momenta between 1280 and 1600 MeV/c are discussed. The energy of the fragment corresponds to laboratory energies of about 90 MeV. The experimental distributions for this case are shown in Figure VI-lb. The maxima of the target-like residue angular and velocity distributions are located at max_ 1 max 2 4° and v2 -l.05 cm/ns, respectively. Calculations of the kinematic quantities defined above were performed as functions of M 0 2; they are shown as solid curves in Figure VI-S. The mass of the total system must be considered in order to interpret the results of the calculations. Undetected light particles evaporated from the heavy residue may be considered as part of the mass M2. The velocities and emission angles of these particles are, on the average, the same as those of the heavy residue, so that v2 is the velocity of the residue both before and after evaporation. If M2 is defined in this manner, the "missing momentum" is not associated with undetected particles originating from equilibrium emission from the ll6 MSU—85—506 500 . - . r . . '1 . . . . 1 . Ag(3zs,c), E/A=22.5 MeV _- 15 . P1=1448 MeV/c v2=1.05 cm/ns « ' €400 _ 91:27.50 —_ 92: 14° ; :1 < OJ ' ‘_ ‘ _ 1 O u E :(c) F . ' 2 .51 . ° mzoo _p - . l.— “ 0.5 1. 0.0 f 0.6 an 1 0.4 Q “U ‘ N \_/ ' :;3 (be: f 0.2 o 5 0.0 Figure VI-S : The results of kinematics calculations for coincidences between heavy residues and carbon nuclei detected at 01-27.5° with an average momentum of 1448 MeV/c are shown as solid curves. The hatched area represents the estimated uncertainties. The dashed curves are obtained when the carbon nucleus is assumed to be a secondary fragment produced in the decay of a particle stable 0 nucleus. A detailed discussion is to be found in Section VI.B. 117 target-like residue. Limits on the mass of the target residue system restrict the range of possible kinematic solutions. In Section V, it was shown that non- equilibrium light particle emission carries away about 10 mass units. Furthermore, there is a non-negligible probability for non-equilibrium emission of nucleons in the form of heavier particles. The average value of M2 should, therefore, be smaller than 120 by several mass units. Inclusive residue mass and velocity distributions from 40Ar induced reactions on Ag at E/A-27 MeV [BORp85] suggest that the distribution probably peaks in the mass region M22 70-110 after evaporation. The present discussion is confined to values of 75 < M2 < 110. The sum kinetic energies, Ek’ extracted from the kinematics analysis are shown in Figure VI-Sa. The values of Ekz 200-300 MeV indicate that, typically, energies between 400 and 500 MeV are dissipated into degrees of freedom other than the nine translational degrees of freedom which are included in the definition of the sum kinetic energy. The multiplicity studies of Chapter V indicate that approximately 100 MeV is associated with the random motion of non- equilibrium light particles. A comparable amount of energy could be carried away by additional intermediate mass fragment emission. Thus, between 200 and 400 MeV is deposited as internal excitations of the residual nucleus or emitted fragments: intermediate mass fragments are emitted in highly inelastic collisions with temperatures of Tz4-7 MeV. The direction of the missing momentum vector is shown in Figure VI- 5b. The missing momentum is directed close to the beam axis. The velocity of the missing mass, shown in Figure VI-Sc, is less than or equal to the beam velocity. A value of about half the beam velocity, 118 characteristic of pre-equilibrium light particle emission, is consistent with a pre-evaporation residue mass of about 110 mass units and the emission of 15-20 nucleons at the early stages of the reaction. This number is consistent with the estimates of the associated particle multiplicities discussed in Section V. The magnitude of the missing momentum is between 20% and 40% of the projectile momentum; see Figure VI-Sd. This value of the missing momentum is larger than the value of the missing momentum, z15%, expected from the systematics of linear momentum transfer measurements for fusion-like reactions [FAT85]. Non-equilibrium light particle and fragment emission account for a major fraction of this momentum, about 25% of the beam momentum. Any remaining momentum may be the result of forward focussed emission not included in the multiplicity estimates of Chapter V. 1 Analyses of the coincidence measurements for other intermediate mass fragments or for different fragment momenta lead to similar conclusions. Table VI-l shows measured recoil momenta and directions for which a value of egax could be determined. Also, values are provided for the inferred kinematic quantities assuming that M2-110. The total kinetic energy ranges from Ek- 200 to 300 MeV. The value of Ek generally increases with increasing fragment momentum. The missing momentum ranges from 1300 MeV/c to 2300 MeV/c, averaging 1600 MeV/c. The missing momentum decreases with increasing fragment energy. While low energy fragments do show larger missing momenta than high energy fragments, there is nothing to indicate contributions from peripheral contributions : the energy losses are largest for lowest energy fragments. The very lowest energy fragments indicate the missing mass is 119 Table VI-l : The observed and calculated kinematic properties of systems for which the peak in the distribution of heavy residue velocities, 32, was within the experimental acceptance when detected in coincidence with an intermediate mass fragment with a momentum at an angle 01. zl vgax egax P3 03 Ek (MeV/c) (cm/ns) (MeV/c) (MeV) 01-27.5° Li 820. 1.20 -5.50 1739. 0.56 209. Li 1082. 1.20 -9.50 1554. 6.71 240. Li 1348. 1.20 -12.50 1376. 11.35 286. Be 836. 1.10 -8.50 2098. 4.74 223. Be 1113. 1.10 -8.50 1846. 1.41 234. Be 1404. 1.10 -12.50 1644. 5.96 266. Be 1685. 1.15 -15.00 1283. 11.04 305. B 832. 1.10 -l7.50 2352. 18.69 254. B 1144. 1.10 -ll.50 1866. 6.96 232. B 1429. 1.05 -13.00 1794. 4.88 258. B 1719. 1.05 -15.50 1579. 6.23 287. B 2009. 1.05 -16.50 1333. 4.22 326. C 1164. 1.10 -l6.00 1971. 14.86 245. C 1448. 1.05 ~14.00 1797. 6.55 252. C 1745. 1.00 -l7.00 1749. 6.56 283. N 1457. 1.10 -17.50 1737. 15.53 251. N 1756. 1.05 -17.50 1596. 9.92 265. N 2056. 0.95 -17.50 1634. 1.16 302. 0 1766. 1.05 -19.50 1650. 13.67 271. O 2069. 1.00 -20.00 1523. 8.34 288. 120 Table VI-l LeontinuedlA;

vg‘ax 93‘” P3 0 3 Ek (MeV/c) (cm/ns) (MeV/c) (MeV) 91-40° Li 801. 1.25 -9.00 1721. 5.26 212. Li 1061. 1.25 -12.00 1570. 7.75 244. Li 1336. 1.30 -15.00 1267. 13.62 291. Be 833. 1.20 -11.50 1913. 8.64 217. Be 1105. 1.20 -12.00 1697. 5.00 234. Be 1380. 1.20 -17.00 1602. 11.51 272. B 845. 1.25 -17.50 1974. 22.31 232. B 1130. 1.20 -13.50 1712. 7.96 229. B 1415. 1.15 -15.50 1685. 5.02 258. B 1692. 1.15 -18.50 1536. 6.26 291. C 1155. 1.25 -l7.50 1682. 19.10 233. C 1433. 1.20 -17.00 1556. 10.58 245. C 1728. 1.10 -20.00 1705. 6.17 288. N 1441. 1.25 -l9.50 1506. 19.75 244. N 1741. 1.15 -21.00 1576. 10.91 274. O 1463. 1.25 -21.00 1560. 22.63 254. 0 1750. 1.20 -23.00 1515. 18.78 272. Table VI-l (continued) : max 121 max Zl v2 02 P3 93 Ek (MeV/c) (cm/ns) (MeV/c) (MeV) 01-52.5° Li 777. 1.25 -9.50 1863. 2.87 220. Li 1036. 1.30 -15.50 1677. 12.85 255. Be 820. 1.25 -11.00 1862. 5.23 218. Be 1083. 1.20 -14.00 1915. 4.17 250. Be 1362. 1.25 -18.00 1675. 8.52 281. B 848. 1.35 -15.00 1659. 18.59 216. B 1109. 1.25 -14.00 1735. 5.30 235. B 1393. 1.20 -18.00 1809. 5.40 273. C 1136. 1.30 -16.50 1637. 13.01 234. C 1415. 1.25 -18.00 1637. 7.23 258. N 1154. 1.40 -20.00 1533. 28.50 241. N 1421. 1.30 -20.00 1560. 14.93 255. 0 1445. 1.45 -23.00 1364. 36.01 261. 0 1731. 1.35 ~21.00 1221. 13.77 261. 122 directed toward the fragment, as discussed in Section VI.A.2. At higher energies, the missing mass is directed between 5° and 15° from the beam direction. VI.B.4. Errors and.uncertainties : In order to assess the sensitivity of the extracted quantities to uncertainties of the input parameters, three of the input parameters have been varied separately : the direction of the residue was varied by i4°, corresponding to the uncertainty in the most probable recoil angle; the velocity was varied by i10%, corresponding to the uncertainty in the most probable recoil velocity; the mass of the carbon ion was varied by i2 amu. The shaded regions in Figure VI-5 indicate the range of values of the calculated quantities which result from these variations. The direction of the missing momentum varies by from 5° to 10°, and the momentum transfer varies by $0.05 P0. These variations do not significantly affect the conclusions of this analysis. VI.B.S. Sequential decay : It is possible that the detected intermediate mass fragments are the decay products of highly excited primary fragments. The dependence of the position of the peak of the residue angular distribution on the fragment momenta indicates that contributions from such sequential decays are expected to be particularly large for fragment energies below the Coulomb barrier. In fact, the missing momenta for low energy fragments of Z-5-8 indicate the preferential emission of matter to the 123 side of the beam axis on which the fragments are detected. Studies of small angle correlations between charged particles also support the existence of such a sequential decay process [POC85a] [POC85b] [POC85c]. This will be dealt with at greater length in Chapter VII. In order to determine the effects of such a correlated emission on the kinematic analysis, the momenta of the undetected products of sequential decay may be excluded from the missing momentum, P3, by including them in the definition of P1, which is then interpreted as the momentum of the primary fragment prior to its particle decay. As a specific example of the effect of sequential decay on the total missing momentum, the undetected sequential decay products are assumed to be emitted in the decay of primary oxygen fragments into detected carbon nuclei, with the velocities of the decay products remaining the same as the velocities of the primary fragments. The dashed curves in Figure VI-5 represent the kinematic analysis in this scenario. The sequential decay has a negligible effect on the sum kinetic energy, Ek’ and the velocity, v3, of the missing momentum. The direction of the missing momentum is pushed closer to the beam direction. The missing momentum along the beam direction is reduced by about 0.1P0, bringing the estimated momentum transfer more in line with ' the pre-equilibrium particle multiplicities and the systematics from inclusive studies of momentum transfer to fissile target nuclei [FAT85]. VI.C, Summary : Studies of intermediate mass fragments and coincident target-like residues provide general information about the energy and momentum 124 balance in fragment producing reactions. In fact, they allow several conclusions about the fragment production mechanism. The velocity distributions of the heavy residues indicate that intermediate mass fragments are produced with low multiplicities from highly damped reactions. For the 32S + Ag reaction at E/A-22.5 MeV, a total energy between 200 and 400 MeV is dissipated into internal degrees of freedom in the target-residue and primary intermediate mass fragment. Significant particle emission occurs prior to full equilibration. More than 20% of the projectile momentum is carried away by particles emitted during the early non-equilibrium stages of the reactions. The mean velocity of these particles is directed close to the beam axis and is somewhat less than half of the beam velocity. This qualitative picture is consistent with the single particle inclusive and coincidence distributions of light particles and intermediate mass fragments. Finally, the residue velocity distributions suggest that the portions of the energy spectra corresponding to emission at energies below the exit channel Coulomb barriers contain substantial contributions from the sequential decay of particle unstable primary fragments. Thus, the number of candidates for the principal mechanism for intermediate mass fragment production can be reduced. One may conclude from both the measured fragment multiplicities and the angular distributions of the target residue that fragments are not produced in reactions with large fragment multiplicities. Peripheral or quasi- elastic collisions can be excluded as an important source of intermediate mass fragments at large angles because the observed energy losses are too large. The fragments do not primarily result from binary 125 emission processes after full momentum transfer to the compound system. This conclusion differs from the interpretations of recent results [MIT85] from 8("Kr + 12C reactions at E/A-35 MeV which were supposed to proceed by complete fusion followed by binary fragment emission. For the present reactions, the peaks in the angular distributions of the recoiling residues never occur at the angles expected for such reactions. This general observation includes reactions producing fragments with masses comparable to the mass of the projectile. It is possible, however, that composite systems in some fraction of the reactions reach equilibrium without significant preequilibrium emission. Some recent studies indicate that this is possible, and that the detection of an intermediate mass fragment in the backward hemisphere is an excellent signature for such reactions [FAT86pc]. Chapter VII Statistical Aspects of Fragment Emission VII,A, Introduction : In previous sections, the importance of dynamics in defining the character of the fragment production mechanism has been emphasized. In this section certain statistical aspects of intermediate mass fragment production will be discussed, specifically the emission of nuclei in particle-stable and unstable excited states. The inclusion of these states in calculations of fragment emission will influence the final distributions both through the increased number of final states available for intermediate mass fragment emission and through the sequential decay of particle-unstable nuclei. A simple statistical formulation which includes excited states of nuclei is presented in Section VII.B. Using this schematic model, the observable effects of the emission of excited nuclei on the mass distributions, relative isotopic yields, and the relative population of excited states are discussed. A comparison of the model calculation with experimental findings will suggest that the emission of nuclei in their 126 127 excited states does have considerable impact on the final distributions of nuclei and nuclear states [FIE86p]. 32 In section C, two distinct measurements of the S + Ag system at ‘E/A-22.5 MeV [XU87] will be compared to the calculations. VII.B Stat'stica emis ion and the o ulation of excited states : VII.B.l. A.sthematic model : In order to investigate the general effect of the emission of excited nuclei, we will use a schematic calculation of statistical emission of charged particles from a compound nucleus. For simplicity, we assume that each available state, i, is initially populated with the weight, Pi’ given by Pia P0(Ai’zi) (231+1) exp(-Ei/T) , (VII-1) where P0(Ai’zi) denotes the population per spin degree of freedom of the ground state of a fragment of mass and charge numbers A1 and 21, respectively; Si and Bi denote the spin and excitation energy of state i; T is the emission temperature which characterizes the statistical population of states of a given nucleus. This temperature is not necessarily the same as the temperature which can characterizes the energy spectra of emitted fragments. In order to evaluate the effect of the decay of particle-unstable states one has to specify the relative populations P0(Ai’zi)' Because it is reasonably transparent, the parametrization [MASBl] used is 128 P0(Ai’zi) « exp ( -VC/T + Q/T) , (VII-2) where VC is the Coulomb barrier for emission from a parent nucleus of mass and atomic numbers Ap and Zp’ and Q is the ground state Q-value: 2 1/3 1/3 VC e Zi (Zp-Zi)/ [ro(Ai + (AP-A1) )], (VII-3) - B A -A.,Z -Z. + B. - B A ,Z . VII-4 Q ((131131) 1) (pp) < > The radius parameter is ro-l.2 fm. The binding energies, B(A,Z), of heavy nuclei are calculated from the Weizsacker mass formula [MAR69]: 2 2 _ _ 2/3_ _Z_ _ A-ZZ _ B(A,Z) COA ClA C2 A1/3 C3 A , (VII 5) -13.0 MeV, C -.595 MeV, and C -19.0 MeV. For the 1 2 3 emitted light fragments we use the measured binding energies, B where CO-14.1 MeV, C i' We restrict our discussion to primary fragments of mass numbers A1520. For these nuclei, all tabulated [AJ286] states with widths smaller than 3 MeV are included in the calculations. The exclusion of broader states was motivated by the necessity that a state live long enough for the nucleus to separate itself from the parent system [FAI82]. Both particle-stable and unstable states are treated explicitly in this calculation. Recent experimental results have demonstrated that nuclei are, in fact, emitted in their excited states. Figure VII-1 depicts the correlation function between two pairs of particles, the a- Figure VII-1 between alpha particles and 7Li nuclei (top) measured in 129 . MSU-853m 5 .2..---, ...... .22,-......--,-. . 9.1330 197 40 7. ‘ 9'27“ ”73 Au( Ar. 11 LUX 4.. :33 E/A-GO MeV _ I lo459711.262: 9 .30, ".444 av ' m ".589 }} lums 3" nnuzss'r 14.04 ‘ ': 'Z9W3MJ4 . . |1|3714555 17.43 . 3 '2“ '22: 12:22. 15 a: 2— II 5“ min-l"! 21.27 . .3? ' lefififirIfiflpT * 1 1- ”NM“ 11111:- 1 . 1 o L - U1 ......... 1 ......... 1.1 o 100 200 300 . q(MeV/c) MSU-85 255 2.5 .1....efl .e..,.. . 47.641 18.9lMev 1 'Me" €49.me 1 : [|9.24Mev I 2.o+- - - I IS 15 +1 .5; 1M1 11+ _ T I M . ‘7 H N E ' III 1,0”) +++--&+ Q1 "0" 1 ’3‘ ""”’ ”I 11+ 303 I". 1- + [I / .- l / D I 4 - 197 40 7. 0.5— [I] AM Ar. 0 L1)X 1 -/ II E/A=60 MeV, 90v=30° -[ 4 J11’.1..1.1....12... O0 50 100 150 200 q(MeV/c) : Correlations between protons and 7Li nuclei (bottom) and 40Ar + Au reactions at E/A-60 MeV as functions of the relative momenta between the two particles. 11 B and 8Be are indicated in the top and bottom figures, Peaks in the correlations which correspond to states in respectively. 130 7Li system (top) and the p-7Li system (bottom), as a function of the relative momentum between the two particles [POC86p]. The resonant structures correspond to the excited states of 113 and 8Be nuclei. In our model calculation, particle unstable states are assumed to decay to available final states through the emission of light particles (n,p,d,t,3He, and a particles). The relative rates to the final states are determined by the statistical model [HAU52] [ST084] j+s J+S r(€,J,j,S) a X X T£(€) 1 (VII-6) S-|j-s| £-|J-S| where s is the spin of the evaporated particle, j is the spin of the daughter nucleus, S is the channel spin, J is the spin of the parent nucleus, 1 is the orbital angular momentum of the emitted particle, and e is the decay energy. T! is the parametrized transmission coefficient for the 2th partial wave. (See Appendix D for details of the decay calculations.) VII.B.Z. Hass distributions : Figure VII-2 shows the primary and final mass distributions for emission calculated according to Eqs. VII-1-6 from a moderately excited xenon nucleus (AP-131, Zp-54, and T-5 MeV). The histogram represents the total primary mass distribution including both bound and unbound states. The dark and light shaded regions represent the contributions from ground and excited particle-stable states, respectively. It is apparent from the figure that at this temperature the contribution to the 131 MSU-86-292 pa 0 N . Ap=131, Zp=54, T=5 MeV 0 Final _J'" Primary—all states Z Bound excited H O H Stable g.s. H O 0 Yield (arb. units) 5] 10—3 10"3 0 5 10 15 20 11 X Figure VII-2 : Mass distributions calculated from Eq. VII-1 for emission from a Xe nucleus at T-5 MeV. Histogram : primary distribution; solid points : final distribution; dark and light shaded regions show contributions from bound ground and excited states, respectively. 132 intermediate mass fragment yield from emission of particle-stable states is substantial and is dominant for heavier fragments. The contribution from particle-unstable states is roughly half of the total yield. The primary distributions are relatively smooth with little structure resulting from variations in level density and binding energies. The solid points in the figure show the final mass distribution after the decay of particle-unstable states. These decays enhance the light particle yields, doubling the 4He yield. The relative yields for the light particles have been used to infer the entropy generated in nuclear reactions [BERT81] [D0885]. The contribution from sequential decay alters the relative yields for the light particles, and therefore must influence such interpretations of the light particle cross sections [ST083] [HAH86]. The yields of heavier fragments, A>4, are depleted by sequential decays. However, they generally remain significantly larger than the primary yields of bound nuclei. Secondary decay products constitute approximately half of the resultant fragment yield. Figure VII-3 compares the calculated final mass distribution (histogram) with the mass distribution measured for proton-induced reactions on xenon at Ep-80-350 GeV [HIR84] (solid points). (The calculated mass distribution includes only nuclei for which cross sections were published in [HIR84], and differs from the final distribution in Figure VII-2 primarily by the exclusion of 6He. The other nuclei not included are 8He, 14Be, 8’17 9’10’18-20C, 12N, 13,140, 17,18 B. and Ne. ) This particular mass distribution has been noted previously [FIN82] [HIR84] for its close approximation to a power law dependence (see section I.A ). However, a power law dependence determined by the distribution for heavier fragments significantly over- 102 Figure VII-3 : 133 MSU-86-29l . 1. . . . . I . . . . l . . . . I . Xe(p,X), Ep=80—350 GeV r '6' 0 Data, Hirsch et a1. '3 U '1' .r Calculation (T=5 MeV)i I'Tl r Frfruul _l Mass distribution from proton induced reactions on Xe (solid points) and final distribution predicted from Eq. VII-l (histogram). 134 estimates the observed cross section in the region from A-6 to 12. The failure of the thermal liquid-drop calculations [HIR84] [MAC85b] to predict this structure was attributed to the inadequacy of the Weizsacker mass formula for light nuclei. However, calculations of ground state yields which use correct masses (dark shaded area in Figure VII-2) do not reproduce the data either. A similar, though less distinct, structure was produced in a statistical calculation [RAN81] which included only particle-stable states, though it became evident only at high intrinsic excitation energies of about 40 MeV/nucleon. The authors of that work estimated that the inclusion of unstable states would wash out the structure. Alternatively, the detailed structure of the mass distributions has been attributed to in-medium effects, specifically to Pauli blocking during thermal freeze-out [R0383] [ROE85]. The present simple calculation reproduces the average slope, as well as the characteristic structures in the experimental mass distribution, using a temperature of T-5 MeV. The detailed structure of the mass distributions could, therefore, be the result of the emission and decay of particle-unbound excited nuclei. This phenomenon is independent of the actual emission mechanism so long as there are significant populations of the particle-unstable states. From the form of Eq. VII-1 it is obvious that the yields of heavier fragments increase with increasing temperature, both because of the behavior of P0(Ai’zi) and because the excited state yields increase with temperature. Figure VII-4 illustrates this temperature dependence of the mass distribution and the contribution from excited states from emission from the A-13l, 2-54 system at temperatures, T=3, 5, 7, and 10 MeV. The -86—426 . ”PHI- MSU 32% o 222222222222 OOOOOOOOOOOO 111111111111 1111111 New Xe. Am.u ts f, S .1 u m e e S a ac r u,o sn)f n s oatm .1 n tmio Mmmm if r de tnir $0.18 .1.10 dsss s(n eeeee FFFFF 136 total primary mass distributions are indicated by the histograms, the mass distributions of nuclei emitted in their ground states are shown by the shaded regions, and the final distributions are shown by the solid points. All mass distributions become less steep at higher temperatures. As the temperature increases, contributions from the primary population of stable ground states become increasingly inconsequential. At the same time the effects of sequential decay on the fragment yields (for instance, the light particle ratios) increases. The sequential decay of the particle-unstable states significantly alters the mass distribution, specifically in the mass region, A-6-ll. It is important to note that the temperature dependence of this parametrization does not agree with the behavior of more complete compound nucleus calculations. These calculation predict that at high temperatures the mass yield curve should become steeper with increasing temperature. Thus, while the present parametrization predicts that the effects of sequential decay become more important with increasing temperature, they will in reality be moderated by more steeply falling mass distributions. ( See Chapter VIII. ) VII.B.3. Isotopic yields : Relative isotopic yields from statistical calculations are also affected by the inclusion of excited nuclear states. Figure VII-5 shows the relative yields of nitrogen isotopes for the calculation shown in Figures VII-2 and VII-3. The distributions are normalized so that the yield of the most abundant isotope is one. The squares represent the relative yields corresponding to the primary population of the stable 137 1 MSU—86-427 1° IIIIrIIIrII N from Xe I llllll l llLl l - primary 0 final D st.g.s. j g...- CD CD I I IIIIII l l lllJll OD U U H CD I H U I I llllll I l llllll H o' ('0 D O 0f I iilllll Relative Yield ..”“A l l 10-3 I LIIIIUI l l llllll I l 10—4 1 l l l I J I l l I I N—Z Figure VII-5 : The calculated isotopic distributions for N nuclei emitted from a nucleus, A-lBl and Z-SA, at T-S MeV. The distribution of stable ground states is shown as squares. The primary and final distributions from calculations which include excited states are shown as the histogram and solid points, respectively. 138 ground states of nitrogen isotopes. The isotopic distribution of primary fragments of all states is shown by the histogram. The larger number of available states around the valley of stability produces a distribution which is narrower than the primary isotopic distributions of the ground states. The sequential decay of particle-unstable states results in a distribution (solid points) which is even more narrowly distributed about the valley of 3 stability. Figure VII-6 compares isotopic distributions (histogram) calculated at T-S MeV with the measured distributions (solid points) for elements, Z-3-8, from proton-induced reactions on xenon [HIR84]. While the calculated isotopic distributions are slightly broader than the measured distributions, the general behavior of the data is described by the statistical calculation. The effect of sequential decay on isotopic distributions has been pointed out previously with respect to distributions from projectile fragmentation [VIY79] [MORR79] and deeply inelastic scattering [BAR78] [BRA78] [LOC82]. It was observed that the decay of proton- and neutron- rich primary fragments towards the valley of 3 stability destroyed significant differences between proposed primary distributions. This can be seen in the calculated temperature and isospin dependence of the isotopic distributions. The temperature dependence of the isotopic distribution results, in this model, from the competition between two effects. As the temperature increases the primary distribution becomes broader. Figure VII-7 shows the primary isotopic distributions for nitrogen nuclei emitted at temperatures of T(MeV)- 1(solid diamonds), 3(open squares), 5(stars), and 10(solid circles). The distribution becomes broader with increasing 102 101 10° 10‘1 10‘2 "U "I -3 .9. 10 >_. m .52. 4..) O (U 10 v—-l £2 10‘1 10"2 10‘3 10‘4 Figure VII-6 : 139 I U I Illlll lilnlnlilnl'lnlnlnl llll [IEIIIII MSU-86—428 -I 'l 'l 'I 'I II 'I 'l 'l 'I 'l ‘l 'I 'l 'l 'I Fl '1‘ r' = A=131, 2:54, T=5 MeV g E Zx=3 21:4 22:5 5 E" 'E F 7 1. E 0 i ‘5‘" “i : . : r a ' I. I. L I. I. I. I. I I. I I. I I. I. I. I EI' l' l' I"I' I l' l' I' I' I I I' I I' II I' I: ; 21:6 21:7 Zx=8 : P 7. : CI 1 r- '1 5 3 E— ).-g 1. 1L -4-20 2 4 65—4—20 2 4 6—4-20 2 4 6 1N¥-Z The calculated isotopic distributions of Zx-3-8 from a nucleus, A-lBl, Z-Sh and T-S MeV are shown as histograms. The solid points represent the measured isotopic ratios for p+Xe reactions [HIR84]. 140 1 MSU—86—429 1° I T I I I I I I I I I a 2 Primary N from Xe 3 - T(MeV)=1(’).3(D).5(+).10(0) ‘ 100 5— 2.2-‘5‘ 1 a \s: a E . 2 cu - . ”—1 >-' 10_1 :— —: a) E 5 .2 ' t " : .4.) _ . 2 (D 10-2 .:_ —:- a: E E 1.0—3 :— _: E II 5 10-4 I I I I I I I I I —4 —2 2 4 6 I o N—Z Figure VII-7 : The calculated primary isotopic distributions for N nuclei emitted from a nucleus, A-l3l and Z-Sh, at temperatures, T(MeV)- 1 (solid diamonds), 3 (open squares), 5 (star) and 10 (solid circle). 141 temperature, and the centroid shifts slightly towards the proton-rich isotopes. The primary distribution at higher temperatures is principally the result of the population of particle-unstable states. However, the decays of these particle-unstable nuclei strongly influence the final distributions. Neutron-rich nitrogen isotopes decay towards stable isotopes, other unstable nitrogen nuclei decay to other elements, and heavier elements decay towards stable nitrogen nuclei. This decay of the unstable nuclei narrows the final distributions. The final isotopic distributions for nitrogen at the four temperatures are displayed in Figure VII-8. At low temperatures (T-l MeV) the distribution, dominated by the emission of nuclei in their ground states, is very narrow and is nearly identical to the primary distribution. As the temperature increases (T-3 MeV), the distribution becomes broader. However, the width of the distribution ceases to change significantly as the temperature increases further (T-5,10 MeV). This follows from the temperature independence of the particle branching ratios. Once the particle unstable states become the dominant contributions to the yields, increases in their populations relative to lower lying states do not alter their patterns of decay to lighter nuclei and thus the final distributions. Note, again, that the behavior of PO(Ai’Zi) at high temperatures is different from what is expected for real nuclei, so that the effect of sequential decay at Tle MeV may be somewhat smaller than calculated. This saturation behavior has also been seen in the energy loss dependence of the isobaric charge distributions of projectile-like 142 1 MSU—86—430 10 5 I I I I I I I I I I I 5 E N from Xe : - T(MeV)=1(’).3(U).5(+).10(0) ~ E / \ E E : 'lll \F-l.“ : .93. " ‘ ‘ >" 10-1 .3- —3‘ Q) E : > _ .. 15' P “- ‘ .9. ‘ I \ ‘ fig 10-2 E— u t' —5 C 1 10—3 :— —: E 3 10—4 I | L l l I I l L -4 —2 o 2 4 6 N—Z Figure VII-8 : The calculated final isotopic distributions for N nuclei emitted from a nucleus, A-lBl and Z-SA, at temperatures, T(MeV)- 1 (solid diamonds), 3 (open squares), 5 (star) and 10 (solid circle). 143 fragments from deeply inelastic collisions [M1680] [LOC82]. The energy loss of the projectile-like fragment is directly related its excitation energy. At low energy losses the widths of the isobaric charge distributions increase with increasing energy loss. The widths cease to increase, however, for energy losses above a modest value of 30-50 MeV [M1680]. In projectile fragmentation reactions, the isotopic distributions do not change significantly between E/A-20 MeV and 2 GeV [BUE76] [GEL78]. In each of these cases, sequential decays may render the initial state of the primary system irrelevant to the final distributions. Figure VII-9 shows the isotopic distributions for the stable ground states of nitrogen nuclei from emitting nuclei of charge, 2-54, and A- 120 (open diamonds), 130 (solid squares), and 140 (stars) at a temperature of T-S MeV. The distributions demonstrate a significant sensitivity to the neutron-to-proton ratio of the emitting nucleus. However, previous arguments have indicated that the final isotopic distributions are strongly influenced by sequential decay. Figure VII-10 shows the predicted final isotopic distributions when unstable states are included. The distributions are narrower than the primary stable ground state populations and exhibit only a modest sensitivity to the neutron-to-proton ratio of the emitting nucleus. VII.B.h. Excited state populations : The mass and isotopic distributions are sensitive to the total population of unstable states, rather than the population of each state. It is, therefore, not possible to conclude from the analysis of the 144 1 MSU—86—431 10 5 I I I I I I I I I I I a : N(st) from Z=54,T=5 MeV: .- A = 120(0),130(I),140(+) - 10° : x“ i3. I o. I 2 — - . 1 CD - v o - .pq ‘0 >" 10‘1 ;— a a) E E > _. .4 .94 - _ 4.2 _ - :9 CD 10‘2 :— 0 ‘E 03' E i 10’3 e" ‘ p : F . 10—4 I I I I I I I L I I I N—Z Figure VII-9 : The calculated primary isotopic distributions for the stable ground states of nitrogen nuclei emitted from a nucleus of charge, Z-Sh, and mass, A - 120 (open diamonds), 130 (solid squares), and 140 (stars), at a temperature of T- 5 MeV. 145 1 MSU—86—432 10 5 I I I I I I I I I I I : N from Z=54,T=5 MeV ; - A = 120(0), 130(l), 140(+) . 100 :— I/ _: "£3 : . \o : Q) I" .I og—l ¢ >-' 10_1 :_' 9 _: <1) E E :> - _ .Fq - a -+-> _ . 2 CD 10-2 '5' —: D: E 0 E I o I 10-3 =- 0 '3 10-4 l I I I I I | I I -4 -2 2 4 6 0 Figure VII-10: The calculated final isotopic distributions of nitrogen nuclei emitted from a nucleus of charge, Z-Sh, and mass, A - 120 (open diamonds), 130 (solid squares), and 140 (stars), at a temperature of T- 5 MeV. 146 previous sections that nuclear excited states are populated precisely according to equilibrium thermodynamical probabilities. Direct measurements of the populations of excited nuclear states would constitute more definitive tests of statistical theories of fragmentation distributions. In thermal models, the ratio of the primary populations of two narrow states in a given nucleus is given by "U (25.+l) —-1-- _ __i_ R - P. Rco exp( AE/T) (25j+1) p exp(-AE/T) , (VII-7) J where AB is the difference in the level energies and R00 is the high temperature limit. This expression was used to extract the effective emission temperatures from the measured relative populations of excited states [MORR84] [MORR8S] [POC85a] [POC85c] [CHI86a] [POC86p]. The populations of low lying particle-stable levels relative to the respective ground states were interpreted [MORR84] [MORR85] in terms of unexpectedly low emission temperatures, T_‘ 1.00 gr 21:4 Zx=7 El " o _._I_ 3 Q) _ ‘_‘ . o . E 0.10 E" _ (t5 “ . - '35 - I“ . I I I I I I D: (1133 I I I I I I I I I I I 0.10 . ‘ T 001 I 1 J l I I I -3—2-10 1 2 3 .—2—10 1 2 3 4 Figure VII-l4: Isotopic distributions from 32$ + Ag reactions at E/A-22.S MeV (solid points) are compared to calculations (histograms) for emission from a nucleus, A-l30, Z-58, at a temperature of T—lO MeV. 155 VII.C.2. Population ratios : Spectra of gamma rays detected in coincidence with five different intermediate mass fragments produced in the 32$ + Ag system are shown as histograms in Figure VII-15. The peaks corresponding to the specified transitions in the coincident intermediate mass fragments are clearly evident. The solid points correspond to the backgrounds estimated from a mixed event analysis for each nucleus. The fractiOn, F7, of nuclei emitting that particular gamma ray is given by the integrated number of counts above background in each peak divided by the product of the detection efficiency for the gamma ray and the number of detected intermediate mass fragments. The measured values for these fractions are indicated by the hatched regions in Figure VII-l6. The ratios expected for the primary distribution, as given in Eq. VII-7, are indicated by the dashed curves. The emission temperatures can be estimated from a comparison of this relationship and the measured values, except for the case of 123. Because of significant feeding from higher lying particle stable states, the 128 ratio is consistent with the entire range of temperatures between 1 and 10 MeV and cannot unambiguously determine an emission temperature. The temperatures estimated using Eq. VII-6 and neglecting sequential decay are shown in Figure VII—17. The temperature estimates 7Be (E*- 429 keV) and 8L1 (E*- 981 keV) are consistent with measurements made with the 14N + Ag system at E/A=35 for the first excited states in MeV [MORR84] [MORR85]. The temperatures estimated from measurements of 1 l 0B (E*- 2154 keV) and 3C (E*- 3854 keV) are somewhat higher, T z 1.5- 156 MSU— 86- 436 1,0z..l....,....1...“...1'”,I 0.9 E_ Ag(328,X) 1°13(2.154— 1._‘740)- 0.9 085 E/A=22.5 MeV 3 - 7 , 70.8 E — Data IIII # III. ' 3 0'7: 0 Background | | TI IIIIIIF ‘10-7 0.6 E I I‘ -20.6 ”d T). O5IIIIFIILIO5 ..—I 08 :__°L1(o(o,,981_g,s)l 128 (0.953-g..s) 08 I)“ ' E (2.621 — 1.674) ' g 0.6illl 0.6 :3 0.4 ,JII-I _WI1.II I 0.4 (U IIH-m-IL 1 .I ll—'I "‘| 'r: cm 0.2 1".02 Q: . J ‘ l L L l l L I P l l l_ l l l j 1 I l l l I l l l 1 l I l l . I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I O .I "Be (0.4 9 - g.s.) ‘30 (3.854 — 3.684) f 0.8 1 0.8 0.6 {0.6 JIM-LI, . I' '1..- I ' 1'1': .1 ' ‘ '. JI ‘ . ”1. 0.4: I .‘"11.}[_¢ |l ' (3J4 . . I 1 .. . I .. . I . . . I. . . .I . I L. I . . -50 0 50 -50 50 E— E (keV)O Figure VII-15: Energy spectra (shown as histograms) of gamma rays in the regions corresponding to transitions in coincident intermediate mass fragments produced in 328 + Ag reactions at E/A-22.5 MeV. The background is indicated by the solid points. 157 MSU— 86— —438 0.8 Tfifl“"l””l""l""_._" I I" mlr' I' -O.8 : A=130, 2:58 i 23 (O 953 ‘ ES) 3 E - Primary 0.4 7 Final 1777 0.4 . / .. 4 . t .. . 0.2 __ //// Da a 1:4 1 0.2 0.0 ::.H Ina}: : :{: ”*4: :H‘%: €01) 0.4 331.1 (0.981 — gs) {—788 (0.429 — g s) —‘ 0.4 I —————— :: _ _‘: 0.3 E- ’ ’ -EE- ’ """ -. 0.3 O 2 E :: / 30 2 . ( - a: 7///////// /: . 0.1 :7 1.— 1 0.1 0.0:::::|L::::%:H.L%:'}::::::::1}1L+LILHHIL %*“::0.0 I 1 _ ‘C 13 __ I 0.15 __ °8 (2.154 1.740) .7 c (8. 854 8884):: 0.15 : I //” : 0.10 :— , “:- / 1 0.10 0.05 W //Zz. ///////////// 0.05 0.00 .1/1LLI.. 1.2/1 111.1II...1J..111..1I:0.00 0 2 0 2 4 6 8 10 Figure VII-l6: The fractions, undergoing specified transitions are indicated as a functions of temperature for calculations from Eq. VII-l both with (solid curves) and without (dashed curves) the contributions from sequential decay. The observed fractions are indicated by the hatched regions. F7, of intermediate mass fragments 158 MSU—86—439 rrllllllllll-IIIIII.IIllll Ag(328,X7), E/A=22.5 MeV 01 F 7Be 8Li 1"B 13c: l l l . + Ollll llllnlllllllLlLlL —o— llllllllllllllllllilllll Illllllllllfillllllll Apparent Temperature (MeV) Figure VII-l7: The apparent emission temperatures of intermediate mass fragments emitted in 328 + Ag reactions at E/A—22.5 MeV, as discussed in the text. 159 2.5 MeV. The transitions in these nuclei are from levels with higher excitation energies than in 7Be and 8Li. This type of relation between the temperature estimates and the level separations was suggested in [POC85a]; it is difficult to measure temperatures higher than the level separation of the states from which the temperatures are inferred. The solid curves in Figure VII-l6 correspond to calculations which include sequential feeding from statistically populated unbound resonances. Feeding from particle-unstable states lowers the observed fraction of the nuclei which undergo the specified transitions. In fact, because the relative yield of heavier unstable nuclei increases in this model, the calculated values of F1 can actually reach a maximum at some temperature and decline with increasing temperature. As was the case with the calculations for unstable states, the observed final populations of excited states relative to the ground states is predicted to be significantly less than for the primary distribution. The inclusion of the decay of particle-unstable states brings the calculation into better agreement with the data at higher temperatures. In order to examine the importance of the shape of the mass distribution on F1, calculations with the quantum statistical model are shown in Figure VII-18 as solid and dashed curves for p/po-O.l and 0.9, respectively. The calculation at p/po-O.l produces the steepest mass distribution, and in that case the relative populations of states is least affected by sequential decay. Because the primary population of excited states increases with temperature as the number of heavier nuclei which may populate states in lighter nuclei decrease, the quantum statistical calculations show a less dramatic dependence on temperature than the calculations of Figure VII-l6. The general behavior of the 160 MSU— 86- 440 Ijiiltiiillrifilirjlll'rii_ 08 ' L. 1 'r r’vT “If I‘rv ‘; 1213 (0.953 - gs.) 0.05/ 0.00 Figure VII-18: The fractions, F7, of intermediate mass fragments undergoing specified transitions are indicated as a functions of temperature for quantum statistical calculations at p/po-.l (solid curve) and .9 (dashed curve). the hatched regions. , 0.8 E — p _ .1 -1- J 0.4 :- ___ p = 9 I /////,Data 0.2 r 0.0 EHH£§$H§HH[HH%HH: 04 5.51.1 (0.981 — g.s.) —:—7Be (0.429 - g.s.) .5 0.4 0.3 ,— —i— —I 0.3 : /‘¥\ . /\ : .. / \\ I- x‘ . CLEB E;)/r:. .. r- . . . )K?” .2ny- /C7’ ()13 c; : WWW/7537“" .///x. .///.". .zfi 0.1 .— r’ 1 0.1 OO:}§f%1l§§§i%‘r§‘r‘r%§ii§}‘fi‘rii ‘r§#§%§%§§%§§§§=%w‘%§ll‘r%h; 00 I 1 _ 13 _ ‘ 0.15 .— °B (2.154 1 740) f c (3 854. 3684) —jo.15 C I; 0.10 0.10 €0.05 0.00 10 The observed fractions are indicated by l6l calculations is similar to that of the calculation from Eq. VII-l. VII,D. Summary of results ; In this chapter, the manifestations of equilibrium populations of excited states in the measured mass, isotopic, and excited state distributions have been investigated. These distributions, when calculated in schematic statistical models, were shown to be influenced by the decay of unstable fragments, and the experimental distributions were shown to confirm the existence of these effects. The detailed shape of the final mass distribution is strongly influenced by the population of excited states in the primary distribution. Factors which determine the overall shapes of the mass and elemental distributions are the number of available states, the Q-values of the fragment emission, and the Coulomb barriers. Although Coulomb effects are neglected in the quantum statistical model, which can also describe aspects of the data, the experimental data clearly indicate the influence of the Coulomb energies. The particle decays of unstable states introduce a characteristic structure in the mass distributions in the region A-6-12, where the yield is suppressed compared to an interpolation of the cross sections between the heavier fragments and light particles. This structure arises from the decay of particle- unstable states. There are relatively few stable nuclear states in that particular region of the mass spectrum, leading to the particle decay of a large fraction of the primary fragments. In addition, heavier fragments decaying into this region will, with a high probability, decay 162 further into light particles, contributing to the diminution of the yields. The general features of the isotopic distributions largely reflect the binding energies of the stable nuclei in their ground states. Calculations of statistical emission which include unstable states provide better agreement with the data than calculations using ground state binding energies alone. The 328 + Ag data appear to require that the effective emission temperature be greater than or equal to 3 MeV; distributions calculated for lower temperatures are more narrow than the observed distributions. The calculated distributions are not very sensitive to increases in temperature above 4 MeV. This is a result of the decay of unstable particles towards the valley of fl stability. Thus, isotopic distributions provide only a lower limit for the emission temperature. This insensitivity to changes in emission temperature has also been observed in fragmentation [BUE76] [GEL78] and deeply inelastic reactions [M1680] [LOC82]. Statistical calculations predict that the observed relative populations of nuclear states are significantly affected by feeding from the decay of unstable nuclei. The present schematic calculations indicate a preferential feeding to lower lying states, reducing the observed relative populations of higher excited states. This feeding from sequential decay leads to considerable complications in attempts to determine the emission temperatures from the relative populations of states. This problem is particularly serious when only low lying particle-stable states are investigated. At higher temperatures, however, similar problems arise for particle-unstable states, with the notable exception of states in 5Li. The present calculations may serve 163 as useful guides to elucidate the magnitude of the influence of sequential decay, but uncertainties in of the calculations prevent precise determinations of these effects. The relative populations of states are sensitive to the charge and mass dependences of the primary yields. This was illustrated in the comparisons between the present calculations, in which difference in the mass spectra were reflected in the temperature dependences of the population ratios. The present parametrizations are inadequate. Improvements in theoretical treatments of the mass and charge dependences of the primary populations of states must also eventually address the angular momentum dependence of fragment production, possible in-medium corrections, and other details of the emission mechanism. In addition, the detailed populations of final states are sensitive to the precise particle decay branching ratios. We have addressed the problem of the particle decay of unstable states in terms of a purely statistical treatment of the branching ratios. The spins, parities, and isospins of many of the tabulated states used in this analysis are poorly determined or unknown; the treatment of their decays is, therefore, not very reliable. Furthermore, nuclear structure effects are expected to affect the branching ratios. These effects have not yet been incorporated, largely because of the information is not available. Nonetheless, the general success of the calculations should encourage further refinements to such models, as well as additional experimental work. Chapter VIII Equilibration and Decay in Nuclear Collisions VIII,AI Model for statistical decay during equilibration : VIII.AN1. Motivation : In prior chapters several features of the data were described as statistical, and in Chapter VII the relative cross sections of intermediate mass fragments and their excited states were calculated with equilibrium statistical formulations. However, both the angular distributions and energy spectra of intermediate mass fragments give clear evidence that particles are emitted prior to, as well as after, the complete equilibration of the composite system. This demands that descriptions of intermediate mass fragment emission encompass the early non-equilibrium stages of the reaction and the evolution of the system towards the fully equilibrated composite nucleus. While this process can be crudely approximated by a two source model, as in Section III.B., the details of the energy spectra and angular distributions can be expected 164 165 to contain information concerning the time scales and mechanisms for particle emission and equilibration. In this chapter a schematic model is presented which attempts to reconcile the equilibrium and non-equilibrium aspects of the data by assuming local equilibrium in a dynamically evolving composite system [FIE84]. In this picture the initial source of emission is assumed to be a localized region of high excitation in the target-projectile system at the overlap of the projectile and the target. It is assumed to be in local equilibrium, to the extent that particles are emitted with equilibrium statistical probabilities. The local region of excitation evolves towards the fusion-like system by accreting nucleons from the cold target spectator. As a result of this accretion, the velocity and temperature of this "hot spot" decrease with time. The assumption of localization is supported by two-particle correlation experiments which indicate emission sources smaller than the compound nucleus [LYN83] [CHI86a] [POC86a]. VIII.A.2. Formulation of the model : The formulation presented here follows closely the model of Friedman and Lynch [FR183a], which calculates the time evolution of the compound nucleus and its emission products. The present model calculates the average number, Ni’ of particles of species, i, with mass, charge, and neutron numbers, ai, 21, and ni, emitted from a source with momentum, Ps(t); mass, charge, and neutron numbers, as(t), zs(t), and * * * * * n (t); and an excitation energy of E =6 2 (t) + e n (t), where e and e s s p s n s n p are the intrinsic excitation energies of the neutrons and protons, 166 respectively. This source is a subset of the total system which has mass, charge, and neutron numbers, a(t), z(t), and n(t); the sizes of both the source and the total system are functions of time, t, and, implicitly, temperature, T. The differential emission rate, dZNi/dEdt, for the 1211 particle type with an energy E is determined by a statistical prescription. The system initially consists of a hot participant region and a cold spectator. The initial size of the participant region is a parameter, which for small projectiles is usually fixed at twice the projectile mass. For initial source sizes less than or equal to twice the projectile mass, the source consists of equal numbers of target and projectile nucleons. For larger initial sources the source includes all of the projectile plus additional nucleons from the target. The source is assumed have an excitation energy determined by the conservation of the projectile momentum and energy. It is assumed to be in thermal equilibrium at the temperature calculated for a nuclear Fermi gas at normal nuclear matter density. From this initial stage, the source evolves by particle emission and accretion. The source accretes spectator nucleons at a rate [das/dt]a, thus slowing and converting its kinetic energy into excitation energy. The system reaches equilibrium at a time te with a temperature Te when as(te)- a(te). After this time, the system continues to emit particles until the excitation energy of the residue falls below the threshold for particle emission. The evolution of the temperature with time is contained in the cooling relation, dt/dT, which is derived from the statement of energy conservation, * dEs dN. d(Ek)i dEa dt— + E (52;) BiD + E -—dE—_ - 32‘ - 0 (VIII-l) The change in the intrinsic excitation of the source is given by differentiation of E:, * dE dz dn ._s __§ * __§ * dt dt €p(T) + dt €n(T) + Q: [nscn + zSCp] at (VIII-2) where dz da dN ._§ _ Z (__s) _ 2 __i z and dt A dt a 1 dt 1 dn da dN __§ Z __s _ __i dt A (dt )a i dt “1 ' (VIII'3) The heat capacity, C, is defined by 66*/8T and is calculated for the protons and neutrons independently. The separation energies, BiD’ is given by BiD- B(a,z) - ( B(a-ai,z-zi) + Bi) , (VIII-A) where the binding energies are defined as in Eq. VII-4 and VII-5. The emission rates as a function of time are given by the integral over the particle energy, E, 168 dE . (VIII-5) dt dE dt 37} . The rate of energy loss due to the kinetic energy of iEh particle 2 d(Ek). d N. 1. _ l 1 . _ dt 0 dB dt E dE . (VIII 6) During accretion, the source momentum is distributed among an increasing number of nucleons, and the source slows. Thus, the kinetic energy of the source is converted into excitation energy at the rate dE P d as(t) __ __ [ dt 1a, (VIII-7) where m0 is the rest nucleon mass. Solving for the cooling relation results in the expression * . * 2 dNi(B - z 5* - n 6*) + d(Ek)i - dEa e z + enn(das dt iD i p i n dt dt a dt dI_-i dt n C + 2 C s n s p (VIII-8) The differential emission rate for the iEh species is given by the Weisskopf formula [WEI37], 169 1 d2 dt ' «2 n3 E M ai+D+P exp(AS) , (VIII-9) where 51 is the spin of the emitted particle, and M is its mass. The cross section for the inverse, fusion reaction i+D~P, ai+D+P is parametrized as 2 E-V ai+D4P- "RiD ( E ) 0(E-V) (VIII-10) where 0(x) is 1 if x>0 and 0 if x50, the radius is given by [(as- ai)1/3+ ail/3 ] r0, for aiz2 RiD- , (VIII-ll) 1/3 rO (as-1) , for ai-l with ro- 1.2 fm, and the Coulomb barrier is given by 2 Ae zi(z - zi) rC- [ai/3+ (a - ai)1/3] ViD- (VIII-12) The Coulomb radius parameter is rC-l.44 fm. The parameter A is adjustable, and can be used to reduce the Coulomb barrier from the value for touching spheres, as reduced Coulomb barriers are observed in fragment energy spectra ( see Section III.B.3. ). The change in the entropy of the system resulting from emission, AS, is given in a constant density approximation by 170 D-JIH * * AS- - [ E + BiD - [zi(ep - Tap) + ni(en - Tan)] (z * + n * E B )2 i‘p i‘n ' ' iD + 2T [ c (z - z.) + c (n - n.)] ] ’ (VIII'13) p s 1 n s i where 0p and on are the intrinsic entropies for the protons and neutrons, respectively, so that S - apn + annn. The time evolution of the parameters is treated in an integral approach. The mass of the source, for example, is given by t dNi daS as(t) - as(0) + JO[-§ df— ai + (dz—)a ] dt te dN. da __$ __§ as(t-w)- as(0) + IO[-§ dt a1 + (dt )3 ] dt 0 dN __i gt + IT (-§ dt ai)(dT) dT (VIII-14) e The kinematics of the emission is determined in this model by the source momentum, which is given by t dP __s Ps(t) - PS(O) + Io dt dt , (VIII-15) where l7l dP (t) Ps(t) dNi -—§E__ - - as(t) - E 52‘ a1.. . (VIII-l6) VIII.B. General features of the calculation : This formulation of a statistical emission model differs from the schematic model presented in Section VII.B. both in its calculation of the instantaneous emission rate and in its inclusion of the time evolution of the emitting system. VIII.B.l. Instantaneous emission rate : The Weisskopf formula is derived from the principle of detailed balance. This is that in equilibrium the transition rate from state 1 to state 2 is the same as the rate from 2 to l. The rate of decay from 1 with an energy E- mv2/2 in the interval dB is expressed as $23+lzmpdE wfil) 1 - (01+Dqu ) ( "2 M3 ) ( w(2) ) . (VIII-l7) The first term on the right hand side represents the rate at which the inverse reaction can take place for specific quantum states of 1 and 2, i.e. the flux times the cross section. The second term represents the phase space represented by the momentum of the free emitted particle. The third term is the ratio of the number of microstates in states 1 and 2. 172 The instantaneous emission rate differs from Eq. VIII-l in three ways. The preexponential term in Eq. VIII-9 has no counterpart in Eq. VII-l. This preexponential term, consisting of the mass, m, of the emitted fragment and the inverse cross section, 0. introduces a 1+D4P’ weak fragment mass dependence. The dependence of this term on ai is depicted in Figure VIII-l. This preexponential term strengthens the yield of heavier fragments, particularly for large sources. The second order, l/T2, term of Eq. VIII-l3, also makes the mass spectrum sensitive to the size of the source. The mass spectra of the instantaneous emission rate of particle stable nuclei from three different sources in the same composite system, a-220, z-89, at T-8 MeV are shown in Figure VIII-2. The spectra are shown as a solid curve for (as,zs) of (24,12), as a dashed curve for (80,35), and as a dotted curve for (200,80). The spectra for the two larger sources are nearly the same, indicating that the mass spectrum is independent of source size when as>>ai. The mass spectrum of the smaller source exhibits a steeper slope than those of much larger sources. Thus, the l/T2 term in Eq. VIII-13 inhibits the emission of fragments which are not small compared to the source. In practice, the requirement that ai< .5aS is imposed in the calculation. The other term in the argument of the exponential is of the form of aif*/T, where the free energy is f*- e*- To. This term strongly favors the emission of light particles from a hot system, resulting in a relative suppression of the heavy fragment yields. The mass spectrum is shown in Figure VIII-3 as a function of temperature for a system of a-llZ and z-52. The instantaneous emission rate of particle stable ground states is plotted as a function of the atomic mass, ai, of the 173 . MSU—86—441 20IIIIIIIIIIIIIIITTTTIIIIT - — -.-‘ y=ai°[(a-a.) 1/ 3+a3/ 3]2 o. d 15 — — a=50 —~ - - - a= 100 .- ' .. ........ a=200 // 10— y (X001) Olllllllkllllll Lllllllll 0 5 10 15 20 25 a1 Figure VIII-l : The dependence on source and fragment mass introduced by the preexponential terms in Eq. VIII-9. 174 MSU—86—442 [ITIIIIIIIIIUTIIIFII 10O T=8 MeV , a,,z,= 2442 ---- 80,35 ......... 23()(),E3() l llll"1__1—T1' _ - - G q - - - d .1 u - q - fl ‘ u q d - - - Relative Yield "5. GO 1 1 l I 1 1 l 1 I 1 1 l 1 7 5 10 15 20 a, 10-6 pa. O. 01 “—I—TTI'HTII l ”Harri llllllll T Iluull l rlmul (:3 .__. F_. Figure VIII-2 : The mass spectrum of the instantaneous emission rate for particle stable nuclei as a function of source size in a composite system of a-220 and z-80. 175 MSU—86-443 I I r’I’ I I I I I I I I ’rrI I I I I I I 100 ~— a=112, 2:52, T= — Relative Yield Figure VIII-3 : The mass spectrum of the instantaneous emission rate of nuclei emitted in their particle stable ground states as a function of source temperature for a compound nucleus of a-llZ and 2-56. l76 emitted particle. The distribution at the lowest temperature, T-3.9 MeV, shown as the solid curve, falls with increasing particle mass, principally under the influence of the Coulomb barrier, At a temperature, T-6.4 MeV, the mass distribution (dashed curve) is somewhat less steep. At a higher temperature, T-l6 MeV (dotted curve), the free energy term becomes important, so that the mass yield drops much more rapidly with fragment mass. As the early stages of the reaction are presumably the "hottest", the non-equilibrium component of the energy spectra should be strongest for the lightest fragments. The mass spectrum is sensitive to the Coulomb barriers for particle emission. If the effective barrier is reduced, for example, by the deformation of the composite system, then the mass spectrum will reflect this change. In Figure VIII-4, the mass spectrum for the instantaneous emission rate of particle stable nuclei from a system, a-200, z-87, T—8 MeV, is shown for calculations with A - 1.0, 0.8, and 0.6 by solid, dashed, and dotted curves, respectively. High Coulomb barriers inhibit the emission of heavier fragments. As the barriers are reduced, the heavy fragment yields are greatly enhanced. VIII.B.Z. Time evolution : The other principal feature which distinguishes the present model from Eq. VII-l is the time evolution of emitting nuclei. In Figure VIII- 5 the time evolution of a compound nuclear system, as(t-0)=a-ll2, zs(t-0)-z-52, is depicted as trajectories in plots of a(t) vs. T(t). These begin at different temperatures, as indicated by the open points, corresponding to incident energies in the Fe+Fe system of E/A=3, 10, 20, Relative Yield Figure VIII«4 : 177 MSU—86—444 - I I IIIIIII l I IITIHI l Ijllllll _ I I I I I I r I I I’ II I IF I I I I I rq a=200, 2:87, T=8 MeV A=1.0 . l I I I lllll 1' I I I lllll l I‘I l IIII (2) particle stable nuclei calculated as a function of the Coulomb parameter, A, as in Eq. VIII-12. The mass spectrum of the instantaneous emission rate of 178 MSU—86—44'7 150 l I I I I I I I I I I I I I I I T I Illj Illl 125 a=112, 2:52 f. llllllllll 100 25 lllllllllllllll \2 (II IIIIIIIIIIIIIIITIIIIIIIII OllllllllllLllILlll 15 10 5 T (MeV) 0 Figure VIII-5 : The time evolution of compound nuclear systems with initial a-llZ and z-52 starting from four temperatures. 179 and 50 MeV. The slope of the trajectories in the early stages are steep, with large mass losses per temperature change. This follows from the energy dependence of the temperature, 'I‘ccEl/2 , as the energy loss is proportional to mass loss. As the initial excitation energy increases the total mass loss increases. The trajectories of Figure VIII-5 are calculated for equilibrated compound nuclei. A major point of this schematic model was to incorporate pre-equilibrium emission into the time evolution of the system. In Figure VIII—6, trajectories for the 12C + 197Au system at E/A-30 MeV are shown for initial source sizes of as(t-0)- 24,50,100, and 209, as indicated in the figure. The accretion rate of the source was fixed to be a constant [das/dt]a- 3 nucleons/(fm/c). This corresponds roughly to the number of spectator nucleons (zl80) divided by the transit time of the projectile across the target (2r(Au)/vpz 50 fm/c, where r is the radius of the nucleus and vP is the projectile velocity). A comparison of Figures VIII-5 and VIII-6 demonstrates that emission prior to the attainment of full statistical equilibrium reduces the total particle emission during the history of the system. Particles emitted prior to equilibration are more energetic, as they carry away the energy associated with the source momentum. As a result, less energy is left for particle emission. In Figure VIII—7, the time evolution of the 12C + 197Au system at incident energies of E/A- 10, 30, 50, and 100 MeV is demonstrated. The accretion rate was fixed to be a constant [das/dt]a- 3 nucleons/(fm/c). Once again, the higher incident energies result in lighter residual systems. However, the ten-fold change in incident energies results in only a 20% change in the residue mass. This is a result of the removal 180 220 MSU—86—448 rIIIIIIIIrIIIIIIIIIIIIIIIII Q) In A ¢rr ll 0 v | 210 24 50 100 209 12C + 197Au E/A=80 MeV 180 llllllllllllllllllllllll IIIIIIIIIIIIIIIIIIIIIII 170 1LllllillllllllllILllllllll 12.5 10 7.5 5 2.5 0 T (MeV) Figure VIII-6 : The time evolution of the 12C + 197Au system at E/A-BO MeV, with four different initial source sizes. The accretion rate used in these calculations was [das/dt]a- 3 nucleons/(fm/c). 181 MSU-86-449 _IIIIIFIIIIIIIIIIIIIIIFIIIIF_ h 220 _ E/A (MeV) = __' ' 100 50 30 10 j I 200 12C + 197Au as(t=0) = 24 160 [daJdt]a=3 (Inn/c)"1 p_s G) O IIITIIIIIIFIIIIII 140 llllllIlLll'lllllllLlIllll 25 20 15 10 5 o T (MeV) Figure VIII-7 : The time evolution of the 12C + 197Au system at four different incident energies, calculated with as(t-O)-24 and [das/dt]a- 3 nucleons/(fm/c). 182 of a great deal of the available kinetic energy by pre-equilibrium emission. If the accretion rate is changed to [das/dt]a- l nucleon/(fm/c), the duration of non-equilibrium emission is extended. The saturation behavior becomes even more evident, as demonstrated in Figure VIII-8, where the trajectories using this slower accretion rate are shown. The emission products are dominated by pre-equilibrium products. VIII.C. Results of the modeli; As demonstrated above, the accretion rate, [daS/dt]a, is an important parameter in this model. While, in general, this would be a time dependent quantity, it is assumed to be a constant in the present calculations. This parameter balances the contributions from equilibrium and pre-equilibrium emission. A calculated quantity which can be used to test whether the chosen parameters achieve the proper balance between post- and pre-equilibrium emission is the momentum transferred to the equilibrated residue. The systematics for momentum transfer [FAT85], as deduced from 12C and 14N induced reactions on 197Au are presented in Figure VIII-9 as solid points. The velocity of the equilibrated system, veq’ as a fraction of the center of mass velocity, vcm, is displayed as a function of the velocity of the projectile above the Coulomb barrier, V. Calculations of the 12C + 197Au system using three different values of the accretion rate ([das/dt]a- l, 3, and S nucleons/(fm/c) ) are shown as solid curves. An accretion rate of [daS/dt]a- 3 nucleons/(fm/c) corresponds approximately to equilibration in the transit time of the projectile across the target. The calculation with [das/dt]a- l/(fm/c) 183 MSU—86—450 IIIIIIFIIIIIIFFIIIIIIIIIIII 220 — E/A (MeV) = — ' 100 50 30 10 I 200 12C + 197Au 160 [dag/dt],=1 (fin/c)"1 ‘ .2 p_a CD 0 I I I I I I I I I I I I I I I I I 140 lllllllllllllllll'llIllllll 25 20 15 10 5 T (MeV) 0 Figure VIII-8 : The time evolution of the 12C + 197Au system at four different incident energies, calculated with as(t-0)-24 and [das/dt]a- 1 nucleons/(fm/c). 184 MSU—86-451 ]_,23 _ r I I I I I I I I I' I I I I I I I.I I I I I I -' : :1£3(:: _+_ :Isaqingtl : 8 : [daa/dtL: g 0.8 E. (fm/c)—1_E O‘ L- .. >0 '_'_ =5 2 0.6 _— _3 -: 2 a 0.4 :— =1 -—_ P I L I I I I I I I I I I I I I I I I I I I I I,I ‘ O 2 4 6 8 1 0 [(E—V)/A]1/2 (Mevl/Z) Figure VIII-9 : The momentum transfer systematics [FAT85] for 12C,laN induced reactions on l97Au are indicated by solid points. The solid curves represent calculations for the 12C + 197Au system using three different accretion rates. 185 is inconsistent with the data. The time to equilibration is too long, allowing for excessive pre-equilibrium emission. The calculations at [daS/dt]a- 3 and 5 /(fm/c) are more consistent with the experimental data. The elemental cross sections predicted by these calculations are shown in Figure VIII-10 for the 12C + 197Au system at E/A- 15 and 30 MeV. The calculations using [das/dt]a- l/(fm/c) are shown as the upper and lower solid curves for E/A- 30 and 15 MeV, respectively, using the same normalization between them, so that apart from the factor of 2 for separating the curves, the energy dependence of the calculated cross sections is preserved. While the 15 MeV curve is generally consistent with the data, the 30 MeV curve falls much too steeply with increasing fragment charge. As a result, the energy dependence for the heavier fragments is not well described. The dashed curves correspond to calculations using the accretion rate [daS/dt]a- 5 /(fm/c) at the two incident energies. The description of both the charge distributions and the energy dependence is improved, though the energy dependence is still not well predicted. The disagreement may be the result of the absence of angular momentum effects in the calculation or of the very simple approximation used to describe the evolution of the system. The normalizations for the case [daS/dt]a- 3 /(fm/c) at E/A=30 MeV is about .5 b. This is approximately a fifth of the total reaction cross section. The calculated multiplicities are low, with only about .03 carbon ions being emitted per reaction. It has been observed in the inclusive data that the cross sections for lighter fragments appear to have stronger non-compound contributions than those of heavier fragments. The pre—equilibrium and equilibrium 186 MSU-86-445 I I I I I I I I 197Au(12c X) o E/A = 15 MeV o E/A = 30 MeV (x2) I IIIIIIII l lllllllI I I IIIIIII l llllllll pa 0 CO I IIIIIIII l lllllllI (pb/sr) pa 0 N I IIIIIIII I llllllll pa O ya I I IIIIII llllllll N Figure VIII-10: The experimental elemental cross sections for 120 carbon induced reactions on 1'97Au at E/A-lS and 30 MeV, shown as solid and open points, respectively, are compared to calculations with accretion rates of [das/dt]a- 1 (solid curve) and 5 (dashed curve) nucleons/(fm/c). 187 charge distributions are compared in Figure VIII-ll. The equilibrium distribution is taken as the sum of the entire emission after equilibrium is reached and is shown as a solid curve. The dashed curve represents the mass spectrum of particles emitted while the source is moving faster than 0.15 of the beam velocity, i.e. during the early pre-equilibrium stages of the reaction. The pre-equilibrium contributions are stronger for lighter fragments. Energy spectra are calculated according to Eq. III-l with a Coulomb width of wx-8 MeV. The emission temperature that is used to calculate the energy spectra includes a contribution from the Fermi velocities of nucleons which make up the source. While the sum of the Fermi momenta of all the target nucleons is zero, the sum of the momenta of any subset, e.g. the nucleons accreted by the source, is not. As a result, the effective temperature, Teff’ is given by k2 ast(a - a t) 4 T - T + -—E a , (VIII-18) eff 2m i 2 0 (a-l) as where ast is the number of spectator nucleons accreted by the source, kF is the Fermi momentum, and m0 is the rest mass of the nucleon. Some fragments are emitted in unstable states which decay by particle emission. While some nuclei decay by emitting more than one particle in sequence, it is assumed here that the decay terminates after the first particle is emitted. The energy spectra for these fragments are calculated by generating the energy spectrum of the primary fragment according to Eq. III-l and transforming this spectrum to that of the detected fragment by 188 MSU—86—446 5 ' l ' I ' l I T 1 O :— 197Au(12C,X) ‘E E o E/A = 30 MeV E I — ‘ Equilibrium I A 104 =_ - - - Preequilibrium _= I-I E E U) - - E : 2 3 3 ' ' A 10 C : : Io .. _ E _ - "d 2 V 10 I I lllllq I I I IIIIII I IIIIIII ’ I lllllll N .p C) CD I---- 0 Figure VIII-ll} The experimental elemental cross sections from 120 + 197Au reactions at E/A-3O MeV, shown as open points, are compared to the yield curves for the pre-equilibrium (dashed curve) and equilibrium (solid curve) stages of the reaction. 189 a a. dN dN p _ 1 dB dE ( a. ) and E Ep( a ) ’ p 1 p (VIII-l9) where ap and Ep are the mass and energy of the primary fragment. The effect of this transformation is to lower the fragment energies and thus make the energy spectra steeper. The energy spectra for carbon nuclei 12C + 197Au reactions at E/A- 30 MeV measured at 9- 30°, 50°, 70°, and 120° are shown in Figure VIII- 12 as solid points. Several important features of the energy spectra are reproduced by the calculations. The calculated differential cross sections exhibit a forward peaking in the laboratory. The energy spectra are characterized by approximately exponential slopes towards higher energies, which become steeper with increasing laboratory angle. The cross sections are somewhat over predicted for the 50° and 70° spectra. In addition, the calculated position of the peak in the distributions are at larger energies than in the observed spectra. When the spectra are decomposed into the contributions from different stages of the collision, it is observed that different laboratory angles are sensitive to different stages of the reactions. Such a partial decomposition of the energy spectra of 12C fragments calculated at laboratory angles of 0-30° and 90° is shown in Figure VIII-l3. At 0-30°, the pre-equilibrium contribution (vs> 0.15 v0), shown as a dotted curve, are dominant, particularly at high fragment energies. The equilibrium contribution, shown as a dashed curve, is a small fraction of the total spectrum, shown as a solid curve, and contributes only at low energies. At 0-90°, the equilibrium and 190 MSU—85 ~452 4 _rI I I II I I I I I I I I I I I I I I I I I I II I-I—I lc) E_- IJy7JXIJI:12%::Iu2(:) g 3 3 E/A = 30 MeV __‘ 2: 1° -' I: E o “a 30°(X30) : Q) :3 ___ ‘.¢» "" -= 2 10 E . ’9. g E : ' + 3 _ ll _ :i 1 'l . ++ _ V 10 :=:_ o I I g c E ; 9 I” I III 50°(x10) E a 100 ‘.-—' + ‘1 Po I I \ : I I. .. b P o .. NU 10-1 :_. 70 (X3) —5 E 1200 3 10-2 I I I I I I I I I I I I I I I I I I I I I I I I l I 0 50 100 150 200 250 E (MeV) Figure VIII~l2z The energy spectra measured in 1'2C + 197Au reactions at E/AF3O MeV at 0-30°, 50°, 70°, and 120°, shown as solid points, are compared to the calculation described in the text. 191 MSU—86—453 103 I-IIII IIII IIII IIII III==IIII IIII IIII IIII IIII IIII: 5 .l I I I z: I, 1 I f I 3 3 120 + Au ff — Total 3 - E/A=30 MeV -- ....... Preequilbrium .- 102 :— [dan/ dt1a=5/ (fm/ 0) 15" - " - Ethbnum ‘3 3 a=3O° 3: a=90° 3 101 .=.— '35“ 1 'U : :: : l—II .. .n. _ m - 4.. .. 0H _ __ _ >~ _ _.__ - 100 =— \\ -EE- ‘—E E l \ :: : I E \ 2: Z - I: I -— d .. r' \ .. - \ E 55 \ I I 2.”. \ I: - -II— ‘ - 10-2 llllIIlllIllllIllllIlLl llllIllllIIllllIlIllIllIl'IIlIl O 50 100 150 200 O 25 50 75 100 125 150 E (MeV) Figure VIII-l3: The calculated energy spectra for 120 + 197Au reactions at E/A-3O MeV at 9-30° (left) and 90° (right), shown as solid curves, are decomposed into a pre—equilibrium component (dotted curve) and an equilibrium component (dashed curve). 192 pre-equilibrium contributions are comparable, but both are dominated by contributions during the late stages of equilibration. This kinematic effect allows for a certain selectivity on the part of the experimenter as to the stage of the collision to be studied. A decomposition of the energy spectra into the contributions from different primary fragments is shown in Figure VIII-14, for a spectrum calculated for 6-30°. The total distribution is shown as the upper solid curve. The lower solid curve represents the primary spectrum of particle stable fragments. Contributions from the n, p, and a decays of primary 130’ 13N, and 16 0 fragments are represented by the dotted, dot-dashed, and dashed curves, respectively. As observed in Section VI.A.2., the sequential decay of primary fragments ought to lower the energy of the maximum in the energy spectrum. The calculations shown in Figure VIII-12 do not show the maxima in the observed locations. However, the spectra in Figure VIII-14 do show that the n and a decay contributions do exhibit maxima at energies below the maxima for the stable fragments. However, because the unstable primary nuclei are produced predominantly during the hot, pre-equilibrium stages of the reaction, their contributions to the spectra have a kinematic boost to higher energies, obscuring the lowering of the Coulomb barrier. In addition, the full touching spheres Coulomb barrier was used in the calculation, which is an extreme assumption, and effects from the recoil of the fragment may need incorporation into the calculation. The differences in the slopes between the spectra are largely the result of the transformation of Eq. VIII-l9, as well as differences in the time dependence of the emission rates . 193 MSU—86—454 —.I I I I I I I I II I I' I I I I I I I I I’I I If I I I I_l 197Au(12C,12C), E/A=30 MeV —— all / stable I llllllll 1 lllIIllI I I I IIIIII I I I I IIIII I l I IIIIII Relative Yield I llllllll 1 IIIIIIII ].()""1. I I l I I l I I l I I l l I I I I l l Ink I I l I, I l l 0 50 100 150 200 250 E (MeV) Figure VIII-14: The calculated energy spectrum for 120 + 197Au reactions at E/A-3O MeV at 0-30°, corresponding to the upper solid curve, is decomposed into the contributions from stable primaries (lower solid curve), neutron-unstable primary fragments (dotted curve), proton- unstable primary fragments (dot-dashed curve), and alpha-unstable primary fragments (dashed curve). 194 VIII.D, Summary ; The object of these calculations was to elucidate certain aspects of the transport phenomenon reflected in the competition between pre-equilibrium and equilibrium emission. The results of the calculations demonstrate the features expected from the decay of a localized region of excitation produced in a nuclear collision which evolves towards the equilibrated fusion-like system. It has been shown that the time scales for equilibration and statistical emission are comparable, and that many features of the data can result from this fact. The phenomena of incomplete momentum transfer, non-equilibrium emission, the evolution towards equilibrium-like emission for heavy fragments and fragments detected at backward angle, the energy and mass dependence of the fragment cross sections, and the kinematic effects of sequential decay are all qualitatively described in this framework. The model may be extended by improving the descriptions of the dynamics of the collision process and of the emission mechanism itself. There is no rotation or angular momentum in the model, while the evidence is clear that these are important aspects of fragment emission (see Chapter IV). The linear evolution with time assumed here is much too simple to mimic what is probably a very complex evolution. The assumption of a hot source surrounded by a cold nucleus is also an extreme assumption. Because of the relation between light particle and fragment distributions discussed in Chapter IV, models calculating the time evolution of the one-body distributions in heavy ion collisions may yield better descriptions of the colliding systems. These may be incorporated in parametrized form into the present calculations. 195 In addition, the formulation for the instantaneous emission rate makes no allowance for in-medium, quantum mechanical, or geometrical and dynamical effects in the emission process. A more complete model for the emission mechanism may replace the present statistical ansatz. Chapter IX Summary and Conclusion IX.A, Summary of present results : The study of the emission of intermediate mass fragments promises to illuminate the more general question of how excited nuclear systems form and decay. The object of the present study was the characterization of relevant single particle inclusive and two-particle coincidence data at incident energies between E/A-lS and 100 MeV and, from this, the identification of the statistical and dynamical aspects of nuclear reactions at intermediate energies. The single particle inclusive cross sections were presented for four systems : 120 + Ag at E/A-15,30 MeV, 12C + 197Au at E/A-lS,30 MeV, 14N + Ag at E/A-3S MeV, and 32$ + Ag at E/A-22.5 MeV. The energy spectra appear to be phase space dominated, showing no structure apart from a peaking at Coulombic energies and an exponential slope towards increasing energies. Though the spectra seem statistical in nature, a two source parametrization demonstrated that a significant fraction of the fragment cross sections can be described as "non-equilibrium", as 196 197 one component of the spectra has an average velocity larger than the center of mass velocity. The charge distributions are characterized by cross sections which decrease with increasing fragment charge. While the slope of the mass distributions have been offered as evidence of a critical phenomenon associated with the liquid-gas phase transition, the distributions from different systems differ in their slopes, and do not indicate a single critical parameter. Instead they indicate a projectile dependence of the emission rates, perhaps through a sensitivity to the entrance channel angular momentum. The isotopic distributions are peaked near the valley of 5 stability. They were described quite well by a Boltzman factor with the binding energies of the fragments and a modest temperature of Tzh MeV. Using the 32S + Ag system at E/A-22.5 MeV, the distributions of light particles detected in coincidence with intermediate mass fragments were investigated. The energy spectra of light particles detected in coincidence with intermediate mass fragments do not differ greatly from the inclusive cross sections. This observation is consistent with uncorrelated emission. The coincident light particles are, however, strongly correlated with the entrance channel reaction plane defined by the intermediate mass fragment and the beam axis. Futhermore, the emission pattern exhibits no distinct preference for the same or the opposite side of the beam axis as the coincident fragment. These angular correlations indicate that dynamical effects are strong and that the nuclear mean field remains a dominant factor in evolution of composite systems formed at intermediate bombarding energies. The associated multiplicities of light particles and intermediate mass fragments was also investigated with the 328 + Ag system. 198 Intermediate mass fragments were found to be associated with the emission of approximately 10 nucleons, in the form of light particles, emitted at approximately half of the beam velocity. These particles carry away a significant fraction of the projectile momentum (zO.2PO) and energy (zl6O MeV out of 720 MeV), resulting in less than full momentum and energy transfer to the equilibrated fusion-like system. Intermediate mass fragments were found to be emitted with low multiplicities, with a total multiplicity for charges Z-3-24 of approximately one. The associated multiplicities show little dependence on other emission and indicate that fragment emission is a stochastic, random process The study of the velocity distributions of target residues in the 328 + Ag and the 14N + Ag systems provided information on the momentum and energy balance in these reactions. The distributions confirm that, in the events which dominate the production of intermediate mass fragments in the forward hemisphere, the momentum transfer to the equilibrated system is incomplete, and the fragment multiplicities are low. An indirect measurement of the undetected momentum and mass indicated that intermediate mass fragments are emitted in highly damped collisions, with 200-400 MeV of excitation in intrinsic degrees of freedom. Nonetheless, a substantial fraction of the projectile momentum is lost to pre-equilibrium emission of nuclei with an average velocity in the beam direction. Finally, the dependence of the residue angular distributions on the momentum of the coincident fragment indicate that emission of fragments at energies below the Coulomb barrier might be associated with additional particles emitted in the direction of the fragment. This can be interpreted as a result of a sizable contribution 199 from the sequential decay of a excited primary fragment into the detected intermediate mass fragment and one or more light particles. While the experimental data provide many indications of non-equilibrium and dynamical effects, they also indicate that the emission mechanism is phase space dominated, and thus may be approximated by statistical methods. A comparison of data with a model calculation of statistical emission of intermediate mass fragments in both their ground and excited states suggests that fragments are emitted in excited states. The decay of particle unstable states produces characteristic structures in the fragment mass spectra and influences the isotopic distributions as well. The relative populations of excited states are also strongly influenced by sequential decay. Because the results of the calculations are sensitive to the precise nature of the branching ratios during sequential decay, the present calculations cannot determine to what extent the initial populations of the states can be approximated by purely statistical distributions among the asymptotic states. A model calculation which incorporates statistical emission with a parametrization of the dynamical evolution of two colliding nuclei was presented. The time scale for statistical emission is found to be comparable to the equilibration time of the composite system, resulting in significant emission prior to equilibration. The model qualitatively reproduces certain features of the data, including non-compound emission, incomplete momentum transfer, the elemental yields, the energy dependence of the cross sections, and certain features of the energy spectra. 200 IX,B, Concluding remarks ; In spite of considerable progress in outlining the phenomenon of intermediate mass fragment emission, there is much room for contributions to the field, both experimental and theoretical. Theoretical studies must concentrate on merging improved descriptions of the dynamical evolution of nuclear collisions and of the mechanism of fragment emission. Solutions to non-equilibrium transport equations promise to elucidate the dynamical features of the reactions. These calculate the one-body phase space densities during the collision. Since the coincidence data suggest that the collision dynamics affect intermediate mass fragment emission in much the same way as they affect nucleon emission, perhaps such models provide a satisfactory explanation of the collision dynamics. Models of fragment emission may have to incorporate a dynamical description of the process. Fragments are, in a sense, complicated multiparticle correlations which must arise from the chaotic nucleon distributions in the emitting systems. At present the emission mechanism can only be approximated with statistical methods, and the new features which might reside in a microscopic understanding of the model remain the object of future study. Progress in theory may be tied to additional experimental clues. Systematic studies of the target, projectile, and energy dependence of the inclusive and coincidence data are needed. Manifold coincidence studies, provided by large detector arrays, will provide a more precise picture of the dynamics of reactions producing fragments in both the forward and backward hemispheres. Studies of particle correlations at 201 small relative momenta promise to provide information on the spatial and temporal character of the emitting regions, in addition to information on the population of excited states. These studies may help investigators to understand the influence of impact parameter, fluctuations, temperature, collective motions, nuclear shapes, in-medium .effects, and other statistical and dynamical aspects of nuclear collisions. APPENDICES Appendix A Kinematic bias in target velocities Systematic errors in extracting information from the residue velocities can result from a biased sampling of the residue velocity distribution by a detector with a small angular acceptance. Consider a distribution in 3 characterized by a mean velocity 30 such that 3 - 30+ 3'. For simplicity, we assume that the distribution of 3', £(3'), depends only on v' so that f(v')-f(v'). By definition the average velocity for all particles is -9 -v < v > - v . (A-l) However, for a finite detector solid angle in the laboratory, 0, the measured average is given by: In 3 f(v')dv'dfl’ + fh 3 f(v')dv'd0' < 3 > - I " (A-2) Q ffl+f(v')dv'dfl' + f0_f(v')dv'd0' ’ where 0+ and 0- denote the solid angles corresponding to the two 202 203 _p kinematic solutions for values of v'. In general, the measured average 0 O 4 is different from v0. As an illustration, we assume f(v') - 6(v'-a). If the detector is A centered about v0 and detects all particles emitted inside a cone of half angle a, we can calculate the average velocity of the detected particles - l /8v a x (A'3) 0 3 O 2 2 2 2 V + a v + a 0 + cosfl 3/2|0-'0+ + ‘Q————— + cosfi 3/zlo-W 2-vo 0-0 2-v0a 0-0_ (c030+ -l) - (cosfl_+l) ’ with v v 9+ - cos.1{";Q sinza i cosa-[l-(—:Q)2 sinza ]1/2}. (A-A) Here, 0+ and 0_ represent the angular limits of the solid angles of 0', 0+ and Q-. For a=0.25v0, the average emission angle in the laboratory is approximately 16°, similar to the one of the experimental distributions. For an acceptance angle of a-6°, the measured average velocity is z 1.1v0. APPENDIX B Gamma ray efficiency calibration The absolute efficiency, 6:, of a detector, i, at an energy, a, is determined from the coincidence rates with other detectors, j, which observe a coincident gamma ray of energy, b by the formula ab bb a b ab 11 J (B-l) 2213’ Ab j J j where Jij represents rate for the coincident detection of gamma ray a in detector 1 with gamma ray b in detector j; K: is the single particle detection rate for gamma ray b in detector i; A? is a term which corrects the efficiency for branching ratios and backgrounds; and Nib is the weighting determined by the angular correlation between the coincident gamma rays. The first term in the numerator is the measured coincident rate for the two gamma rays. The second term is a correction for random coincidences. This correction is calculated from the rate of random 204 205 coincidences between two gamma rays of the same energy, corrected for the detection rates of the two gamma rays. The difference between these two terms is the estimated real coincidence rates. The term A? is the product of two contributions. The first is the correction for background counts which would otherwise be included in the singles rate, K?’ The second factor, B, is the correction resulting from the specific branching in the decay scheme of the source. It is equal to the average number of gamma rays, a, which are emitted from the nucleus in coincidence with gamma ray, b. Both numbers, and thus A, are less than one. The term WI? -l/w(a,b;fi) where w is the angular correlation between the two gamma rays and fl is the angle between the two gamma rays. This correlation is given by [GIL75] L +L' com-2 P (cost?) 2 H) 1 1 6 6 , RK 2’ 2 where PK is the kgh Legendre polynomial, L1 is the lowest multipole of the ith transition, L1 is the next highest multipole, and 6L is the mixing ratio defined by Zill'lfil'ii . (3-3) 1' o ' 2 and L0 is the lowest multipole. The angular momentum algebra is found in 206 ' _ _ 1+J1-J2+L'-L-K “ “ “' RK(L1,L2,J2,J1) ( 1) J1 L L x (L L l -l | K 0) . W(J1 J1 L L | K J2), (B-h) where W is the Racah coefficient [RAC42] and L- (2L+l)1/2. In practice, the angular correlations are tabulated [TAY71] in the form 00(fi)‘ E A2kP2k(COSB)’ (B-S) where the A2k are tabulated coefficients. The efficiency calibration in the present experiment was executed with measurements with three sources, 88Y, 60Co, and 75Se. The characteristics of sources are summarized in Table B-1. 207 Table B-1 : The properties of the sources used to provide absolute efficiency calibrations. The gamma ray source is given and the nucleus which emits the gamma ray is given in parenthesis, E is the transition energy, J: is the initial and J; is the final spin and parity of the transition, L is the multipole, 6 is the multipole mixing ratio, B is the number of partner gamma rays emitted in coincidence with each transition, and A2 and A4 are the angular correlation coefficients for the transition pairs. Source E Ji Jf L B A2 A4 (keV) 60Co (6ONi) 1173 4* 2* E2 1. .10204 .00907 1333 2+ + E2 88y (88Sr) 898 3' 2* E1 1 - 07143 0 1836 2* 0* E2 92 7SSe (75As) 136 3* %' E1 98 -.0328 o 264 g‘ %' M1,E2 .95 (6--.045) 1+ 2- ‘ 121 2 2 £1 1 o -.413 o i“ i’ 279 2 2 M1,E2 8S Appendix C Jacobian for relativistic transformation The efficiency for detecting a gamma ray emitted from a nucleus which is moving in the laboratory inertial frame must be corrected for the relativistic transformation of the detector solid angles. The Jacobian of the transformation for the detection efficiencies is given by the ratio of the solid angle in the laboratory inertial frames, 0, and the solid angle in the inertial frame of the emitting nucleus: [J]- d0 _ d cosQ * * ' (C-l) d c050 where 0 and 0*are the angles between the directions of the gamma ray in the two inertial frames and the axis defined by the velocity of the emitting system, B, and 0 and 0* are the solid angles subtended by the detectors in the two frames. For photons, p-E, and the Lorentz transformation [HAG73] between the two frames is pTI- 7 ( pII- fiE ) and (C-2) 208 209 * E - 1 ( E - fipll)’ (C-3) where 12- (l-fiz) and pll - E-cosfl is the component of the momentum parallel to the velocity fl. Thus * * _ p|| _ c030 - fl * c050 E 1_ Bcosfi and (C'4) d cosQ (l-Bcosfi)2 (l-QcosQ22 ‘ (l-pcoso) + fl(cosfi-fi) ‘ (C-S) <1-82> * d cosfl so that |J|- 72 (1 - 8 cosO)2. (c-s) Appendix D Sequential decay In the schematic statistical model described in Chapter VII, nuclei are assumed to be emitted in particle unstable states. These nuclei are then assumed to decay statistically by the emission of a light particle, n,p,d,t, or a. Each nucleus is allowed to decay only by the particle decay channels listed in [AZJ86]. Thus, if proton and deuteron decay are specified, then only decay by these channels is calculated. If no decay channels are listed for a given state, then all energetically allowed decays are calculated. The relative decay rates to final states are calculated for all allowed decay channels according to the statistical model [HAUS2] [ST084]. The initial state |JM> decays to a final state |jm>|sms>|£m£>, where J is the spin of the parent with projection M, j is the spin of the daughter with projection m, s is the spin of the emitted light particle with projection ms, and £ is the orbital angular momentum of emission with projection mg. The relative branch to each final state is given by 210 211 r(c,J,j,s) a 2 z T£(E) s 2 x 2 ||2 (D-l) where E is the Q value of the decay and S is the channel spin resulting from the coupling of j and 5. Eq. D-l can be rewritten as r(e,J,j,s) a 2 2 T£(E) X 2 ||2 S 2 s 3 ms,m x 2 ||2 (D-2) m 2 By orthogonality . 2 2 || - 1 (D-3) m ,m s z ||2 - 1 (D-4) m 2 With the restrictions on S and £, the final relation is j+s J+S r(e,J,j,s) a E: E: T2(E) . (D-S) S“ll-SI £=lJ-S| Thus, all of the angular momentum dependence of the branching ratio is contained in the two summations. The transmission coefficients, T3, can be calculated in the WKB approximation [HAH86] for emission through a barrier to be 212 T£- x-exp(-2w), (D-6) where 2 V+V-E x _ 23 + (1+,5) _ 1 _ ( C L )1/2, (D-7) kR k2R2 E is a measure of the barrier from the Coulomb potential, V and the CI effective centrifugal potential, VL; in addition, 2 2 2 1/2 _ g_ gc 1/2 _ ( 29c E ) _ a 2m 2122 he ( 2E ) , k “C , (D 8) R - (1.3 +1.25 A1/3) fm , and (0-9) -2w - (2er + "[n + 25in-l( 2 n - kR 2 1/2)] ( n + (£+.5) ) _(2,+1).1n x(£+.5) + n + (2+.5)2/(k2R2) (D_10) 2 2)1/2 (n + (3 + .5) In these expressions, E is the transition energy, Z1 and 22 are the charges of the daughter nuclei, A is the radius of the parent, p is the reduced mass of the daughter nuclei, and fl is the partial wave of emission. Transmission coefficients calculated for partial waves, £=0-8, as a function of emission energy are shown in Figure D-l for three decay channels of 12C : n (bottom), p (middle), and a (top). The plots are truncated where the transition energy is greater than the barrier, at which point the transmission coefficients are taken to be 1. 213 MSU-86—457 :rlrr..[1...]....[..fi.r.... ~ on + 8Be (1:0-8) . 10D :— -: lo-Zb‘f%‘:%.::.%::::::.:1l:e::: . p + 1113 (1:0—6) - ICC) f’ '1 ..... -1 _. E—* 10 10‘2 _ I l.: ::l.:: :I::.: l::::: - n + 11c (1:0—5) - 1130 f’ I -? 10-1 E’ -€ 10-2 ll. . .1.. 1 ..|. 0 5 10 15 20 25 Energy (MeV) Figure D-l : Transmission coefficients calculated as a function of 12 ener with the form T -x-exp(-2w) for three deca channels of C : 8y 3 Y n(bottom), p(midd1e), and a(top). The calculations are shown as solid curves for partial waves 2-0 to 8 (from left to right). 214 This formulation has the unfortunate property that the coefficients reach a maximum and then decline with increasing energy, and the coefficients do not approach unity. This can be avoided by dropping the pre-exponential term, so that Tg’ exp(-2w). (D-ll) Transmission coefficients calculated in this manner are shown in Figure D-2. This is the formulation of the transmission coefficients used in the calculations in Chapter VII. Finally, results using a sharp cutoff approximation are investigated. In this approximation 9 2 1 ifx2<0 T2 - _ , (0-12) 10 ifx>0 where x2 is the sum of the Coulomb and the effective centrifugal potential barriers. Figure D-3 depicts the mass distributions calculated with transmission coefficients calculated with Eq. D-6-10 for the case of emission from a nucleus, A-l3l, Z-SA, and T-5 MeV. The histogram represents the distribution using Eq. D-ll. The calculation using Eq. D- 6 is represented by the open circles and using Eq. D-12 by solid squares. The mass distributions show little dependence on the form of the transmission coefficients. 5 The ratios of populations of levels in AHe, Li, 6Li, and 8Be are shown as functions of emission temperature in Figure D-4 for each of 215 MSU—86—458 . l . . if. I . 1 . . l . . . . I . . . . I r . . r4 . a + 8Be (1:0—8) . 100 g- 10‘1 :— 10‘2 :1: 1:: ::I:%. :%:+. 1.: ‘ - p + 1113 (1:0-6) 4 100 :— —a "1 _. E-' 10 E L. 10—2 :::* .: :. :‘l‘1.: 10-1 g- 1()-J3 . . . 1 111 11. . . . . _ 15 20 25 Energy (MeV) Figure D-2 : Transmission coefficients calculated as a function of energy with the form TI-exp(-2w) for three decay channels of 12C : n(bottom), p(midd1e), and a(top). The calculations are shown as solid curves for partial waves 2-0 to 8 (from left to right). 216 2 MSU-86—459 10 IT'V'IIIIITTIIIIIIIIII IIIIIII IIIIII A=131, 2:54, T=5 MeV I I :2 101 ;- -——-T=:eXp(—2w) ‘3 .2 C O T: x-exp(—2w) >" ' 63; I sharp cutoff ‘ CD _ . >’ 0 L3 10 E' 1 £2 E E m - I Cr: r I I H C) | H I 1IIIIIr J ILIIIII I I 10-2 I I l I I I I I I; I I I I I i4] I I I l 0 5 10 15 20 .Ax Figure D-3 : Final mass distributions calculated for the system A-l31, Z-Sh, and T-S MeV using three forms of the transmission coefficients : Tfl- exp(-2w) (histogram), Tz-x-exp(-2w) (open circles), and a sharp cutoff approximation (solid squares). 217 these calculations. The calculation used in the text, using Eq. D-ll, is depicted by the solid line. The dashed line depicts the calculation using Eq. D-12. The dotted curves depict results using Eq. D-6. The comparison suggest that the calculations are not very sensitive to the precise formulation of the transmission coefficients. It would be wrong to surmise from this that the decay calculations are insensitive to changes in the branching ratios. Instead, the effect on the branching ratios resulting by changing the transmission coefficients as above is insufficient to alter the final distributions significantly. Direct alterations in the branching ratios can produce significant effects. In order to demonstrate the possible effects, results of calculations using fixed branching ratios are shown in Figures D-5 and D-6. In these calculations, if the branching ratios are not specified in [AJZ86] then the particle branches are assumed to be equal for the allowed decay channels. The effect of this assumption on the mass distribution is minimal. Figure D-S shows the results using calculated branches as a histogram and the results using fixed branches as solid points. The variations are of the order of 10%. The general features of the mass distribution seem to be rather insensitive to these changes in the branching ratios. The population of low lying bound states relative to the total ground state yield for states measured in the 32$ + Ag reaction are shown as the hashed regions in Figure D-6. The solid curves represent the results of statistical calculations as presented in Figure VII-16, where the branching ratios for the available particle decay channels are calculated. The open points represent calculation with fixed branching ratios. Small differences are noticeable between the calculations for 218 MSU- 86- 460 .f.,...,.,..j,..., , . WWII A=131,Z=54 “(i—01 —)x2 "5L1 35%) ‘ U II'II‘ fir I IIIUIUIQ I I I l I I I IITIII RP/R- : 2: : I l I : I : |- qb I- J I . 2. 9 I8 :8 3.04 3 b Ll~ : [0 . Q) ' - E 100 :— _5 CU : 2 F4 P .. (D : - m d 10‘1 3‘ ‘2 2 o I I . : 1.0—2 l J I I I l I I I I l I I I I I'I I I I I I Figure D-S : A comparison of the mass distributions calculated according to Eq. VII-l using branching ratios calculated from transmission coefficients (histogram) and using branching ratios that are fixed to be equal for all allowed decay channels (solid points) for a system A-l3l and 2-54 at T-S MeV. 220 MSU ~86 ~462 0.8 T .ln..rnrrpfinl..1.45-.1..1...r1..rr]4.fir....<0.8 0.6 __ 1,: (2.621 - 1.674) '1 0.6 — Tl=exp(-2w) I /’ '2 0.2 __ ///. Data 0.2 ()1) E%%%¢% :{e%%:%: .2 0.0 0.4 i—“Li 10.4 0.3 E— “0.3 0.2 ’7 0.2 0.1 I- ‘0.1 ()0: :li' I]:TH%: _ l. 0.0 : 1 p _ j: 13 _ 3 0.15 _ °B (2.154 1.740) ‘__ c (3.354 3.684) 1 0.15 0.10 3- —:— €0.10 . o d .9. .o. . r : 0.0C) I .Il....lI.IJI.IIIlI. ‘ .JIJIIIIIILIII..I.I.II “(3130 O 2 4 6 8 O 2 4 6 8 10 T (MeV) Figure D-6 : A comparison of the fractions, F7, of nuclei in their excited states as calculated according to Eq. VII-l with branching ratios calculated according to statistical rules (solid curves) and branching ratios fixed to be equal for allowed channels for a system of A-l30 and Z-58. The fractions observed in 328 induced reactions on Ag at E/A-22.5 MeV are indicated by the hashed regions. 221 all the state ratios; in the case of 13C, the change in the branching ratios results in a 25% suppression in the yield of the excited state relative to the ground state. This is the result of large differences in the a/n branching ratios in 170 between the two calculations. The other ratios that are not affected here are not necessarily immune to problems with the branching ratios. Thus, while the calculations predict the general magnitude of the effects of sequential decay on these ratios, they cannot be expected to be in full agreement with the data simply as a result of uncertainties such as these. Appendix E Quantum Statistical Calculations The quantum statistical model [ST083] [HAH86] describes the mass distribution in an infinite nuclear system in thermal and chemical equilibrium at a fixed temperature (T), density (p), and isospin (N/Z). The distribution is calculated from the number of available states and their occupation for point like particles of mass, mi, spin degeneracy, gi-28i+1’ where 31 is the spin, and chemical potential, pi, in a volume v - v - 2 n.V. , (E-l) where Vext- (N+Z)/p. This is the total volume of the system less the volume excluded by the real volumes, Vi’ of the nuclei. The number of the ith species of particle, ni, is given by l “i' eXP(-pi/T) - l + (giV/Ai)°FBE(-#1/T) (Ii-2) for bosons and by 222 223 ni - (giV/Ag>(2/«1/2>-FFD -' : 1 a) 10‘3 E‘ E 3 n l :. % 4 '23 102 5' ‘a '5 E T 5M V ° T 10M v E . = e = e . D: 101 :- 1. 5 ° 2 10° :- '3 10'1 é- 1. 10’2 r ‘a 10"3 r 3 5.1. ..1....1....1....1 .l....l....l....|....J.: O 5 10 15 20 0 5 10 15 20 Ax Figure E-l : The primary and final mass distributions for the fragmentation of a A-lBl, Z-Sh system at a density of p/po-.4 and at temperatures, T- 1,3,5, and 10 MeV, are shown as histograms and solid points, respectively. 225 MSU—86-456 2 -1....1....l....T....l. .l....l....l.....l..e.,. 1° A=131, 2:54 —Pr1mary ' T=5 MeV 0 Final 3 101' . . . ’ 'j P/Po =-05 P/Po =3 3 100 1, o . ; 0 I 10‘1 j E : CD 10-2 j ..—1 2 >>* I a) 10-3 . '3 > El I 1 l 1 1 1 1* «F4 e % : § : . L::l:: #l% 1.. I _,_, 102 .r I I l I I 1: a: 101 5’0 P/Po = . P/Po = 9 j i o : :10() E" O I? 10‘1 _r 1. 10_2 r' 1 E a 10-3 :- 1 :.l....l....l....l....l..l....l....l....l....l: O 5 10 15 20 O 5 10 15 20 Ax Figure E-2 : The primary and final mass distributions for the fragmentation of a A-13l, Z-SA system at a temperature of T-S MeV and at densities, p/po- .05, .3, .6, and .9, are shown as histograms and solid points, respectively. 226 p/po=.9, the distribution rises with increasing fragment mass. As the density decreases, the relative yield of lighter fragments increases, and, by p/po-.OS, the distribution falls steeply with increasing fragment mass. At a fixed temperature, the effect of sequential decay on the relative populations of excited states will increase with increasing density. This results simply from the larger populations of the heavier parent nuclei relative to the primary populations of the daughters. At a fixed density, the effect of increasing temperature is mixed. The number of excited states in the primary distribution increases with temperature, but the relative population of the heavier, parent nuclei is diminished. As a result, the quantum statistical calculations of excited state populations are affected more gradually by increases in temperature as compared to results from Eq. VII-l. There are a number of weaknesses in the quantum statistical model. The quantum statistical calculation is a nuclear matter calculation and neglects the effects of the nuclear surface and the long range Coulomb interaction. The energy spectra of fragments produced at lower incident energies, E/AS 100