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MICHIGAN GANSTATEU I IIIIIIII IIIIIIIIII IIIIII IIIIIIII L 00916 3241 II This is to certify that the dissertation entitled THE EMISSION TEMPERATURE AND THE NUCLEAR EQUATION OF STATE presented by Hongming Xu has been accepted towards fulfillment of the requirements for PhoDo degreein PhYSiCS Wééav’é Major professor Date Feb. 19, 1991 MSU i: an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY Michigan State, University PLACE IN RETURN BOX to remove We checkout from your record. TO AVOID FINES return on or before due due. DATE DUE DATE DUE DATE DUE I—T I MSU I: An Affirmative Action/Equal Opportunity Institution CWMa-ni THE EMISSION TEMPERATURE AND THE NUCLEAR EQUATION OF STATE By Hongming Xu 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 1991 ABSTRACT THE EMISSION TEMPERATURE AND THE NUCLEAR EQUATION OF STATE BY Hongming Xu The relative populations of a large number of particle stable states of intermediate mass fragments were measured with the Oak Ridge Spin Spectrometer for 32S induced reactions on "“‘Ag at an incident energy of E/A=22.3 MeV The measured relative populations of these states were compared to those calculated from a thermal model which include sequential feeding from higher lying particle unstable states of heavier nuclei. This comparison indicated an average emission temperature of T2 3 —4 MeV. To study whether emission temperatures can provide information about the nu- clear equation of state and the in-medium nucleon-nucleon cross section, dynami- cal calculations based on the Boltzmann-Uehling-Uhlenbeck (BUU) equation were performed. Calculations for 4"Ca+‘"’Ca and ‘°Ar+27Al collisions indicate the cross sections for heavy residues are rather sensitive to both the equation of state at sub- nuclear density and the in-medium nucleon-nucleon cross section. This dual sensi- tivity may be reduced or eliminated by measurements of the emission pattern of the coincident light particles. Excitation energies and total angular momenta were also calculated for the residues formed in ”Ar + 2"Al collisions. These calculations sug- i gest that reaction dynamics, not Coulomb or thermal instabilities, plays the most important role in limiting the production of fusionlike residues at energies E/ A z 30 MeV. From the emission rates of nucleons and the thermal excitation energies of heavy residues produced in ”Ar +27Al and “Ar +‘2‘Sn collisions, consistent thermal freeze- out times were obtained. The total excitation energies and temperatures predicted by BUU calculations are comparable with those obtained from experiments. These . predicted values for the excitation energies and temperatures are quite sensitive to the equation of state and the impact parameter. Surprisingly, These two observables show little sensitivity to the in—medium nucleon-nucleon cross section. ii To my Parents M eihua and Shujing iii ACKNOWLEDGMENTS I am very happy to express my appreciation to my dissertation advisor, Prof. Bill Lynch, for his guidance, encouragement, support, and numerous discussions through- out the course of this dissertation study. His scientific belief and his persistent efforts constitute the essential part of my dissertation, and his knowledge, ability, and stan- dards as a scientific researcher (both experimental and theoretical) have earned my great respect. Hearty thanks go to Prof. Konrad Gelbke for his advice, suggestions, continued interest and valuable contributions to my dissertation. His understanding of people and profession as a researcher have left a deep impression on me. I am very grateful to Dr. Betty Tsang for her advice, friendship and technical help, particularly during the initial stages of my data analysis. I would like to thank Prof. Pawel Danielewicz and Prof. George Bertsch for many fruitful discussions and encouragement during my theoretical project that forms the second half of my dissertation. I am also indebted to Prof. Mitsuru Tohyama for helpful discussions. It is my great pleasure to thank my fellow graduate students at the cyclotron and the physics department, Pi Bo, Young Cai, Ziping Chen, Yibing Fan, Wenguang Gong, Baoan Li, Yeong Duk Kim, Tapan N ayak, Xun Yang, and Fan Zhu, and to Drs. Nelson Carlin, David Fields, Dave Mikolas, Tetsuya Murakami, and Romualdo deSouza, for their friendship and enjoyable discussions. iv I would like to acknowledge the excellent support of the National Superconducting Cyclotron Laboratory and the physics department of Michigan State University. I am particularly grateful to Prof. J .S. Kovacs for bringing me to MSU through the CUSPEA program, and for handling many necessary documents. Special thanks also go to Jackie Bartlett, Rilla McHarris, Babara Pollack, at the N SCL; and Stephanie Holland, at the physics department, for their kind and valuable help. My disserta- tion committee members, Profs. Bill Lynch, Maris Abolins, George Bertsch, Pawel Danielewicz, Konrad Gelbke, and Mike Thorpe deserve recognition for their services. Finally, I would like to acknowledge the understanding, support and love of my parents, Meihua and Shujing, who played a special role in raising me up to the point where I was able to select my career. Special thanks also go to my brother and sister, Goumingand Weihong, for their love and friendship. Contents LIST OF TABLES LIST OF FIGURES 1 Introduction I Motivation ................................. A Complex Fragment Emission and the Emission Temperature . B Disappearance of Fusionlike Residues .............. II Organization ............................... 2 Experimental Details I Experimental setup ............................ II Background Subtractions ......................... III 7-ray calibrations ............................. A Line Shape Calibration ...................... B Absolute Efficiency ........................ C Coincidence Summing ...................... 3 Data vi xi 10 13 13 15 26 26 27 29 33 I Single fragment Inclusive Cross Sections ................ II 7-Ray Spectra From Decaying Fragments ............... A 7-Ray Spectra From Germanium Detectors ........... B 7-Ray Spectra from N aI Detectors ............... Sequential Feeding and the Emission Temperature I Feeding from Higher Lying States .................... A Levels and Level Densities .................... B Primary Populations ....................... C The Decay Branching Ratios ................... II Elemental and Isotopic Yields ...................... III Mean Emission Temperatures ...................... IV Summary and Conclusions ........................ BUU Equation in the Lattice Hamiltonian Approximation I The Formalism .............................. A The BUU equation ........................ B The Lattice Hamiltonian Method ................ II Numerical Realizations of the BUU with the LHM ........... A Initialization ............................ B Density evaluation ........................ C Equation of Motion ........................ D Two-Body Collisions ....................... vii 36 36 56 57 57 61 62 64 67 74 79 80 84 85 87 91 III E Pauli-blocking ........................... 92 Ground State Stability and Conservation of Energy .......... 93 The Disappearance of Fusion-Like Processes and the Nuclear Equa- tion of State 98 I Fusionlike Cross Sections and the Equation of State .......... 99 A 40Ca+4°Ca Collisions at E/ A =40 MeV ............. 99 II Entrance Channel Effects and the Formation of Hot Nuclei ...... 112 A Decomposition of the Excitation Energy ............ 112 B Freezeout Conditions ....................... 117 C Collisions at E/A=30 MeV .................... 124 D Limiting Angular Momenta ................... 127 E Limiting Excitation Energy ................... 131 III Conclusions ................................ 133 Nuclear Temperature and Nuclear Equationof State 137 . I Freezeout Conditions ........................... 138 II The Excitation Energy at Freezeout . . . .i ............... 148 A Excitation Energies ........................ 148 B Massive Transfer Models ..................... 159 III Nuclear temperatures of the Residues .................. 163 A Formalism ............................. 163 B Sensitivity of Fusionlike Cross Sections to The Equation of State103 viii B Results ............................... 164 IV Summary ................................. 170 8 Conclusion A 173 APPENDICES 176 A Correction of Finite Statistics to the Collective Excitation Energy 176 I The Goldhaber’s Problem ........................ 176 II Correction of Finite Statistics to the Collective Excitation Energy . . 177 A Thomas-Fermi Approximation .................. 179 B Local Momentum Analysis .................... 180 B Massive Transfer Model ‘ 182 I Complete Fusion ............................. 182 II Incomplete Fusion ............................. 183 LIST OF REFERENCES 184 ix List of Tables 3.1 3.2 3.2 6.1 6.2 6.3 Parameters used for the fits of the inclusive cross sections. ...... 35 Extracted fractions, F,, of observed fragments which were accompanied by the designated 7-ray transition. ................... 54 (continued) ................................ 55 The critical parameters for fusionlike reactions in 40Ca.+“°Ca collisions. 107 Parameters used for the isoscalar nuclear Mean Field ......... 108 The critical parameters for fusionlike reactions in 40Ar+27Al collisions. 111 List of Figures 1.1 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Differential cross section of fission products as functions of both the incident energy and the folding angle between the two fission fragments for 40Ar-l-232Th collisions .......................... Relative time spectrum between a particle detector ( at GIMF = 20°) and a NaI(Tl) 7-ray detector (at 0., = 138°). .............. Coincidence and background spectra for 11B fragments ......... Coincidence and background spectra for 11C fragments ......... Coincidence and background spectra for 12C fragments ......... Background subtracted coincidence yield attributed to 7-ray decays of excited 12C fragments and corrections due to coincidence summing at high 7-ray multiplicity ........................... Background subtracted coincidence yield attributed to 7-ray decays of excited llB fragments and contributions from individual transitions. . Background subtracted coincidence yield attributed to 7-ray decays of excited 11C fragments and contributions from individual transitions. . Calibrations for the 7-ray response function .............. Measured values of the normalization function 17.,(Eo) ......... xi 16 18 19 20 23 24 25 32 3.1 Inclusive differential cross sections for lithium and beryllium isotopes. 3.2 Inclusive differential cross sections for boron isotopes. ......... 3.3 Inclusive differential cross sections for carbon isotopes. ........ 3.4 Inclusive differential cross sections for nitrogen isotopes ......... 3.5 Inclusive differential cross sections for oxygen isotopes. ........ 3.6 Spectra of 7-rays detected in coincidence with isotopes of 8Li, 7Be, 10B, 12B, and 13C from Germanium detectors ................ 3.7 7-ray spectra measured in coincidence with loBe fragments ....... 3.8 7-ray spectra measured in coincidence with 12B fragments. ...... 3.9 7-ray spectra measured in coincidence with 13C fragments. ...... 3.10 7-ray spectra measured in coincidence with MC fragments. ...... 3.11 7-ray spectra measured in coincidence with 14N fragments. ...... 3.12 7-ray spectra measured in coincidence with 15N fragments. ...... 3.13 7-ray spectra measured in coincidence with 160 fragments. ...... 3.14 7-ray spectra measured in coincidence with 180 fragments. ...... 4.1 The level density of 20N e as a function of excitation energy ....... 4.2 Element yields summed over all measured energies and angles. . . . . 4.3 Isotope yields summed over all measured energies and angles. 4.4 Comparison of the calculated 7-ray fractions, F,, with measured ones: 4.5 Comparison of the calculated 7-ray fractions, F1, with measured ones: 11. ..................................... xii 37 38 40 41 42 46 47 48 49 50 51 52 53 60 65 66 4.6 4.7 4.8 5.1 5.2 5.3 5.4 5.5 6.1 6.2 6.3 6.4 Comparison of the calculated 7-ray ratios, R” with measured ones. 71 Results of the least squares analysis of 7-ray fractions F.7 and ratios 51,. 73 Summary of the emission temperature extracted from recent experiments. 77 The density distributions as functions of the radius for 40Ca and 124Sn nuclei. ................................... The binding energy per nucleon for the mass range 30 S A S 200 initialized at the beginning of the BUU calculations for both the stiff and the soft equations of state. ..................... The time evolution of “Ca and 124Sn ground state nuclei projected in the a: — 2 plane in step of 40 fm/c. ................... Stability tests and the conservation of energy for the ground states of. ”Ca and 12“Sn nuclei. .......................... The step size dependence of the conservation of the total energy for 4"Ca+‘“’Ca collisions with the soft equation of state. ......... The time evolution of test particles for 4°Ca+40Ca collisions at E/A=40 MeV and b=2 fm with the stiff E08 and am, = 41 mb projected in the 1: — 2 plane in step of 20 fm/c ....................... The time evolution of test particles for 40Ca+4°Ca collisions at E / A=40 MeV and b=2 fm with the soft E08 and 0..., = 41 mb projected in the x — 2 plane in step of 20 fm/c ....................... Observables calculated for the 40Ca+ “Ca system at E/A=40 MeV assuming am. = 41 mb. ......................... Residue cross sections for 40Ca+ “Ca and 40Ar+27Al collisions. xiii 88 89 95 96 97 100 101 Chapter 1 Introduction I Motivation N ucleus-nucleus collisions have proven to be an excellent laboratory for the study of statistical and dynamical properties of highly excited nuclear systems. The properties of such systems evolve with incident energy. At incident energies of a few MeV above the Coulomb barrier, the formation of a fully equilibrated compound system (commonly referred to as ‘complete fusion’) and its subsequent statistical decay is the dominant process for central collisions [More 72, More 75, Frie 83, Sobo 83, Sobo 84]. The statistical decay by emissions of 7, neutrons, and light charged particles as well as fission has been well described-by statistical models of compound nuclear decay. At incident energies above E/ A “at: 15 MeV, however, the situation becomes more complicated. First, complete fusion of projectile and target becomes less likely, and one observes the onset of preequilibrium emission mechanisms. Second, the limits of stability and the mechanisms for decay of very hot nuclei are not known. This latter issue provides a strong stimulus for the investigations of energetic nucleus-nucleus collisions despite their complexity. Investigations of nucleus-nucleus collisions have focussed either on the properties of hot fusion-like composite residues consisting of significant fractions of the pro jec- tile and target nucleons or on the statistical and dynamical aspects of the hot, but non-equilibrium, initial stages of the reaction. In a practical sense, such a separa- tion is artificial because one can not address issues concerning the properties of hot residues without considering the mechanisms by which they are formed. In this dis- sertation, both the preequilibrium processes of the initial stages of the reaction and the thermal properties of the composite residues are considered. In the first part of the dissertation, the intrinsic excitation of intermediate mass fragments emitted dur- ing the non-equilibrium initial stages of the reaction is determined by measurements of the 7 rays from the decay of particle stable states of the fragments. Further dis- cussions of the physics motivation of these measurements are given in subsection A of the introduction. These measurements have shown that the intrinsic excitation at freezeout is surprisingly small. A theoretical investigation of the factors which may affect the intrinsic excitation is the topic of the second half of the dissertation. There we also explore the properties of the residues produced in these reactions. Additional background concerning the properties of such residues is given in subsection B of the introduction. A Complex Fragment Emission and the Emission Temper- ature The emission of low energy intermediate mass fragments (IMF), 3 3 Z S 20, in processes distinct from fission has been observed for a large variety of nuclear reac- tions [Lync 87, Gelb 87a, Gelb 87b, Cass 89, Guer 89]. In general, the energy spec- tra of these fragments exhibit broad maxima at energies close to the exit channel Coulomb barrier and exponential slopes at higher energies. The fragment distri- butions [Gelb 87a, Mini 82, Finn 82, Hirs 84, Chit 83, Troc 86, Fie186a, Fiel 86b, Faty 87b, Sang 87] follow an approximate power-law dependence on fragment mass, A", in both proton and heavy ion induced reactions, possibly indicating that the mass distributions are determined by a common physical process. For reactions at in- termediate energies, E/A=20-500 MeV, the angular distributions are forward peaked indicating that appreciable emission occurs prior to the attainment of statistical equi- librium of the composite pro jectile-target system [Chit 83, Jaca 83, Fiel 84, Mitt 85, Troc 86, Fiel 86a, Fiel 86b, Kwia 86, Faty 87a, Faty 87b, Poch 85a, Poch 85b, Chit 86, Xu 86, Xu 89, Poch 87, Chen 87a, Chen 87b, Chen 87c,Sain 88] . At backward angles, the angular distributions, particularly for heavier fragments, become more isotropic, consistent with significant contributions from the statistical emission by equilibrated heavy reaction residues that could, for example, be formed in incomplete fusion reactions [Sobo 83, Sobo 84, Kwia 86]. At present, there is no consensus concerning the origin of these fragments. F rag- ment production has been calculated within statistical [Mini 82, Finn 82, Hirs 84, Fiel 84, Boal 84, Lope 84b, Snep 88, Boal 88b, More 75, Gros 82, Frie 83, Rand 81, Fai 82, Bond 84, Ban 85, Hahn 87, Fiel 87, Come 88] as well as purely dynamical [Schl 87, Vice 85, Lenk 86, Baue 87, Aich 88, Boal 88b, Sura 89a, Sura 89b, Sura 89c] models. Most models reproduce selected observables such as the fragment mass dis- tribution. Differences between the various fragmentation models reflect, to a great ex- tent, differences in assumptions concerning the densities, internal excitation (charac- terized by emission temperature) and degree of thermalization which characterized the system at thermal freezeout. For example, IMF emission has been related to the oc- currence of adiabatic instabilities [Bert 83, Schl 87, Snep 88, Boal 89a] which may lead to the liquid-gas phase transition of highly excited nuclear matter [Lope 84b, F in 82, Jacq 84]. Other statistical models [More 75, F iel 84, Tsan 88, Hahn 87], as well as dynamical models [Boal 89, Boal 89a, Sura 89a, Sura 89b, Sura 89c, Sura 90, Peil 89], which do not incorporate a phase transition have been equally successful at reproduc- ing many features of the fragment data. To distinguish between different statistical models, one needs to measure experimentally the freezeout densities as well as the emission temperatures, to constrain the assumptions of these fragmentation models. Since the fragment kinetic energy spectra are sensitive to collective motion, the temporal evolution of the reaction, as well as Fermi motion and Coulomb barrier fluctuations, they do not provide quantitative information concerning the internal excitation energy at the freezeout stage of the reaction [Fie184, Frie 83, Bond 78, Ban 85, Stoc 81, Siem 79]. Information about the intrinsic excitation and the degree of thermalization at freezeout may be better obtained from the relative populations of nuclear states of the emitted fragments [Morr 84, Morr 85, Poch 85a, Poch 85b, Chit 86, Xu 86, Poch 87, Chen 87a, Chen 87b, Chen 87c, Bloc 87, Galo 87, Sain 88, Xu 89, N aya 89]. Emission temperatures of T=4-5 MeV were extracted‘from pairs of widely sepa- rated (AE 2 T) particle unbound states in ‘He, 6Li, and 3Be fragments at angles significantly greater than the grazing angle where contributions from projectile frag- mentation are negligible [Poch 85a, Poch 85b, Chit 86, Poch 87, Chen 87a, Chen 87b, Chen 87c, Chen 88a, Sain 88]. The relative populations of these states were found to be surprisingly insensitive to the incident energy over the range of E/A=35-94 MeV [Chen 87a, Chen 87d]. Moreover, these measurements revealed little sensitiv- ity to the gates placed upon the linear momentum transfer to the target residue [Chen 87c] or the associated multiplicity of charged particles emitted at forward an- gles [Sain 88]. Slightly lower values, T=3 MeV, were extracted [Bloc 87] from the neu- tron decays of excited states of 8Li emitted in the 1“N+ "“‘Ag reaction at E/A=35 MeV. In contrast, significantly lower values, T=l MeV, were extracted [Galo 87] from the neutron decays of excited 13C nuclei emitted close to the grazing angle in the l“N+ 165Ho reaction at E/A=35 MeV. Finally, measurements involving the 7—ray decays of both low lying [Xu 86, Morr 84, Morr 85, Morr 86] and high lying [Game 88, Sobo 86] particle stable states have been performed. Before this disser- tation started, the emission temperatures obtained from the decay of particle stable states [Morr 84, Morr 85] were reported to be much lower than those obtained from the decay of particle unstable states [Poch 85a]. Some of earliest work done in the dissertation study [Xu 86] demonstrated that these low-lying particle stable states are more difficult to interpret due to sequential feeding from higher lying particle unbound states [Poch 85a, Xu 86, Hahn 87, Fiel 87, Gome 88, Sobo 86]. When the sequential feeding is considered, these low-lying 7-ray measurements are not in contradiction with the emission temperatures extracted from the decay of particle unstable states [Xu 86]. If the populations of excited states can be described in terms of thermal dis- tributions corresponding to a single emission temperature, this temperature can be unambiguously determined by measuring the relative populations of just two states. Indeed, prior to the measurements undertaken in this dissertation and the disserta- tion of Tapan N ayak [N aya 90], emission temperatures were generally extracted from relative populations of just a few states. On the other hand, the degree of thermaliza- tion and the internal consistency of this thermal assumption can only be investigated by measuring a large number of states. To perform such tests, a large number of particle stable states of intermediate mass fragments were measured in this dissertation for 328 induced reactions on ““‘Ag at an incident energy of E/ A=22.3 MeV. These measurements were performed at angles back of the grazing angle to avoid large contributions from peripheral processes. Previous particle correlation experiments [Fiel 86b] on this system established that fragments are emitted with a low average multiplicity, M [MF 5 1, for a broad class of violent projectile target collisions representing about 60-70% of the total reaction cross section. In these reactions, large amounts (200-400 MeV) of energy are converted into intrinsic excitation, and a significant fraction of intermediate mass fragments are emitted prior to the attainment of statistical equilibrium of the composite system [Fiel 86b]. In this dissertation work, the measured relative populations of excited states of intermediate mass fragments are compared with those calculated from a thermal model which includes sequential feeding from higher lying states. The comparison indicates an average emission temperature of T: 3 — 4 MeV. These measurements, combined with previous measurements of particle unstable states, provide a picture of a constant or gradually increasing emission temperature with incident energy. The maximum observed emission temperature, is rather small. There are other indications from neutron multiplicity measurements, high energy hard 7-ray measurements, and measurements of charged particle spectra which suggest a similar limitations to the intrinsic excitation of composite residues. Such observations may relate to the characteristic fragmentation temperature pre- dicted to occur by multiparticle phase space models [Gros 88, Bond 85] when the hot system expands to sufficiently low density. Questions concerning how hot systems expand and cool may be better addressed by dynamic calculations. Surprisingly, these models also predict a low and nearly constant intrinsic excitation at freezeout [Lenk 86, Schl 87, Snep 88, Frie 88, Boal 88a, Boal 88b]. Within these calculations, Both the particle emission and the expansion play roles in cooling the system. To study whether the observed low and slowly varying emission temperatures can teach us anything about the nuclear equation of state or in-medium nucleon-nucleon cross section, we have performed dynamical calculations based on the Boltzmann-Uehling— Uhlenbeck (BUU) equation, which is a theory based on the one-body density matrix. Such calculations have the disadvantages that they do not properly describe many body correlations and they do not contain sufficient fluctuations to properly predict the emission of intermediate mass fragments, We therefore investigated excitation energies and emission temperatures of the heavy residues predicted by the BUU calculations. We are encouraged to try this approach by the results of molecular dynamical calculations [Lenk 86, Schl 87] which indicate that all reaction products, regardless of their masses, have about the same emission temperature. This suggests that the excitation energies of heavy residues calculated in our study may provide insights concerning the emission temperatures of intermediate mass fragments. Our BUU calculations also allow us to address questions concerning the fusion cross sec- tions, excitation energies and angular momenta for the heavy residues which have not been studied in previous dynamical studies. Further discussion of these questions was given in the following subsection. B Disappearance of Fusionlike Residues On rather general grounds, one expects fusion of the projectile and the target nuclei to become less likely with increasing incident energy. Most experimental investigations of the energy dependences of fusion or fusionlike processes have concentrated on mea- surements of the traditionally well understood residue decay channels leading to the production of the evaporation residues or fission fragments. Such measurements indi- cate that fusionlike processes, particularly for Ar [Lera 86, Auge 86, Nife85, Fahl 86, Jacq 84, Fabr 87, Bour 85] or Si [Deco 90, Grif 90] induced reactions, decrease rapidly with incident energy when E/ A Z 20 MeV, and eventually vanish at around the Fermi energy E/A 2' 35 - 40 MeV. Fig. 1.1 shows a example of the fission fragment folding angle distribution (the angle between the two fissioning fragments) measured for 4oAr+232Th collisions. In such measurements, one sees two peaks in the folding angle distributions at E/A=31 MeV: a large peak at OFF 2 170°, from the decay of target residues in peripheral collisions, and a small one at an 2 110°, from central fusionlike collisions. As the energy increases from E / A = 31 MeV to 44 MeV, the peak due to fusionlike reactions decreases with energy and eventually vanishes at E/A=44 MeV. Total excitation energies and emission temperatures were also extracted from the velocities of the fusion-like residues [Lera 86, Auge 85, Nife85, Bour 85, Gali 88] and coincident light particle spectra [Goni 88, Gali 88, Wada 89, Deco 90], respectively. Light particle evaporation spectra have also been analyzed [Goni 89, Bohn 90, Grif 90] to extract the temperatures and the excitation energies of the residues. These analyses suggested that the maximum excitation energy that a nucleus can sustain, decreases with the mass of the composite system, from E" / A z 5-6 MeV for light systems with total masses AS 100, to a value of E“ / A z 3 MeV for a total mass A 2 200. [Guer 89, Lera 86, Auge 85, Bohn 90, Fahl 86, Bour 85]. If one assumes a level density of a = A/ 8 MeV, the residue temperatures for these heavy systems are comparable to the emission temperatures extracted from the emission of non-equilibrium IMF ’s that were discussed in subsection A. The disappearance of fusionlike cross sections has been most frequently interpreted to be a consequence of the instability of hot nuclei at high temperatures [Finn 82, Lope 84b, Bert 83, Schl 87, Snep 88, Boal 89a, Bord 85, Levi 84, Besp 89, Gros 88]. For example, the static model of Levit and Bonche [Levi 84] predicted a limiting temperature of T 2 5 — 10 MeV, above which nuclear matter becomes unstable against hydrodynamic expansion. If one assumes a soft nuclear equation of state and a level density of a = A/ 8 MeV, this ‘limiting temperature’ is consistent with the observed disappearance of fusion-like processes with Ar induced reactions at E/AZ 35 MeV [Auge 86]. These analyses assume the existence of an equilibrated residue at an excitation predicted by incomplete fusion. To address the validity of this assumption, 700 “Ac-Jun ' 3 coo die/1111,1111, 1a m p UI O o o N o 10 ”HOV/U . . I . 35mm. . i 1 / 5 .' A ”NW/u “HOW“ I 91.11» 10111“ 016 . 11. - 1 Figure 1.1: Differential cross section of fission products as functions of both the inci- dent energy and the folding angle between the two fission fragments for 40Ar+232Th collisions [Conj 85] . 10 one must also consider the processes which govern the formation of a highly excited composite residue. Within the presently available microscopic dynamical models, the formation of a fusionlike residue depends on the interplay of the nuclear mean field and the in-medium nucleon-nucleon scattering cross section. Clearly within these models, whether two nuclei fuse or not depends on these ingredients. For example, a larger nucleon-nucleon cross sections would give individual nucleons in the medium more chances to collide with the others. This would result in more stopping and a larger cross section for the formation of heavy composite residues. Therefore a study of, for example, fusion cross section with models having these ingredients may allow one to place limits on these model parameters. Similar questions can be asked about the sensitivity of the residue cross sections or the excitation energies to the nuclear equation of state. Dynamical models based on the Boltzmann-Uehling-Uhlenbeck equation [Bert 84, Bert 87, Aich 85, Bert 88, Rema 86, Baue 87, Baue 88, M011 85a, Moli 85b, Krus 85,Aich 87, Aich 88, Gale87, Welk 88, Gale90, Peil 89, Cass 88, Cass 89] allow the possibility of investigating these questions. Prior to this dissertation study, however, these issues had not been investigated. II Organization This dissertation contains two distinctive parts: The first four chapters deal mainly with an experimental determination of emission temperatures for a large number of particle-stable states. The remaining three chapters describe a theoretical project, designed to address whether one can learn anything concerning the nuclear equa- tion of state at sub-nuclear density from measurements of emission temperatures or heavy residue cross sections. Final conclusions are given in chapter 8. Both parts of 11 this dissertation are motivated by the idea of testing the concept of thermal freeze- out by emission temperature measurements and by the desire to learn whether such measurements can provide information concerning the nuclear equation of state. In particular, this dissertation is organized as follows: In Chapter 2, the ex- perimental details, including an overall description of the experimental setup, the background subtraction of coincident 7-rays, as well as the 7—ray calibrations, will be given. In Chapter 3, the inclusive fragment cross sections are presented and fitted with simple parameterizations. The bulk of the particle 7-ray coincidence data are also presented in this chapter. In Chapter 4, a detailed sequential decay calculation is described. The results of this model calculation are compared with the experimental data and a mean emission temperature is extracted. Summary and conclusions concerning the experimental study are also made in this chapter. In Chapter 5, a comprehensive description of the improved Boltzmann-Uehling- Uhlenbeck (BUU) equation is provided. In this dissertation study, the BUU equation is improved with a Lattice Hamiltonian method. Such a method gives an excellent conservation of total energy which therefore allows us to study the thermalization and energy deposition which would otherwise be impossible. In this chapter, we will give a detailed descriptions of formal equations, numerical solutions of these equations, and numerical tests of the ground state stability and the conservation of energy. In Chapter 6, the improved BUU equation is applied to 4°Ca+‘°Ca and 4oAr+27Al collisions in order to address the following two important questions: 1) which observ- ables are most sensitive to the nuclear equation of state at sub-nuclear density; 2) what are the dynamical limits to the formation of heavy residues. We will first discuss the fusion cross sections and their sensitivities to the nuclear equation of state and 12 the in-medium nucleon—nucleon cross section. We then discuss how one can design an experiment to disentangle these sensitivities. Finally, we examine the various aspects which limit the calculated fusionlike cross sections. In Chapter 7, we return to address the emission temperature and its possible relation to the nuclear equation of state. For such purposes, we investigate ‘OAr+27Al and 40Ar+"“Sn collisions to determine the sensitivities to the equation of state. A short summary is given in Chapter 8. Chapter 2 Experimental Details After a brief description about the experimental setup in section I, we discuss in detail how to understand the coincident spectra and the background spectra obtained with the NaI(Tl) detectors of the Spin Spectrometer. Since the background yields measured with these detectors are much higher than the yields of the discrete transitions of interest (in some case, background is more than 20 times larger), it is crucial to understand the line-shapes of the spectra. For this purpose, we present, step by step, the calibrations of the line shapes using 7 — 7 sources, as well as proton inelastic scattering. The determination of the absolute efficiency and the correction of the line shape distortions due to double hits are also discussed. I Experimental setup The experiment was performed at the Holifield Heavy Ion Research Facility of Oak Ridge National Laboratory. Silver targets of natural isotopic abundance were irradi- ated with328 ions of 714 MeV energy. Intermediate mass fragments were isotopically identified with five AE - AE' — E surface barrier detector telescopes, positioned at the laboratory angles of 0”” = 20°, 25°, 30°, 45° and 50°. The telescopes subtended solid angles of A0 = 9.8, 10.1, 15.4, 36.3, and 28.6 msr, respectively. Each telescope con- 13 14 sisted of two planar AE—detectors with thicknesses between 50 and 100 mm and an E- detector with thickness of 1.5 mm. Cross contaminations between adjacent isotopes were reduced by restricting the analysis to fragments that stopped in the E-detectors of the telescopes thus permitting two independent particle identification gates. This introduced energy thresholds at about E/A=8 MeV for 10B at 9”” = 20°,25°,30° and at about E/A=7 MeV for 9111117 = 45° and 50°. In order to reduce computer dead time and speed up data acquisition, a hardware gate was set during the experiment which suppressed triggers of the telescopes generated by light particles (p, d, ..,a). These particles are emitted with significantly larger cross sections than intermediate mass fragments which were the focus of the present experiment. In order to make sure that no nuclei with Z _>_ 3 were rejected, the gates were set such that a small fraction of a-particles were recorded on tape. About 85% of all light particles were rejected by this method. Coincident 7-rays were detected with the Spin Spectrometer [Jaas 83]. Six of the NaI(Tl) crystals of the Spin Spectrometer were replaced by Compton shielded Germanium detector modules. In addition to the particle 7-ray coincidence events, the Spin Spectrometer was triggered by the detection in one Germanium detector of 0.898 or 1.836 MeV 7-rays from an 8‘BY source positioned close to the Ag target. With a high probability, a 0.898 (1.836) MeV 7-ray detected in the Germanium ensures the interaction of the companion 1.836 (0.898) MeV 7-ray elsewhere in the the Spin Spectrometer. Using this additional source data, it was possible to monitor the gain shifts of the photomultipliers of N aI(Tl) detectors and make corrections for these gain shifts, run by run, in the off-line analysis. 15 II Background Subtractions In the backward hemisphere of the Spin Spectrometer, neutrons could be suppressed by time-of-flight discrimination. The time-of-flight separation of neutrons frdm 7-rays for detectors in the forward hemisphere of the Spin Spectrometer was considerably worse due to the large cross sections for fast, noncompound neutrons at forward angles. To reduce the systematic errors arising from background subtraction, we consequently restricted our analysis to data taken with the N aI(Tl) modules in the backward hemisphere (0., 2 90° ) of the Spin Spectrometer. To illustrate neutron suppression in the backward hemisphere, we show in Fig. 2.1 the relative time spec- trum obtained between a solid state particle telescope located at 011m:- = 20° and a NaI(Tl) 7-ray detector located at 9., = 138°. The time spectrum clearly exhibits a sharp peak due to prompt 7-rays and a long tail caused predominantly by low energy neutrons emitted from excited target residues. Significant background reductions could be achieved by selecting prompt 7-rays with a narrow time gate. The lower and upper limits of the time gate employed for this particular detector pair are shown by the arrows marked as t; and t1, ,respectively. The energy spectra of coincident 7-rays were transformed, event by event, into the rest frames of the detected fragments using relativistic Jacobians and Doppler shift corrections. Since these transformations shift and broaden 7—ray transitions of the target residues, particular attention was paid to identifying and correcting for such effects. For this purpose, background spectra were generated by performing similar transformations to 7-ray spectra measured in coincidence with 9Be nuclei which have no strong transitions at the 7-ray energies of interest. The background spectra were then used in the fitting procedure to extract the yields of 7-rays from the decay of the detected intermediate mass fragments. 16 MSU-89-023 105 w E em=20° . . 9,=138° « COUNTS ' V V‘jn' 101i 1.L.n1..+.lrmn.1.... 60 7O 80 90 100 At (ns) Figure 2.1: Relative time spectrum between a particle detector ( at OIMF = 20°) and a NaI(Tl) 7-ray detector (at 0, = 138°). The limits of the time gate used for the analysis is indicated by the arrows marked as t; and th. 17 For illustration, the transformed 7-ray spectra measured in coincidence with 11B, 11C, and 12C fragments are shown in the left hand panels of Figs. 2.2-2.4, respectively. The spectra were summed over all measured particle emission angles and energies and over all 7-ray detectors located in the backward hemisphere. The dashed lines show the corresponding background spectra. On this scale, individual transitions are barely, if at all, visible. A better visual comparison of coincidence and background spectra is possible when smooth analytical functions are subtracted from both of them. The dotted curves correspond to functions of the form f(E‘7) = A ' exp(—E.,/a) + Bexp(—E.,/fl) + 0 (2'1) where A, B, a, 6 are constants adjusted by fitting the background and C is a constant offset. The solid and open points in the right hand panels show the coincidence and background spectra after subtraction of these functions. On these scales, the individual 7-ray transitions are clearly discernible. Moreover, spurious structures of the experimental background spectra are small in comparison with the identified peaks of the coincidence spectra. (The subtraction of the function f (E,) cancels in the final data reduction; this intermediate step only facilitates a detailed comparison of the coincidence and background spectra in regions of high background.) The yields of 7-rays from the decay of the detected intermediate mass fragment were fitted by folding the detector response function with the energies of known transitions of the detected fragment and adjusting the strengths of the individual transitions and the normalization of the background spectrum. The detector response function was calibrated over the energy range of E 2' 0.5- 7 MeV with 7-rays emitted from 2°Na,°°Co,°°Y, and 207'Bi radioactive sources as well as 7-rays produced by the inelastic scattering of protons from 12C and1°O target nuclei. The response function includes detailed descriptions of the photo-, first and second escape peaks, as well as line shape corrections due to coincidence summing. Finally, the inclusive fragment 18 ”sum-020 60] 1x103 X'O3 [ET Ag(azsnliB‘y) ‘_ te—(2.12"8~s‘) -: 3 ! ‘. ’ . _ 4—(4,44-vg.s.) 4O ’ * (2.12"3 8) ’ I f (5.024g.s.) z: . 2/ ‘ 1* ‘ 2 z r g (6.74-7.98 ‘ 8 i .5 —-(4.44-1g.s.) Afi‘ & § -. g.s.) 0 f1. ] It . . . t it . b, ’ ¢ 0. 20 - t‘ ‘- 1* 11* . (6.74—7.98 ' -v g.s.) E (MeV) , E (MeV) Figure 2.2: Left hand panel: Coincidence (solid line) and background (dashed line) spectra for 11B fragments. The dotted line corresponds to the function of Eq. (2.1). Right hand panel: Coincidence (solid points) and background (open points) spectra after subtraction of the function of Eq. (2.1)The locations of specific 7-ray transitions in 11B are marked by arrows. 19 MSU-89-022 8:"""I”‘rrvn'1""1""T**'*I"J‘*'1'“'1""I""l"*‘l""1"" 0-75 1: , . )(IO3 :} “(328.1107) :r 4——— (2.00-og.s) ‘ x103 ’ I} " l ‘— ($32-15) 6 P '1 ‘ (D . y ‘ [ (LEO-1.8.) - 0 50 E ’ :¢———— (ZOO-1.3) w ++ f+ ° 8 . y . t (63457.) .50 1 o F 3' i 4 ’ ‘3 (4-33"8-3) ‘ hi HHSAH $0“ “‘6'.“ . 3" [39’1” . 3," [1]} WW“ ”0: 0.25 Figure 2.3: Left hand panel: Coincidence (solid points) and background (dashed line) spectra for 11C fragments. The dotted line corresponds to the function of Eq. (2.1). Right hand panel: Coincidence (solid points) and background (open points) spectra after subtraction of the function of Eq. (2.1). The locations of specific 7-ray transitions in 11C are marked by arrows. 20 MSU-88-073 20 rt‘"I""'l""I"f'T""I""l7"'lf"'I""I""I"" 4 ‘ 1; 4.44MeV-og.s 1 3 xIOS: {As(°°3.“c7) f: [ ‘xlo m ’ 3' ‘ " i F 15 .. ‘11 .1... .t .1 3 z [ lg " t ‘ 8 ~ .1 o 4.44MeV-vg.s , . Figure 2.4: Left hand panel: Coincidence (solid points) and background (dashed line) spectra for 12C fragments. The dotted line corresponds to the function of Eq. (2.1). Right hand panel: Coincidence (solid points) and background (open points) spectra after subtraction of the function of Eq. (2.1). The locations of specific 7-ray transitions in 12C are marked by arrows. 21 yields and the fragment-“pray coincidence yields were summed over angle to extract the fraction F, of observed fragments which were accompanied by the designated 7-ray. Technical details of the response functions and corrections for the effects of coincident summing are presented in the next section. Here we present a less detailed overview of the analysis procedure. The effects of coincidencesumrning are illustrated for the simple case of the 4.44 MeV 7—ray transition of excited 12C fragments, see Fig. 2.4 for the coincidence and background spectra. The final coincidence yield after background subtraction is shown by the solid points in Fig. 2.5. The dashed curve shows the detector response as calibrated via the 1"’C(‘y,‘y’)’2C reaction for which the gamma ray mul- tiplicity is one. This calibration underpredicts the high-energy tail of the line shape for the spectrum measured in the "“‘Ag(328,12C7) reaction in which the average 7-ray multiplicity is high. Due to this high '7-ray multiplicity, there is a non-negligible prob- ability that two coincident 7-rays or a 7-ray and a neutron are detected in a single N aI(T l) module. We denote this effect as ‘coincidence summing’; it depends on the associated 7-ray and neutron multiplicities and on the geometry of the experiment, but is independent of the beam intensity. The calculation of the line shape distortion due to coincidence summing will be described in Section III(C). The corrections are illustrated in the lower part of Fig. 2.5. The dashed line shows the response of the detector to a given number of 4.44 MeV 7-rays in the absence of coincidence summing. A fraction, p(: 0.28), of these 7-rays, will interact with the detector in coincidence with a second 7-ray or a neutron from the same reaction. The summed response to the 4.44 MeV 7-ray plus the second 7-ray or neutron is shown by the dashed-dotted curve. The remaining fraction, 1 — p, of the 4.44 MeV 7-rays will interact individu- ally with the detector with the response function measured at low multiplicities and shown by the dotted curve. The total response function, corrected for coincidence 22 summing, is shown by the solid lines in Fig. 2.5; it corresponds to the sum of the yields represented by the dotted and dashed-dotted curves (See also Eq. (2.19)). This parameter-free correction reproduces the measured coincidence yield rather well. All fitted spectra include corrections due to coincidence summing. Coincidence summing corrections were also required to extract the 7-ray yields from measurements obtained with the Compton suppressed Germanium detectors of the Spin Spectrometer. Due to the superior resolution of the Germanium detectors, individual 7-ray hits are well separated from summed events and corrections to the line shape are not required. In the Compton suppressed operating mode, however, additional 7-rays or neutrons detected in the Germanium detector or the Compton shield result in a multiplicity dependent loss of efficiency of about 20%. This loss of efficiency are corrected and the data are also included in Figs. 4.4 and 4.5. Figures 2.6 and 2.7 give examples for more complicated coincidence spectra. The solid points show the final coincidence yields fromuB ( 2.6) and 11C (2.7) 7-decays, after background subtraction. (The original coincidence and background spectra were already shown in Figs. 2.2 and 2.3.) The photopeak locations of the most important 7-ray transitions are marked by arrows. The lower panels show individual contri- butions from the most important transitions used in the fits. The most important transitions and branching ratios used in the final fits are shown in the inserts. Clearly, the individual populations of states above about 6 MeV excitation energy are not well determined. In these and other ambiguous cases, we have used the summed strengths of the groups of states indicated in the upper parts of the figures to provide informa- tion about the emission temperature. 23 MSU-88-O74 3 'TTI'I'rI"rTIITTTI""I"'rI'f"l'T o - 4 L . “03’ Ag(323,12C7) 1 «(4.44 MeV-03a.) j F 1 1 2 " ‘ —-Corrected Response '- "Orisinach-pom 1 COUNTS Coincidence Summing Corrections -— Corrected Response -- Original Response --- Double Hits xlO3 L A A E (MeV) Figure 2.5: Upper panel: Background subtracted coincidence yield (solid points) at- tributed to 7-ray decays of excited 12C fragments. The location of the photopeak for the decay of the 4.44 MeV states is marked by an arrow. The dashed line shows the original response function determined from the calibration at low 7-ray multiplicity. The solid line shows the final line shape which includes corrections due to coincidence summing. Lower panel: Corrections due to coincidence summing at high 7-ray mul- tiplicity. The solid and dashed lines are the same as in the upper panel. The dotted curve corresponds to the calculated response due to the simultaneous detection of two 7-rays; the dashed-dotted line corresponds to the undistorted response when only the 4.44 MeV 7—ray is detected. The solid line corresponds to the sum of the dotted and dashed-dotted lines. 24 MSU-89-Ol9 l ‘ l ' IT I l I T I ] 3000 Ag(3ZS,“B'y — _ (2.12-1g.s) , 1 I [— (4.44-vg.s) 2000 - (5.02-+g.s.) . (6.74—7.98 1000 7 3° ’ U) E1 . Z O . l . i ' l l l ' D . Individual Contributions . O + U 46% 53% 7.98, 3/2 . - 7.29, 5/2’ - 8'77. , 2000 . 67.57. 25.57. '79' 1/2_ I 70% 30% '74' 7/2_ 867. “147. "02' 3/2 1 100% r 4 1000 Figure 2.6: Upper panel: Background subtracted coincidence yield (solid points) attributed to 7-ray decays of excited 11B fragments. The solid line shows the fit used for the extraction of the 7—ray fractions,F.,, listed in 3.2. The locations of several strong transitions are shown by arrows. Lower panel: Contributions from individual transitions. Important transitions and branching ratios are given in the insert. 25 MSL-BQ-OZ! 600 '"I""I""I""i""l""i""l 400 _ . O 200 ) _ m 4 ‘2 , 1 . 1 . :3 400 ' ' _ O O 7.50. 3/2’1 5.90. 15/2+ 5.45. 7/2‘ 5.34. 1/2* 4.80. 3/2’ 4.32. 5/2‘ 2.00. 1/2" 200 3.3 3/2" Figure 2.7: Upper panel: Background subtracted coincidence yield (solid points) attributed to 7-ray decays of excited llC fragments. The solid line shows the fit used for the extraction of the 7-ray fractions,F.,, listed in 3.2. The locations of several strong transitions are shown by arrows. Lower panel: Contributions from individual transitions. Important transitions and branching ratios are given in the insert. 26 III 7—ray calibrations A Line Shape Calibration The response functions for the individual detector modules of the Spin Spectrometer were calibrated with 7-rays emitted from 2°Na,""Co,°‘°Y, and 2°7Bi radioactive sources as well as 7-rays produced by the inelastic scattering of protons on 12C and 160 nuclei. In total 12 calibration points were measured over the energy range of E, a: 0.57 — 7 MeV. After gain matching of the individual detector modules, the 7-ray spectra were summed over the detectors located in the back hemisphere of the Spin Spectrome- ter. The summed spectra were then fitted with a parameterized response function. Examples of calibration fits are shown in Fig. 2.8. The fitted response function was parameterized as: 51(E1 E01 A0) = i Aklak(E1 Eh) + 16k(E1Ek)l1 (22) k=0 with exp[(Lk/2a’2)(Lk + 2E — 2Ek)], forE S E}, — L1, ak(E, Ek) = exp[—(E — Elf/20,2], forEk — L1, < E < E], + U], (2.3) exp[(Uk/2ai)(Uk — 2E + 21131)], forE 2 E], + U); and ,Bk(E, El.) = 31,625+ arctan[ak(E — E1, — bk)] +Tk arctan[ak(E — E1, - bk - c)] - TK arctan[ak(E — E], — bi, + c)]} (2.4) In Eqs. (2.2)-(2.4), the indices k = 0, 1, and 2 denote the photo-, first and second escape peaks, respectively; E0 and E denote the original 7-ray energy and the detector response in MeV; the functions ak( E, Er) and ,61,(E, E1.) parameterize the line shapes of the individual peaks and the Compton backgrounds, respectively. The positions of 27 the first and second escape peaks were given by: E), = E0 — 0.511k. (2.5) The photopeak amplitude, A0 , was fit to the measured spectrum, and the relative normalizations of the amplitudes, A1 and A2 , were determined from the calibrations and could be expressed in the functional form: A], = CkAo{1 — exp[(1.56 — Eg)/3.0]}6(Ek — 1.56), (k = 1,2), (2.6) where 9(3) is the unit step function, 9(2) = 0 for a: < 0 and 9(2) = 1 for a: > 0, and CI = 0.90 and Cg = 0.18. The energy dependence of the line shape parameters was determined by the calibrations and could be represented by the functions: L), = [0.19 + 3.52 exp(—E,.)]a,,, (2.7) U1. = [0.47 + 1.22 exp(—Ek/1.91)]a’;., (2.8) a), = (6.8 + 3313;”) x 10-3, _ (2.9) s _ —0.012E,, + 0.075, for 13,. < 4.44MeV (210) " " 0.0034513. + 0.0063, for E, 2 4.44MeV ' T]; = 29.50}, (2.11) a). = —1.0/(78.7a,,), (2.12) b, = —2.0 — 377.05., + 50.0E1/(1.0 + 9.8E), (2.13) C = 72.000. (2.14) Apart from the 7-ray energy E; , the calibrated detector response function contains only the adjustable parameter, A0, which determines the normalization. B Absolute Efficiency The relation between the fitted amplitude A0 and the total number of 7-rays of energy E0 was calibrated at low energies, E, = 0.57 - 2.75 MeV, via 7-7 coincidence mea- surements using radioactive sources with coincident transitions. At higher energies, 28 E, 2 4.4MeV, the absolute efficiency was obtained from p - 7 coincidences measured for the inelastic scattering of protons on 12C andl°O. If the observed 7-ray peak can be attributed completely to the particle 7-ray coincidence yield, we can define the normalization function 17,(E'o) as: 771(30) = A0(EO)/N(EO)1 (2-15) where A0(Eo) is the amplitude fitted to the spectrum of a 7-ray of energy E0 (defined as A0 in Eq. (2.2) and N (E0) is the total number of emitted 7- rays. For two coincident 7-rays of energies E0 and E0 ’, the efficiency for the detection of 7-rays of energy E0 with the N aI(Tl) detectors of the Spin Spectrometer can be calibrated by determining the amplitude A0(Eo) for the spectrum measured in coin- cidence with 7-rays of energy Eo’ detected with a Compton suppressed Germanium detector module of the Spin Spectrometer. When gated on the 7-ray peak of energy E0’ in one (Ge) detector, non-negligible contributions, A0(Eo’), of the same energy E0 ’ are observed in the other (N aI) detectors (Fig. 2.8). These contributions,Ao(Eo ’), are entirely due to random coincidences and thus allow us to correct for the random coincidence contributions to the true peak Ao( E0). Since Ao(Eo) is determined from the spectrum summed over all detectors contained in the backward hemisphere of the Spin Spectrometer, angular correlation effects are effectively integrated out. When such random coincidence effects are important, r],(Eo) is not given by Eq. (2.15). Instead, making the random correction, one has, 777(E0) P7(E0) 1 "7(E0) = [A0(E0) _ A0(E0 I) 7]»,(Eo ,) P,(Eo ,) NGe(EO ’)P77(E0) ° (2.16) Here, Na,(Eo’) denotes the total number of 7-rays of energy Eo’ detected in the Germanium detector; P,,(Eo) denotes the conditional probability that a 7-ray of energy E; is emitted in coincidence with the detected 7-ray of energy Eo’; typically, P,,(Eo) = 0.8 — 1.0; P,(Eo) and P,(E'o’) correspond the single inclusive emission 29 probabilities for 7-rays of energies E0 and Eo’, respectively. The second term in the square brackets corrects for random coincidences. When gated on the other 7-ray peak E0 (Fig. 2.8, the upper and lower figures in the left-hand column), the true coincidence amplitude for 7-rays of energy 130' and random amplitude for 7-rays of energy E0 can be extracted. Thus, a corresponding equation for r],(Eo ’) can be established with E0 and E0’ interchanged in Eq. (2.16), allowing the unambiguous determination of n,(Eo) and 17,(Eo’) by an iterative procedure. For the case of 7-rays emitted in the 12C(p,p’7) and 1"O(p,p’7) reactions, one places a gate on the respective peak in the proton spectrum to determine the number, Npi, of inelastically scattered protons. Summing over the NaI(Tl) detectors in the backward hemisphere, one obtains: 717(E0) = A0(E0)/[NP’PP7(E0)I1 (2°17) where the amplitude A0(Eo) is determined from the coincident 7-ray spectrum, cor- rected for random coincidences; Pp,(Eo) is the conditional probability that a 7-ray of energy E0 is emitted in coincidence with the detected inelastically scattered proton. For the transitions of interest, the conditional probability PP,(E0) is unity. Figure 2.9 shows individual points measured for the normalization function. The solid line shows the analytical interpolation used in our analysis, r],(Eo) = 4.11E0"°'°°° exp[-0.0143(3.912 + 1n E0)2 — 0.010(3.912 + 1n Eo)°]. (2.18) C Coincidence Summing For the average event analyzed in the present experiment, the individual NaI(Tl) modules in the backward hemisphere trigger with a coincidence probability of about p = 28%. This high probability is due to the rather large multiplicity of 7-rays (and neutrons) emitted from highly excited target residues. Only a fraction, 1 — p, of 30 recorded fragment 7-rays will be correspond to single 7-ray interactions for a given detector module; there is the probability, p, for the coincident interaction of a second 7-ray with the same detector module. This ‘coincidence summing’ effect leads to considerable, multiplicity dependent line shape distortions, see Fig. 2.5 of the main text. These line shape distortions were evaluated by folding the fraction, p, of the original calibration function, Eq. (2.2), with the normalized background function, B,(E), obtained from an energy spectrum in the backward hemisphere of the Spin Spectrometer which reflects the pulse height distribution for neutrons and 7-rays emitted from target residues. This latter spectrum was gated by the detection of a 9Be fragment in a particle telescope to avoid introducing structures due to discrete 7-rays emitted from the detected fragments. The corrected response function has the form: 5:1(E1 E01A01p) = A010 "' p)€—7(E, E0) + pLE dE’[€1(E " Eb E0)B‘Y(El)l}1(219) with [1115' ,(E’) = 1. (2.20) where s,(E, E0) is the result of Eq. (2.2) with A0 = 1. 31 MSU-88-062 XIOS- 1:11fi I I T T TrilltfivlifiivIIVT'I'FIJ—a : Response Function for NaI Detectors : 3 20 r_ ] xlO ’ 207 -e 207 . . - - Bl -2 Bi+27 12c . ._ C (0.57uev gate) (p,p7) -1 6 15 _ —-Response 4 : randoms .. -Compton : 10 — i m I i 62-. 5 f.e _‘ :3 . ° 3 O ' '- ‘ " i O [ l ‘ I" ‘ o _ 1 I .L 1 .L 1 | 1 1. 1 1 x 103: 2°“’Bi'-»2°"Bi+27 .- (1.06MeV gate) 15 :- 10 :— C randoms 5 i l O 1 ~ 'I " — — - A 0.5 1 E (MeV) Figure 2.8: Calibrations for the 7-ray response function, s,(E,Eo,Ao), given by Eqs.(2.2)-(2.4). The solid lines show the fitted line shapes; the dashed lines show the calculated Compton background. 32 MSU-88-053 10 — I I I I I { I I I r I I I I : 020781 : 5 : 31‘ “CO : .. 0”? q - DuNa ‘ g; 2 _ .P-t-Mylar _1 v >~ w 1 _— ‘3 0.5' O 2 I l L Lll l 1 1 1 l I 1 ’ 0.5 1 2 5 10 E (MeV) Figure 2.9: Measured values of the normalization function 1).,(Eo), defined in Eq. (2.15)-(2.17). The solid line corresponds to the analytic interpolation given by Eq. (2.18). Chapter 3 Data In this chapter, the data for single particle inclusive spectra and the coincidence 7- ray spectra are presented. In the first section we present the single fragment kinetic energy spectra, and fits to the spectra using a ‘moving source’ parameterization. The coincident 7-ray data are discussed in the second section. I Single fragment Inclusive Cross Sections The inclusive differential cross sections, measured at 0mm = 20°, 25°, 30°, 45° and 50°, are shown in Figs. 3.1-3.5 for isotopes of lithium, beryllium, boron, carbon, nitrogen, and oxygen, respectively. Consistent with previous measurements [Fiel 86a, Fiel 86b], the spectra exhibit broad maxima at energies close to the exit channel Coulomb barrier and rather featureless, nearly exponential slopes at higher energies. These slopes become steeper at larger angles. In the center-of-mass system, the cross sections are peaked at forward angles, indicating emission prior to the establishment of full statistical equilibrium of the composite nuclear system. In order to obtain analytical interpolations of the inclusive cross sections to unmeasured angles and energies, the data were fitted by a parametrization employing the superposition of three Maxwellian distributions (‘moving sources’): 33 34 £735 = ZN‘l/E — Uc exp{—[E - Uc + E; - 2V Ei(E — Uc) COS 0]/T.-} (3-1) 5:1 Here, U6 is the kinetic energy gained by the Coulomb repulsion from the heavy reaction residue assumed to be stationary in the laboratory system; N; is a normal- ization constant and T.- is the ”kinetic temperature” parameter of the i-th source; E.- = mug, where m is the mass of the emitted fragment and v,- is the velocity of the i—th source in the laboratory system. This choice of parametrization was chosen for simplicity. Fits obtained with this parametrization are shown by the solid lines in Figs. 3.1-3.5; the parameters are listed in Table 3.1. Because of the small angular range covered by the data, substantial ambiguities exist for the individual parameters. 35 Table 3.1: Parameters used for the fits of the inclusive cross sections with Eq. 3.1. The Coulomb repulsion energies U6 and the temperature parameters T,- are given in units of MeV, and the normalization constants N.- are given in units of pb/(ereV3/2). T1 Ul/C N1 T2 02/6 N2 T3 U3/C N3 Uc 6Li 0.156 5.1 73.5 0.095 9.6 219.9 0.025 5.0 226.4 35.0 7Li 0.143 8.6 229.7 0.081 9.6 369.7 0.007 5.0 270.6 34.6 8L1 0.160 5.0 0.1 0.090 11.0 34.3 0.009 13.7 17.1 34.3 7Be 0.161 7.7 65.7 0.085 10.1 73.1 0.018 5.0 41.3 45.4 9Be 0.154 6.8 74.6 0.086 10.0 107.9 0.000 5.0 121.5 44.5 l"Be 0.180 5.5 57.7 0.105 10.2 48.9 0.043 10.4 29.7 44.1 103 0.172 2.5 5675.6 0.097 10.5 92.8 0.001 12.7 42.6 47.0 11B 0.167 4.0 977.7 0.101 10.4 190.3 0.008 11.7 106.8 46.6 12B 0.173 2.7 1494.4 0.102 11.0 21.9 0.014 12.5 12.0 46.3 13B 0.167 3.3 127.4 0.097 12.0 4.8 0.013 13.1 2.3 46.0 11C 0.164 3.5 260.2 0.100 11.2 26.0 0.013 9.0 17.2 58.0 12C 0.165 3.8 1608.1 0.102 10.1 156.9 0.022 8.4 139.0 58.8 ‘30 0.162 3.5 1787.7 0.100 10.1 107.1 0.037 6.7 98.6 66.8 “C 0.164 3.5 930.5 0.102 10.6 37.5 0.038 10.2 23.7 58.0 13N 0.165 3.6 125.5 0.102 12.3 3.8 0.001 11.4 4.6 69.3 14N 0.169 2.7 11904.2 0.105 10.5 49.5 0.051 7.4 37.0 68.9 15N 0.164 3.2 8122.8 0.106 9.6 123.1 0.051 8.0 79.0 68.5 16N 0.159 3.8 211.4 0.102 11.4 9.6 0.043 10.4 9.5 68.5 17N 0.155 4.5 40.3 0.104 11.9 3.6 0.048 11.6 3.5 67.8 150 0.164 2.8 2206.9 0.106 11.6 5.6 0.052 7.5 7.0 76.9 160 0.159 4.2 1254.1 0.107 9.8 66.3 0.057 7.8 55.9 76.5 170 0.158 3.8 1254.4 0.107 10.1 33.6 0.054 8.3 32.4 76.1 180 0.158 4.0 557.1 0.103 10.4 19.8 0.051 8.1 19.5 75.7 190 0.149 5.5 22.1 0.099 12.6 2.0 0.055 11.8 3.1 75.4 36 II 7-Ray Spectra From Decaying Fragments A 7-Ray Spectra From Germanium Detectors Spectra of 7-rays detected in coincidence with isotopes of 8Li, 7Be, 1"B, 12B, and 13C are shown by the histograms in Fig. 3.6. To obtain these spectra, the energy spectra of coincident 7-rays were transformed into the rest frames of the coincident particles using relativistic J acobians and Doppler shift corrections. Since these transformations shift and broaden 7-ray transitions of the target residues, particular attention was paid to identifying and correcting for spurious structures in the 7-ray background. Similar to the analyses of the N aI(Tl) spectra, background spectra were generated by performing Doppler shift transformations on raw 7-ray spectra measured in coinci- dence with 8Li, 9Be, and 11B nuclei. These nuclei have no strong 7-ray transitions at the 7—ray energies which could be measured with the Germanium detectors; however, these background spectra contained discrete transitions from target residues common to all spectra. The Doppler shifted background spectra are indicated by the solid dots in the fig- ure. The following transitions were analyzed: 8Li(1"‘, 0.981 MeV)—+ 7+8Li(2+, 9.3.), 7Be(%-, 0.429 MeV) -+ 7+7Be(%-,g.s.), 10B(l"‘, 2.154 MeV)—> 7+1°B(0+, 1.740 MeV), 13C(gdi', 3.854 MeV)—) 7+13C(%-, 3.684 MeV), and overlapping transitions: 12B(2+, 0.953 MeV)-+ 7+”B(l+, 9.3.), ”3(1: 2.621 MeV)—+ 7+"B(2-,1.674 MeV). We did not analyze the transition, 7Li(%-, 0.478 MeV)—» 7+7Li(%-, 9.3.), because the pile-up of two coincident a-particles in the telescopes is misidentified as a 7Li, [Wohn 74], nor the long-lived transition, loB(1"’, 0.718 MeV)—) 7+1°B(3"‘, g.s.; 1' = 1.02113), [Ajze 86a], because this decay occurs at a considerable distance from the target re- sulting in major uncertainties in the efficiencies of the 7—ray detectors. The data in Fig. 3.6 were summed over all detectors; the individual detectors provide comparable 37 4 MSU-BB-OSS 10 , 102E ’8 ‘9 :> Q) . 2 1 E a 3 102 C £3 E "d 1 . \ . b N 2: '5 10 r I 1 !' X=9Be X=1°Be I O 100 200 O 100 200 E (MeV) Figure 3.1: Inclusive differential cross sections for lithium and beryllium isotopes; the laboratory detection angles are indicated in the figure. The solid lines represent fits with Eq. (3.1); the parameters are listed in Table 3.1. 38 MSU-BB-OSG 104 ....,c..-,...-,.,-fi.,.a..,....r 32 : Ag( S,B) E/A=22.3 MeV 1 g ...... _ 1 ' 11 I 102 ‘i:. B 1. '1 1. 0 20° 1 E o 30: . 1:1 45 _ i- 0 50° 1 o A: AAAJ AAA ..... . H O N 'V'V'1 dza/dEdQHQub/MeV-sr) F ..... I. II -. “a 1 2 E" .. ’9‘ 1 10-2f.1..1... .'.1.,4 -L .1.1..-.1.‘ ‘_ 0 100 200 O 100 200 300 E (MeV) Figure 3.2: Inclusive differential cross sections for boron isotopes; the laboratory detection angles are indicated in the figure. The solid lines represent fits with Eq. (3.1); the parameters are listed in Table 3.1. 39 104. 'I""l"'.[. MSU-88-057 .,.. E A= 2.3 MeV . 2. ,\10 r L. I f.” r > a) . 2 1r \ E “Q r 5% 1 C40"“ "d f "[3102? \ Nb 1- "U 1F 1 10-2: .1 1.... I... 0 100 200 300 0 100.200 300‘” E (MeV) . Figure 3.3: Inclusive differential cross sections for carbon isotopes; the laboratory detection angles are indicated in the figure. The solid lines represent fits with Eq. (3.1); the parameters are listed in Table 3.1. 40 MSU-BB-OSB I“ l 717* l I 32 s S,N 102r ”N A8( ) : E/A=22.3 MeV A r o a=20° iii 1!’ o a=30° >’ : D o=45° a) f o a=50° a -. "".."""""'= -° 102' 15m c: : 1 "U 1 1 £11 I . “U 1. \ 1.1L..L1..H1; ‘ b - 'l I I' 1 N “U 102 1. I...1-..11. .1 1...s1.f1.1.m1 O 100 200 300 O 100 200 300 E (MeV) Figure 3.4: Inclusive differential cross sections for nitrogen isotopes; the laboratory detection angles are indicated in the figure. The solid lines represent fits with Eq. (3.1); the parameters are listed in Table 3.1. 41 MSUoBB-OSS ....,....,....,.fi.,-- 32 102} 150 Ag( 3.0) : E/A=22.3 MeV A 1' o a=20° iii 1!" o a=30° :05) !’ D aa4s° . o 111-50° E ‘P Ffffij'rl'r"l""1". .0102)- 170i 3 : = c F 1. -o 11 1. 5 ‘ 1 \ 1 1 Nb : 1 fr 1. 1! 1 'r 1 ;... “41 ...... 1.14.1...-11-..1. ‘ O 400 200 400 E (MeV) Figure 3.5: Inclusive differential cross sections for oxygen isotopes; the laboratory detection angles are indicated in the figure. The solid lines represent fits with Eq. (3.1); the parameters are listed in Table 3.1. ll Relative Yield 42 1.0 . Ulvvvilvrvv 1.0 -_. Ag<°ZS.X) '1 I'II'fi 1213 (0.953 - gs) 0.8 . 0.8 E/A=22.5 MeV (2.621 - 1.674) 0.6 :- 0.6 E _. Data 0'4 :— 0 Background 0'4 0.2 I— 0.2 00211111111111111111511.11r111111111l11r1200 : “1.1 (0.981 - g.s.) 7Be (0.429 - g.s.) : 0.8 .— -: 0.8 0.6 Z.— -—;0.6 0.4 : —‘. 0.4 0.2 ' ‘ 0.2 0.0 ”-411111111-111 111 111+1l1111}111.10.0 1 13 __ 1 0.8 °B (2.154 — 1.740) c (3.854 3.684): 0.8 0.6 3 -I 0.6 0.4 '1 1 L. -i 0.4 ‘ ‘- n -1 i 0.2 ' F 11‘“ 3., 1*02 0.0. .....1. W111 0.0 —100 -50 0 50 —50 Q 50 100 E—E7 (keV) fiigure 3.6: Spectra of 7-rays detected in coincidence with isotopes of 3Li, 7Be, 1° B, and 13C produced in 323 from Compton Shielded Germanium detectors. 3. 43 numerical contributions to the sum. The inclusive fragment yields and fragment-7-ray coincidence yields were summed over angle and combined to extract the fraction,F.,, of observed fragments which were accompanied by the designated 7-ray. Spin alignments were assumed to be zero. This introduced a spin alignment dependent uncertainty in F, of about 3%. Values for F, obtained for these transitions, after correction for the efficiency loss due to the coincidence summing effects are presented in Table 3.2. B 7-Ray Spectra from NaI Detectors The 7-ray spectra detected in coincidence with 10Be, 12B, 13C, 1“C, 1‘N, 15N, 16O, and 180 fragments are shown in the left hand panels of Figs. 3.7- 3.14, respectively. The respective background spectra are represented by the dashed lines. The right hand panels show the yields obtained after subtraction of the background spectra. These yields are associated with 7- ray transitions in the detected fragments. The solid lines show the fits used for the extraction of the 7-ray fractions, 17,. The inserts in the left hand panels show the most important transitions and branching ratios. Photopeak locations of important transitions or groups of transitions are indicated by arrows in the right hand panels. For the actual fits, we used the complete set of transitions and branching ratios from the compilation of ref. [Ajze 8488]. The 7-ray yields associated with decays of excited 11B, 11C, and 12C transitions were already presented in Figs. 2.5-2.7. For 12C, 13C, and 10Be fragments we have investigated whether the measured values of F, depend on the fragment kinetic energy or scattering angle. Within the experimental uncertainties, no dependence of the F., on either quantity was observed. Values for F7, listed in Table 3.2, were obtained by combining the data for the various intermediate mass fragment kinetic energies and scattering angles. The coincident 7-ray spectra can be well understood in terms of known transitions 44 in the detected fragments. The good agreement of the measured and fitted spectral shapes justifies, a posteriori, our treatment of the background associated with emis- sions from target residues. The only case which shows noticeable deviations from our standard calibration and background subtraction procedures corresponds to the width of the 0.95 MeV 7-ray peak measured in coincidence with 12B fragments, see Fig. 3.8. This peak results from the superposition of the decays 12B(2"‘, 0.953MeV) -vy+”B(l+,g.s.) and 12B(1", 2.621MeV) -+7+uB(2‘, 1.674MeV). For this low en- ergy 7-ray, the line width was somewhat larger than expected from the overall calibra- tion of the response function, suggesting that the resolution of the Spin Spectrometer was slightly worse during the experiment than during the calibration. This degrada- tion of the resolution could possibly arise from the coincidence summing of low energy 7-rays and x-rays which lie below our experimental thresholds and therefore are not taken into account by the coincidence summing corrections described in Appendix C. This resolution problem made the background determination and subsequent sub- traction more difficult for 7-ray energies below about 1 MeV. In order to extract the strength of this peak more accurately, the spectrum was fitted by folding the calibrated response function with a Gaussian of 0.14 MeV FWHM (while conserving the integral normalization of the spectrum). The 7-ray fraction extracted from this peak agrees within 10% with that extracted [see Table 3.2] from the 7-ray spectra measured with the Compton shielded Germanium detectors for which the background subtraction was less problematic. For a considerable number of transitions, the energy resolution of the NaI(Tl) detectors was insufficient to allow reliable determination of the individual 7—ray frac- tions. In such cases, the 7—ray fractions are only given for groups of transitions which could be determined with good statistical accuracy. Transitions contained within a particular group are identified in Table 3.2 by 2 1, E2, etc. 45 There are several sources for the uncertainties in the 7—ray fractions listed in Table 3.2. Because the resolution for low energy 7-rays could not be accurately assessed from the fragment 7-ray coincidence data, the extracted 7-ray fractions have asso- ciated uncertainties which could be as large as 10% for 7-ray energies significantly below 3 MeV. Above 3 MeV these uncertainties are less than 2%. Additional uncer- tainties are associated with low counting statistics, uncertainties in the interpolation of the 7-ray efficiency (see Appendix B), ambiguities in the fitting procedure, and the possibility for misidentification of the mass and charge of the intermediate mass frag- ment detected in the particle telescope. These uncertainties were estimated and for simplicity, were combined in quadrature to provide the uncertainties listed in Table 3.2. It was particularly difficult to estimate the uncertainty associated with possible errors in the functional form of the background. Upper limits on this uncertainty were obtained by fitting with different background assumptions. With extreme back- ground assumptions, the experimentally determined yield varied by less than 8%. This extreme error estimate, however, was not incorporated into the uncertainties listed in Table 3.2. 46' MSU-88-07 7 25 x IO3 ’ 20 COUNTS 10 Figure 3.7: 7-ray spectra measured in coincidence with 10Be fragments. The left hand panel shows the raw coincidence spectrum with the background indicated by the dashed line. The right hand panel shows the spectrum associated with 7-ray decays of excited 10Be fragments. The solid line shows the fit used for the extraction of the 7-ray fractions, F, , listed in Table 3.2. The insert shows important transitions and branching ratios used for the fit. Photopeak locations of important transitions are indicated by arrows. 47 MSU-88-297 osfl ' F ' ' l ' ' ' ' I ' T ' ' l ' ' ' ' ' I ' ' ' i T T ' f ‘ 1 3 XI . 3 12 » 32 12 ‘XIO 15 -" Ag( 23. B7) . Ag( 8. 137) ‘ . 0‘ 1 '\ 2.72. 0+ 1 g *‘f 35% _ z > ‘0 2.62, 1 4’ ' 'l 2 ,- 80% lax 14:: l o.95-»g.s. D ’ , 1 6‘7 2‘ 1’ ‘—§ 1 8 101. p 96.8% ° ' 1 2.624157 Y. 0.95. 2+ ’ 1 l s 100% + t “l g.s., 1 1 . a 1 5 _ l l.67+g.s. . 362-1095 ‘ O ‘ ‘ ‘ '0 1 Figure 3.8: 7-ray spectra measured in coincidence with 12B fragments. The left hand panel shows the raw coincidence spectrum with the background indicated by the dashed line. The right hand panel shows the spectrum associated with 7-ray decays of excited 12B fragments. The solid line shows the fit used for the extraction of the 7—ray fractions, F7, listed in Table 3.2. The insert shows important transitions and branching ratios used for the fit. Photopeak locations of important transitions are indicated by arrows. 48 MSU-88-3OO 25 b . I fit Y fl’ Y I Y r I T 1 v 111 I I I It 1 T '”1’ I I I ‘l I T I I 1 I T] V V le—rfr' '7 I I U 2.5 32 13 3 ~ \ 32 13 1’ Ag( 31 C7) ‘ xIO3 X|O : Tb Ag( 8, C7) :' 3.68-’g.S. : 20 -- \ 3.35. 5/2’4— 3.85»g.s. — 2.0 m - ' 63% 367. «- « E—- - " 3 68. 3/2' 3.09-tg.s. ~ 2 » ‘- 1007. - :2 * t ‘ O 15 — c 100% — 1.5 o L V. ‘ .. \. u- \. ’ 4 _ \o J 10 f ‘1" -* 1.0 1. \\.O. .- \.‘. _ \ o. 5 —- \1 0.5 OllllllLllllllllLllLlllLllll 1111111111411411; 00 2 3 4 5 6 7 2 3 4 5 6 E (MeV) Figure 3.9: 7-ray spectra measured in coincidence with 13C fragments. The left hand panel shows the raw coincidence spectrum with the background indicated by the dashed line. The right hand panel shows the spectrum associated with 7-ray decays of excited 13C fragments. The solid line shows the fit used for the extraction of the 7—ray fractions, F1, listed in Table 3.2. The insert shows important transitions and branching ratios used for the fit. Photopeak locations of important transitions are indicated by arrows. 49 MSU-88-078 5 V"'I""I""l""l"r*Ifi"I""’f*"1""1""1'"VIr"'I*'r' ’5 32 14 ‘ 3 x103; ‘, Ag( S. C7) 1’ Ag(3zs,”07) ””0 r 1, .* 6.734g.s. ‘ 0.6 b‘ --0 1 4. f 49% “£7.34. 2* 4 6.09-1g.s. E03 5 90% 7.01. 2_ 3 ‘ z x, 100% 6.90. 0- 4. , D 31 r 93:: 3’3- 3.) l 70143.3. « O ' 99% ' ' - 'I " 04 o ’ t‘ —— 0.09.1‘ ‘ ' - . . 100% ., . t . . X 1 2- fix «i I ' as. 0” t . . 1 1" ‘ 0.2 1" \\ ‘1 l L \ .. x‘ . ‘ L ‘~\‘ ' “ 0’---1---.14LL----l.-..l-.‘.-l..---.L.lxk..1---.L--.-l..xm_l-.... 2 3 4 5 6 7 8 9 4 5 6 7 8 9 E (MeV) Figure 3.10: 7-ray spectra measured in coincidence with MC fragments. The left hand panel shows the raw coincidence spectrum with the background indicated by the dashed line. The right hand panel shows the spectrum associated with 7-ray decays of excited 1“C fragments. The solid line shows the fit used for the extraction of the 7-ray fractions, F.” listed in Table 3.2. The insert shows important transitions and branching ratios used for the fit. Photopeak locations of important transitions are indicated by arrows. 50 MSU-88-079 Ilvvvvlvvllllvlvlvivr Ag(3zS.“N'/) "l x ID3 x l03 fi—ZBI-vgs. COUNTS 4.92—7.03 — 0.4 -+ 3.3. ‘ p ll‘lllllllll 111111 AlllllllllllllllllllllllllllLL 00 . 0..l..1LL... 2 3 4 5 6 7 82 3 4 5 6 7 8 E (MeV) Figure 3.11: 7-ray spectra measured in coincidence with 1“N fragments. The left hand panel shows the raw coincidence spectrum with the background indicated by the dashed line. The right hand panel shows the spectrum associated with 7-ray decays of excited 14N fragments. The solid line shows the fit used for the extraction of the 7-ray fractions, F1, listed in Table 3.2. The insert shows important transitions and branching ratios used for the fit. Photopeak locations of important transitions are indicated by arrows. x l03 COUNTS 10 O 7.57. 7/2* f 7.30, 3/2” «_ 7.16, 5/2“ _ 3.32. 3/2‘ . 5.30, 1/2+ ‘. 5.27. 5/2*- g.s.. 1/2- .- . 4 lllLlLll‘lllfiIlllllllllllll I 11 Illlllll‘llll Illlllllllllllllll 1.0 0.5 MSU-BQ-OIB 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 I I I I-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 Iq 2.0 r 32 15 . . A S, N (5.27+5.30)->g.s. 3 l g( 7) *- 7.16—6.27 ””0 "o 9.13, 5/2 - l - l 8.57. a/z‘t-r 7.57-*5.27 . — 15 1 3.31. 1/2’ ' ‘ 2345678 2 3 E (MeV) 5678 Figure 3.12: 'y-ray spectra measured in coincidence with 15N fragments. The left hand panel shows the raw coincidence spectrum with the background indicated by the dashed line. The right hand panel shows the spectrum associated with 7-ray decays of excited 15N fragments. The solid line shows the fit used for the extraction of the 7-ray fractions,F.,, listed in Table 3.2. The insert shows important transitions and branching ratios used for the fit. Photopeak locations of important transitions are indicated by arrows. x l03 COUNTS 52 MSU-88-OBI .UTIIIIII—vvvr'rtrv'rrfirl—vvIvlvvrfi‘hrIHITT11Ir1IIII!Ivlrrfiv'vvr‘ 3 Dr E 1 7 ‘ 6.13»g.s. _. 0.6 1 0.37. 2‘-r 1 j" 100’? 113% 7.12. 1: :. Ag(azS,1807) l . - b 100% ”2' z 1* 1 * ‘ 100x 643- 3‘ ~. 6.92»g.s. +- r s l' " al— ' J 0.4 : 1.2 88” 0+ 3 t 7.124“. : ' ‘2. _. ll . \ :r . . 3. «r- fit . — 0.2 L \\ i + ill 1 : ‘~.\ 1 .l g . x“ l . L LillllllljlllljlllLLILLLLJJJll Ill]lllllllLllllJllJlllll‘lll 00 3 4 5 6 7 8 9 4 5 6 7 8 9 E (MeV) Figure 3.13: 7-ray spectra measured in coincidence with 160 fragments. The left hand panel shows the raw coincidence spectrum with the background indicated by the dashed line. The right hand panel shows the spectrum associated with 7-ray decays of excited 160 fragments. The solid line shows the fit used for the extraction of the 7-ray fractions, F.” listed in Table 3.2. The insert shows important transitions and branching ratios used for the fit. Photopeak locations of important transitions are indicated by arrows. 53 MSU-88298 5 Ing'l""l"‘fI'"'[*""'rv’Ivvvvt.jj-',,rvfifva‘ ‘* . + . . . “03: l ,_ 355 3654193 «1.0 32 13 ‘> . 4L bl» Ag( S, 07) “t 32 a i “'03 * 1’ A3( 8.107) ‘ m r ‘l 1' - 0 3 L \° ' 3.03.0* c v « D \. 99.77. . +0 * O 3." 1.93-»g.s. . , U » 3—1 ~ 0.6 . 13.92-21.98 ‘ 2r o . I ~ 0.4 1.— 01-2 - . .. ..L*Lmrr+l--n-l--.- - l 2 3 4 5 6 1 E (MeV) Figure 3.14: 7-ray spectra measured in coincidence with 180 fragments. The left hand panel shows the raw coincidence spectrum with the background indicated by the dashed line. The right hand panel shows the spectrum associated with 7-ray decays of excited 120 fragments. The solid line shows the fit used for the extraction of the 7-ray fractions, F.” listed in Table 3.2. The insert shows important transitions and branching ratios used for the fit. Photopeak locations of important transitions are indicated by arrows. 54 Table 3.2: Extracted fractions, F." of observed fragments which were accompanied by the designated 7-ray transition. Values marked by t are fractions obtained from Germanium detectors corrected for the effects of coincidence summing. For transitions which could not be resolved experimentally, the F7-value is given for the summed strength. These transitions and lib-values are identified by ‘21”, i = 1, 2, ..., 14. Fragment Transition (J ’, E ‘) F, 21' 7Be (11‘, 0.429) _. (g‘, 0.0) 0.222 a; 0017* | 8L1 1 (1+, 0.981) —. (2+, 0.0) | 0.207? :1: 0.03311: ] 1°Be (2+, 3.37) _4 (0+, 0.0) 0.61 :1: 0.03 L (2+, 506) _. (2+, 3.37) 0.16 :t 0.02 (z: 1) (1-, 5.96) —. (2+, 3.37) [ 10B T (1+,2.154) —. (0+,1.740) [0.082 30015T 11B (52.12) .4 (320.0) 0.110 a: 0.015 (314.44) —. 310.0) 0.143 :1 0.016 (3‘, 5.02) —. (3100) 0.059 :t 0.008 g‘, 6.74) .4 (g‘, 0.0) 0.135 :1: 0.028 (2; 2) (£36.79) _. (3:00) (31.37.29) -+ (g ,0.0) (g ,7.98) —. g’, 0.0) llC g, 2.00) —. (i, 0.0) 0.151 3: 0.028 3‘, 4.32) —. (230.0) 0.133 :1: 0.016 (2: 3) (526.34) —. @3200) (3‘, 4.80) —. g‘, 0.0) 0.062 :1: 0.013 (1+, 6.34) —. (g‘, 0.0) 0.219 :1: 0.032 (2; 4) (3:, 6.48) —) at, 0.0) (g ,6.90) —. (g— ,0.0) 3+,750) —+ (3100) 128 (2+, 0.953) _. (1+, 0.0) 0.415 :1: 0.054f (z 5) (1-, 2.621) —4 (2-, 1.674) 0.43 :t 0.09 (z: 5) ((2216331) —.( 21:00:15); ) 0.28 :1: 0.04 (2: 6) 1‘, . -+ , . L 12C [ (2+, 4.44) —. (0+, 0.0) D1406 :1: 0.030 1 1 13C (3+, 3.854) _. (3.33.684) 0.070 a: 0.0091 (3*, 3.85) —. (y, 0.0) 0.370 3: 0.029 (2; 7) (323.68) —4 (50.0) (523.09) —. (5,00) l 55 Table 3.2: (continued) [_Fragment [ Transition (J",E") [ F. 21') 14C (l’,6.09) -9 (0+,0.0) (3',6.73) —+ (0+,0.0) (2+, 7.01) .—+ (0", 0.0) (2', 7.34) —+ (0+,0.0) 0.481 :1: 0.040 (2 8) 14N (0+,2.31) -+ (1+,00) 0.165 406% (1+,3.95) -—» (0+,2.31) 0.062 :1: 0.040 (0‘,4.92) —1 (1+,0.0) (225.11) -+ (1+,0.0) (125.69) —. (1+,00) (325.83) —> (11200) (1126.20) —> (1+,0.0) (3+,6.44) —» (1+,00) (2*, 7.03) —» (1420.0) 0.216 :1: 0.021 (E 9) 15N (325.27) —» (320.0) 325.30) —» (320.0) 0.391 :1: 0.042 (210) 327.16) -» (325.27) (3*, 7.57) —+ 325.27) 0.165 :1: 0.040 (211) 160 (326.13) —. (0+,00) 0.220 i 0.025 (226.92) —. (0+,00) (1-,7.12) —. (0+,00) 0.146 :h 0.024 (212) 180 (2+, 1.98) —. (0+,00) (2+, 3.92) —. (2121.98) 0.75 :1: 0.07 (E 13) (4+,3.55) _. (2131.98) (0+,3.63) .4 (2121.98) 0.27 :1: 0.06 (214) Chapter 4 Sequential Feeding and the Emission Temperature The relative populations of states of the emitted fragments provide a measure of the intrinsic excitation energy of the emitting system at freezeout. It is important to know whether this excitation energy is thermally distributed. This can be explored by direct measurements of the relative populations of exited states. However, the observed populations of excited states are influenced by the sequential decay of heav- ier particle unstable nuclei [Poch 85a, Xu 86, Hahn 87, F iel 87, Come 88, Morr 86] and the populations and decays of many of these unbound states are not known experimentally. Since one does not usually know the feeding corrections experimen- tally, they must be calculated. These calculations [Xu 86, Hahn 87, Fiel 87, Chen 88] usually make the simplifying assumption that the states [Ajze 84, Ajze 85, Ajze 86a, Ajze 86b, Ajze 87, Ajze 88] of primary fragments are populated according to a ther- mal distribution characterized by a temperature, T. The accuracy of this assumption must be checked by comparing the calculations to the experimental data. This chapter is organized as follows: In section I, we describe the essence of the sequential feeding calculation and how various fragments and their excited states are included in the calculation. We then present a method for choosing unknown 56 57 spectroscopic factors of low lying states, the primary populations of these states, as well as the branching ratios used in the decay calculations. In section II, we compare the results of the calculations to the inclusive elemental yields and to the isotopic yields of the detected fragments. In Section III, the calculated and measured values of the coincident 7-rays are compared. We first compare results for individual transitions. We then discuss a least-x2 squares fit method to extract an average emission temperature. Finally, in section IV, the experimental results are summarized and put into perspective with other similar measurements. I Feeding from Higher Lying States A Levels and Level Densities To determine the feeding corrections to the measured 7-ray fractions, we performed sequential decay calculations for an ensemble of nuclei with 33 Z _<_13. To facil- itate the actual numerical calculations, a lookup table containing excitation ener- gies, spectroscopic factors and different decay channels with corresponding branch- ing ratios for approximately 2600 known levels for isotopes within this charge range [Ajze 84, Ajze 85, Ajze 86a, Ajze 86b, Ajze 87, Ajze 88] was constructed. Since the spins, isospins and parities of many low-lying particle bound and un- bound levels of nuclei with Z S 11 are known, the information for these lighter nuclei was used in the sequential decay calculations. For known levels with incomplete spec- troscopic information, values for the spin, isospin, and parity were chosen randomly according to primary distributions obtained from the non-interacting shell model [Brow 88, N aya 90]. These calculations were repeated with different initialization for the unknown spectroscopic information until the sensitivities of the calculations to these uncertainties could be assessed. The results of the calculations appear to be 58 insensitive to details in the sampling algorithm, and essentially the same results were obtained in simpler calculations where spins of 0-4 (1 / 2—9 / 2) were assumed with equal probability for even A (odd A) nuclei, parities were assumed to be odd or even with equal probability, and the isospins were assumed to be given by the isospin of the ground state. For later reference, this latter distribution of unknown spins is termed a ‘flat spin distribution’. The low-lying discrete levels of heavier nuclei with 2212 are not as well known as those of lighter nuclei. To calculate the decay of these heavier nuclei for low excitation energies, E“ S 60(A5, Z), we used a continuum approximation to the discrete level density [Chen 88], modifying the empirical interpolation formula of ref. [Gilb 65b] to include a spin dependence: (215 + 1)eXPl-(Ji + $720.31 2(2J; + 1)exp[—(J,- + 3)’/2a,3]’ for E" S to, ME", 1:) = iexp[(E‘ - E1)/Tll T1 (4.1) where a? = 0.0888[a,-(co - Bani/1,3 , (4.2) and a.- = A; / 8; J5, A5, and Z.- are the spin, mass and charge numbers of the fragment, and the values for 60 = 60(A,,Z,-),T1 = T1(A.-,Z.-), and E1 = E1(A,-,Z,-) were taken from Gilbert and Cameron [Gilb 65b]. For Z Z 12, E0 = E0(A,-, Z,-) is determined by matching the level density at 60 provided by Eq. (4.1) to that provided by Eq. (4.3) given below. [Note: In Eq. (4.1) and also in Eq. (4.7) below, we match the density of levels rather than the density of states because the spins of many of the discrete levels are not known.] For higher excitation energies in the continuum for all nuclei, we assumed the level density of the form P(E‘:Ji) = p1(E‘)p2(J.',O’,'), (4'3) 59 where . _ eXP{2[ae(E‘-Eo)l‘/’} ”‘(E l ‘ lzfilaxv—Eorlma.’ (4'4) p2(J.-,a.-) = (2J:+1)exp[‘2-‘;(?Jg+302/203], (4.5) a? = 0.0888[a.-(E‘—Eo))]‘/’A?/3. (4.6) For Z.- Z 12, E0 = E0(A,-, Z5) is determined by matching the level density provided by Eq. (4.1) at £0 to that provided by Eq. (4.3). At smaller values of Z.-, E0 is adjusted for each fragment to match the integral of the continuum level density to the total number of tabulated levels according to the equation: f; dE‘fdJ p(E‘,J) = [0° dE‘ 25w —- E‘), (4.7) where 60, for these lighter fragments, was chosen to be the maximum excitation energy up to which the information concerning the number and locations of discrete states appears to be complete. An example [Chen 88a] of determining so for the isotope 20Ne is given in figure 4.1. To reduce the computer memory requirements, the populations of continuum states were stored at discrete excitation energy intervals of 1 MeV for E" 515 MeV, 2 MeV for 155 E“ _<_30 MeV, and 3 MeV for E“ 230 MeV. The results of these calculations do not appear to be sensitive to these binning widths. In this way, the total number of discrete energy bins including the discrete states came to be about 38,000. Parities of continuum states were chosen to be positive and negative with equal probability. To save both space and time, the isospins of the continuum states were taken to be equal to the isospin of the ground state of the same nucleus. 60 4o frrrTYY rIfTY V V Y Yfr‘ V V Trrff : 2°Ne Jr A L l A L_‘ A 30)- > p :3. r U! _ , a l 0 p G zor : '76 ’ ‘ > h 4 Q) _ 4 ...a _ . 10)- " :lfi' r p r P r<}-— , m _0 r In 5 10 15 20 25 30 Ex (MEV) Figure 4.1: The level density of 20Ne as a function of excitation energy [Chen 88a]. The histogram gives the number of known levels whereas the solid curve shows results of level density predicted by eq (4.3). 61 B Primary Populations For the ith level of spin J,- we assumed an initial population P,- given by P; O( P0045, Z.‘)(2J,‘ + l)exp(—E‘/T), (4.8) where P0(A.-, Z5) denotes the population per spin degree of freedom of the ground state of a fragment and T is the emission temperature which characterizes the ther- mal population of states of a given isotope. (This temperature is associated with the intrinsic excitation of the fragmenting system at breakup and is, in general, different from the “kinetic” temperature which may be extracted from the kinetic energy spec- tra of the emitted fragments.) The initial populations of states of a given fragment were assumed to be thermal up to excitation energy of Ezutofi = pA. This cutoff was introduced to explore the sensitivity of the calculations to highly excited and short-lived nuclei, some of which may be too short-lived to survive the evolution from breakup to freezeout. Calculations were performed for cutoff values of p = 3 and 5 MeV corresponding to mean lifetime of the continuum states of 230 fm/c and 125 fm/c, respectively [Stok 77]. The calculations were qualitatively similar for the two cutoff energies. For simplicity, we parameterized the initial relative populations, Po(A.-, Z,-) by Pom, Z) o< eXP(-fVc/T+ 62/7"), (4.9) where Va is the Coulomb barrier for emission from a parent nucleus of mass and atomic numbers A, and Z? and Q is the ground state Q-value Va = z.(z, — z.)eZ/{ro[A}/3 + (A, — A.)1/3]} (4.10) and Q = [3(Ap - Ag, Z, - Z.) + 3.] — B(Ap, Z,). (4.11) 62 We used a radius parameter of ro=l.2 frn, A,=122, Z,=54 (these values were assumed for compound systems due to incomplete fusion). The binding energies, B(A, Z), of heavy nuclei were calculated from the Weizsicker mass formula [Marm 69]. z2 C(A-ZZ)’ AV3— 3 A ’ with 00:14.1 MeV, (31:13.0 MeV, 02:0.595 MeV, and 03:19.0 MeV. For the emit- B(A, Z) = 00A — (1142/3 — 02 (4.12) ted light fragments we used the measured binding energies, 8;, of the respective ground states [Waps 85]. At each temperature T, the parameter, f in Eq. (4.9) was adjusted to provide optimal agreement between the calculated final fragment distribu- tions (obtained after the decay of particle unstable states) and the measured fragment distributions. This constraint reduced the possibility of inaccuracies in the predicted primary elemental distributions at high temperatures [Hahn 87, F iel 87]. The values of f obtained for different T are discussed in the last section of this chapter. C The Decay Branching Ratios The branching ratio for a state to decay by different channels has to be known for decay calculations. If known, tabulated branching ratios were used to describe the decay of particle unstable states. If unknown, the branching ratios were calculated from the Hauser-Feshbach formula, with additional constraints on isospins and pari- ties. The branching ratio for a channel 0 in the original Hauser-Feshbach formula is [Haus 52], I‘c G’C “I:- = 2 G.- (4.13) where Z=|S+jl l=|J+Z| Ge: 2 Z T7(E). (4.14) Z=]S—j] l=|J—ZI Here, J and j are the spins of the parent and daughter nuclei, Z is the channel spin, 3 and I are the intrinsic spin and orbital angular momentum of the emitted particle, 63 and T1(E') is the transmission coefficient for the 1th partial wave. By incorporating parity and isospin conservations, we can write Go as CC = < TI,DTI,FT(3)I,DT(3)I,F[T1,PT(3)I,P >2 Z=IS+jI l=lJ+ZI x Z: Z {[1+1rp1rp1rp(-1)']/2} T1(E). (4.15) 2:15- j| l=|J-Z| The factor, [1 + 1rp7r07rp(-1)']/2 enforces parity conservation and depends on the parities 1r = i1 of the emitted fragment and the parent and daughter nuclei. The Clebsch-Gordon coefficient involving T1,p,T 1,1), and T1], the isospins of the parent nucleus, daughter nucleus, and emitted particle, likewise allows one to take isospin conservation into account. For decays from states for which the kinetic energy of the emitted particle is less than 20 MeV and I S 20, the transmission coefficients were interpolated from a set of calculated optical model transmission coefficients [Brow 88, N aya 90]. For decays from continuum states when the kinetic energy of the emitted particle exceeds 20 MeV, the transmission coefficients were approximated by the sharp cutoff approxi- mation; T¢(E) = 1, for ($10 = 0, otherwise, (4.16) with to = (zw/hmlAE” + (A. — A.)‘/31\/24(E - v5), (4.17) where p is the reduced mass, and h is Plank’s constant. The calculation was restricted to the particle decays via 11, 2n, p, 2p, (1, t, 3He, and or channels. The 7-ray decay of particle stable states was taken into account in the calculation of the final particle stable yields. 64 II Elemental and Isotopic Yields The measured fragment elemental and isotopic distributions and the calculated distri- butions for p = 3 MeV are compared in Figs. 4.2 and 4.3. The solid points correspond to the fragment yields summed over all measured energies and angles. The dashed lines in Fig. 4.2 show the calculated elemental distributions of primary fragments (summation of all particle stable states for all isotopes of a given element) assumed for the temperatures T = 2, 4, and 8 MeV; the parameters, f, are indicated in the figure. The solid lines show the calculated final elemental distributions obtained after the statistical decay of particle unbound fragments. The parameter, f, was adjusted at each temperature so that the calculated final elemental distribution closely follows the trend of the measured elemental distribution. (After choosing appropriate but different values for f, very similar results were obtained for p = 5 MeV.) Since these parameters, f, have been adjusted to reproduce the elemental yields measured in this experiment, one must be very cautious in applying the results of these calculations to other reactions. The dashed, solid and dotted histograms in Fig. 4.3 represent final isotopic dis- tributions obtained for the three temperatures, T = 2, 4, and 8 MeV, using the parameters, f, given in Fig. 4.2. In general, the isotopic distributions are fairly well reproduced. For T=2 MeV, however, the calculated isotopic distributions are some- what narrower than the measured ones and for T=8 MeV, the calculated distributions are somewhat broader than the measured ones. The agreement is slightly better for calculations in the neighborhood of T=4 MeV. 65 MSU-88—29l b..gamflln..T--..,-...,.-..T... 10113 S+Ag E/A=22.3 MeV ‘g i ----- Initial i 5 Final 1 109g 1 i- T = 2.0 MeV 1 7: f= 0.74 10 r w .1 EU}. 10:;‘:5%].:.1] ‘4 ]~ ]::%:]::::]Afia:; z 10 _r “a 3 '~- T = 4.0 MeV ‘ O l— _ '1 Q 7: f— 1.3 10 g' ‘t‘ 1 f — -1—--- 1035441441: +5 -; . 1444345 10 g 1 i T = 8 MeV 1 101' l-..“- 1 E- |W 105.41 :1 “.TTT.‘ 1. 1 ..41. :1 .L‘ 2 3 4 5 6 7 8 Figure 4.2: Element yields summed over all measured energies and angles. The dashed and solid histograms show the primary and final fragment particle stable yields for the feeding calculations described in the text. The three panels show the results for T = 2, 4, and 8 MeV, respectively. The adjusted values for the parameter, f, in Eq. (4.9) are given in the figure. 66 COUNTS Figure 4.3: Isotope yields summed over all measured energies and angles. The dashed, solid, and dotted histograms show the final fragment distributions for the feeding calculations at T = 2, 4, and 8 MeV, respectively. 67 III Mean Emission Temperatures Starting from the initial distribution, Eq.(4.9), we have calculated the fraction, F.y , of 7-rays emitted in coincidence with a given fragment as a function of the emission temperature, T, which characterizes the ensemble of emitted fragments. The results for an excitation energy cutoff of p = 3 MeV are presented in Figs. 4.4-4.6 for the transitions given in individual panels. The range of calculated fractions, F.y , for individual 7-rays are bound by the solid curves in Figures 4.4 and 4.5 for transitions measured in this experiment. The range of calculated values for the relative 7-ray intensities, R, = F71 /F.,2, are shown by the solid lines in Fig. 4.6 for those fragments for which more than one 7-ray transition were measured. The corresponding calcula- tions for an excitation energy cutoff of p = 5 MeV are qualitatively very similar and in some cases, indistinguishable. The spread in calculated values of 171’s and 34’s shown in Figs. 4.4-4.6 reflects primarily uncertainties in the spins, isospins and parities of many low lying particle unstable levels which directly feed the particle stable states of interest. The range of calculated values was determined by repeating calculations with different spectro- scopic assumptions until the sensitivity of the calculation to those uncertainties could be assessed. In order to illustrate the modifications due to feeding from particle unbound states, the dashed lines in Figures 4.4-4.6 show the results of calculations which include feeding from higher lying particle stable states, but not from particle unstable states. In all cases, both F,’s and [74’s are predicted to be sensitive to feeding from particle unbound states for temperatures higher than approximately 2-3 MeV. (For example, for 11C with p: 3 MeV and T = 3 MeV, 55% of the yields of the 4.32 MeV excited state and 59% of the ground state yields are predicted to proceed through the sequential 68 decay of heavier particle unbound nuclei). The ratio R, has the advantage of being independent of the total feeding to the ground states of the observed fragments. Since the ground state is fed more strongly than the excited states, the ratios R, are slightly less sensitive to the uncertainties in the sequential decay corrections than the fractions F, shown in Figures 4.4 and 4.5. The shaded horizontal bands in Figs. 4.4, 4.5 and 4.6 correspond to the experimen- tal values of F, and Rh respectively, that are obtained when data for different interme- diate mass fragment kinetic energies and scattering angles are combined. In general, the experimental data are larger than the calculations for emission temperatures less than 2 MeV. For most transitions at temperatures of about 3 to 4 MeV, the range of calculated values lie within 20% of the range of experimental values permitted by our estimate of the experimental uncertainties. However, at these and higher tempera- tures, the calculations are not very sensitive to the temperature, making it impossible to extract reliable upper limits based on individual cases. Some of the transitions, e.g., the fractions F, for 8Li(0.98 —> g.s.), 15N(5.27 + 5.30 -> 9.3.), 16O(6.13 -i g.s.), and the ratio R, (=F,1/F,g) for 11C(6.34 - 7.50 -+ g.s. /4.32 —* g.s.), deviate sig- nificantly from the overall trends, with the ranges of calculated and measured values in disagreement by more than 20% at temperatures of 3-4 MeV. Such discrepancies could be due to inaccuracies in the spectroscopic information that influence strongly the calculations for these nuclei, or could be indicative of non-thermal excited state populations either in these nuclei or in heavier nuclei which feed these transitions by sequential decay processes. To provide a more quantitative comparison between calculations and experimental data, we have performed a least squares analysis. For each initial temperature in the 69 ‘33U-53- 3‘24 323+Ag E/A/é 22 3 MeV 02 WW/ I/ ;‘ "’Be(0.429-»g.s.) 0.0 P::¢:]:::;%::::%:::r+. :;:;I If: r+:;_;-I: 000 O 5 }///////.//I/ ’ I ///fl//// W ’ O ' 2 ' 10 0.1 Be(3 3733 8) 1°Iae(5. 95-»3. 37) 0.0 _+¢C¢]¢:Cf%¢t—t:%:t4¢ ‘ ‘¢‘:::‘l*‘~"‘*“% 00 0.10 W 0'2 //////////////// ///////// /////////////////// 0 25 8Li(0. 981-1. 5.) 11B(2.12:g.s) 0.05 0.1 0.00 . . 0.0 0.2 a: /////////// 0.1 0.1 0.0 - . 0.0 0.2 0.2 0.1 0.1 0.0 . 0.0 0.2 0.10 0.05 ‘ - 0.00 0.5 V///////l/// 12C(4. 44g. 5. ) 0‘ A2 4 6 80192 4 6 3 T(MeV) Figure 4.4: The solid curves indicate the range of calculated fractions, F,, for frag- ments decaying through the designated q—ray transition as a function of the emission temperature, T, which characterizes the ensemble of emitted fragments. (The values for F, on the curves are one theoretical standard deviation from the average value of F, provided by the calculations.) The dashed lines show the fractions calculated when feeding from particle stable states is included, but not feeding from particle unbound states. The horizontal bands indicate the measured values. 5 LL. Figure 4.5: The solid curves indicate the range of calculated fractions, F,, for frag- ments decaying through the designated 7-ray transition as a function of the emission temperature, T, which characterizes the ensemble of emitted fragments. (The values for F, on the curves are one theoretical standard deviation from the average value of F, provided by the calculations.) The dashed lines show the fractions calculated when feeding from particle stable states is included, but not feeding from particle 70 VSU-89-025 32 S+Ag E/A=22.3 MeV 05 -.,. «“3443. vvvvvvvvvvvvvvvv 0.50 ' %7/////f W ////// , """""" / (i “13 (0.953 -. gs.) ; 0-25 _ (2.621 a 1.674) 12B (1.674-2g. .), 00 ’--%:%t :t:].:. :1::;;1T; 243:]::::f:::4i:3:fi:jf*‘ 000 0.1 ~ I” IIII ”’ _‘ g;y////‘///' , , _ ‘ // my” ./7..:::::‘..::.:. .......... :::“.:::::‘ 0'5 t 13C(3.354»3.534) 13C(3-09-3-85) 1 00 ::¢,L.¢:.]::::3::4:IL¢ ..rt3;::r%.:::];fit%:‘o.0 0.5 0.0 f O . 25 / // (finial/”I/l/ll/fl/l/II/l/fl/l ' ’ , //////. 0.00 W“ ______ ¢¢:%‘*wa::t¢:::t‘ 0.0 * x" I” 0.5 0.2 //// W” 277W 2 ' ’2'“? // / // / "’N(7.16+7.57 15N 5.27+5.30)4 0.0 ,,-- --- ______ ;0.4 0-25 WWW/W /,..// // ///J 0.2 I is “—”"—‘ i . 0(6.13»g3 1 0 6.92+7.12 0.00 2 / (/ /% 00 """""" ' /",’/// // f// . ‘ 0(1.98»g.s.) 1°0(3.55»1.95) ~ (39241.98) (3.63-4.98) O shrirrnr -.--11--.1. +141-.1...11--.-1.70.0 O 2 4 6 8 0 2 4 6 8 T (MeV) unbound states. The horizontal bands indicate the measured values. 71 VEU-69-026 323+Ag 133/11: 22.3 MeV \\\\‘\s\ \g 5455“}- \g 1 5 \\\\\\ \\; \\\\‘ $\\ 10 - 0.5 5’ xxxxxxxxx xxxxxx xxxxxxxxxxxx 11C(6; .3-4 257.5.)141:1(4—3——2 703) 0.5 0.0 2. 31 0.50 " 4:4455253235 ' 3 W3\ 55N<——-—;:2:;§3 o_75 \\\\ 4414:...” \-" ‘\\\\ \l\\\\\\\\\\\\\\\ 0.4 05° ,6 6.92+712 180(555563) 0.2 0'25 O . 3 298196 0.000 2 4 6 8 0 2 4 6 s T (MeV) Figure 4.6: The solid curves indicate the range of calculated fractions, 17,: F,1/F,g, of designated 7-ray transition probabilities as a function of the emission temperature, T, which characterizes the ensemble of emitted fragments. (The values for R, on the curves are one theoretical standard deviation from the average value of F, provided by the calculations.) The dashed lines show the ratios calculated when feeding from particle stable states is included, but not feeding from particle unbound states. The horizontal bands indicate the measured values. 72 calculation, we compute the function, 1" e ,i- cal,i2 X:=;z(yxp y ) (4.18) 2 0'; i=1 where ye,”- and 316.1,,- are the i-th experimental and calculated values of the 7-ray fraction, F,, or the ratio of 7-ray fractions,R,,; V is the number of independent data points (11:28 for Fq’s, and u =12 for R,’s); and 0;, given by a? = 03”,.- + 0351.1": is an uncertainty associated with the comparison for the i-th measured quantity. In the latter expression, O'cxpd is the experimental uncertainty; 0d,,- reflects the range of calculated values corresponding to the different assumptions for the spins, isospins and parities of low-lying states where this information is incomplete. cred..- was computed as the variance of the calculations indicated in Figs. 4.4-4.6. Values for X: calculated for the 7-ray fractions in Figures 4.4 and 4.5 are shown on the right side of Fig. 4.7. Values for x3 calculated for the ratios of 7-ray fractions given in Fig. 4.6 are shown on the left side of Fig. 4.7. The solid and open circles depict the values of x?, obtained for excitation energy cutoffs of p = 3 and 5 MeV, re- spectively, when the unknown spectroscopic information for low-lying discrete states was chosen according to the noninteracting shell model. The open squares depict the values of X3 obtained for an excitation energy cutoff of p = 3 MeV when the unknown spectroscopic information for low-lying discrete states was chosen accord- ing to the simpler ’flat spin distribution’ described in section V. For all calculations, minimum values for X: are observed in the region of T = 3-4 MeV for both sets of measurements. The comparison involving ratios of 7-ray fractions may be slightly more accurate because such ratios are insensitive to uncertainties in the ground state yields, for which much more sequential feeding contributions are observed in these and other similar calculations [Xu 86, Hahn 87, Fiel 87, Chen 88]. This argument is supported by the reduced values of x?, indicated in the figure. For the comparison involving ratios of 7-ray fractions, the minimum value of X3 approaches unity, cor- Ififrcc 73 MSU-89-Ol6 32 S+Ag E/A=22.3 MeV 3 “I""InninnIfinrnrl"WI... ...I..nIW'finl..nl.n.ln"In. r 5 . .5 ‘5 " . 1: ‘ b I 1. F 2r ‘1 L «i N A "' N A >< . “ b T if p- 1?..- ‘L 4 4r 1 ‘ o a i m : 0 soft EOS : t Ostiff E03 5 2 — —- Liquid Drop Formula 5". Figure 5.2: The binding energy per nucleon for the mass range 30 _<_ A S 200 initialized at the beginning of the BUU calculations for both the stiff (circles) and the soft (squares) equations of state. The solid line indicates the results from the liquid drop mass formula. 90 = i“,(t)6t + 0(513) + . .. = 511;}? + 0(6t3) + - - - (5.29) a similar exercise can be performed to check the accuracy of Eq. (5.26) as well. The force F,, given in Eq. (5.26) must be derived from Eq. (5.17). It has the form, for example, for x-component F“, 762 0+2 7'93 +2 Fil = " 2: Z 216fi(2- [riu — KulliU(r?1 _11K2’K3) [(3:73 -1 K3=r°3 —216l‘=2 +U(7‘?1, K2, K3) — U(T‘?1 +1,K2, K3) — U(7‘?1 + 2, K2, [(3)] (5.30) where U is the mean field potential given by Eq. (5.2). The other notations were given in the previous subsection. By similar equations, one can compute the other two components 17:2, 17,3. clearly, F 75 —V.-U as was the approximation used in early calculations [see also (5.7)]. The precise calculation of Eq. (5.30) requires information about the mean field from 64 neighboring lattice points and this slows down the calculation of the force term considerably. During the simulation, the positions 1"} and momenta 13',- of the test particles are known at times which differ by 6t / 2. On the other hand, when one wants to calculate the total energy and various contributions to the excitation energy, one needs to know the positions and momenta at the same time. To achieve this in our simulation, we branch out from the main flow of the test particle propagation and move the position one half time step forward using a equation similar to Eq. (5.25) to match the time at which the momentum is known. Then we calculate the quantities of interest. The dependence of the conservation of energy on time step size 6t, as well as the stability of the nucleus propagated by the mean field, is discussed in section III. 91 D Two-Body Collisions Collisions between two test particles are only allowed to occur for test particles within the same ensemble. This reduces the number of computations and allow us to' use the collision cross section 0,". without reduction [Bert 88]. Successful collisions are allowed if the two test particles are close enough and if their momenta after scattering are not Pauli-blocked. If one of the test particles are Pauli-blocked, the original momenta of both test particles are restored. Details of the Pauli-blocking will be discussed in subsection E. To check whether particles are close enough, let us consider two test particles at (F1,fi'1;1"2,fig). The two test particles follow straight line trajectories between succes- sive time steps. For a collision to occur, the two test particles must pass by each other the point of the closest approach within a collision radius defined by r,m = (0,", /1r)1/’. This condition can be expressed by the following two equations 65.21 - 61721 5t _ < . — . 6v21 | _ 61221 2 (5 31) 2 6" «5" m, r21 021 '2 S 1.2 = 9;: (532) 6" 2— |7‘21I I 5021 7171 Where 671.21 = 172 — Fl, 61721 = ‘62 — 171 and 61221 = I172 - 171'. If the pair of test particles satisfies both Eq. (5.32) and Eq. (5.31), the momenta of the two particles are changed from (1')};13'2) to (133’ , [7'2 ') with“p1'and"pg’ given by 51' = 5.... .+ 2612;. (5.33) 52' = 131-m — $6223. (5.34) where 13;”, = %(p1 + 133) is the nucleon c. m. momentum and 6p’ 2, is the relative momentum assigned randomly according to an isotropic distribution with a magni- tude 6p21=|p2p"1|. This algorithm clearly conserves both momentum and energy. 92 Although it violates in principle the conservation of angular momentum for each in- dividual pair of test particles, our numerical simulations indicate that its influence to the total angular momentum is negligible. E Pauli-blocking After the pairs get their respective new momenta, it is checked whether the collision violates the Pauli principle. To do this, we build of sphere of radius 1' around F1 and a sphere of radius p around [2'1 ’ so that no test particles inside the phase-space means full occupation. Scaling with ground states (a phase-space of (47,1)2R3P2. is fully occupied by NmtA test particle) and with the relation r/ p = R/Pp one can get r and p, respectively, once no is given. Thus the occupation probability is calculated by 5071,51) = a: (5.35) no where 711 is the number of test particles inside the phase-space volume not including the test particle being checked at (1'1, p1 ’). Similarly, one can calculate the probability f2. The probability for this pair to collide successfully is calculated by P = (1 - f1)(1- f2)- (5.36) In our calculations, we choose no = 4 and thus, the radii of the spheres are r = 0.60 f m and p = 0.904 f m‘1 , respectively. This yields about a 80% blocking probability for the ground states of “Ca and l2“Sn nuclei. Probability closer to 100% would be preferred. Most of the collisions allowed for the ground states of 40Ca and 12‘Sn nuclei occur between test particles near the phase space boundaries of the nuclei. There, the Pauli blocker samples regions of phase space in which no test particles are found. Clearly, additional work on the Pauli blocking algorithm is needed. 93 III Ground State Stability and Conservation of Energy To check the stability of ground states produced by the Lattice Hamiltonian Method, we have performed extensive calculations for ”Ca and 12“Sn nuclei. We start the cal- culation by providing a nucleus in its ground state using the algorithm as described in section II (A). We then let the individual test particles (nucleons) propagate ac- cording to their mean field for a period of 300 fm/c with the step size 6t = 0.5 fm/c. Figs. 5.3 shows projections of the nuclear densities in the a: — 2 plane for ”Ca (left two columns) and l2“Sn (right two columns) ground state nuclei as a function of time in steps of 40 fm/c. In all our calculations the full potential includes the Coulomb field and the symmetry terms in addition to the isoscalar mean field. As shown in these figures, only a few test particles escape from the mean field over a period of of 160 fm/c. A more quantitative analysis of the calculations is shown in Fig. 5.4. The left column is the results obtained for “Ca and the right is that for 124Sn. The top windows display the binding energy per nucleon predicted by both the soft equation of state (open circles) and the stiff equation of state (open crosses) calculated from Eq. (5.15). One can see that both equations of state produce an effective binding energy B/ A 2 8 MeV. The binding energy obtained for calculations with the soft equation of state is somewhat larger since the corresponding potential is deeper at low densities. The number of escaped particles Acmpcd and the root-mean-square radius Rn,“ for both nuclei are plotted, respectively, in the bottom windows. The escaped particles are defined as those having a local density lower than 7%po. The root-mean-square radius includes all test particles. Once nucleon-nucleon collisions are turned on, spurious emission of nucleons oc- 94 curs due to the insufficient Pauli-blocking at the nuclear surface. With a nucleon- nucleon cross section of am, = 41 mb, the average emission rate is less than 8% over a period of 160 fm/c. The total energy is, nonetheless, well conserved. The dependence of the conservation of energy on the time step size 6t has been extensively investi- gated. In Fig. 5.5, we show the results for 4"Ca-l—‘mCa collisions using the mean field given by Eq. (5.2) with the soft EOS parametrization. AH is the energy difference calculated between t = 140 fm/c and t = O. For 6t 5 0.5 fm/c, the total energy changes by less than 0.1 MeV/ A during this time interval. We have chosen 6t = 0.5 fm/c in all our calculations. 95 MSU-Sl-O3O Soft eos Stiff 1505 Soft eos Stiff 50$ 40 40 124 124 . Co . Ca . Sn . Sn t=0fmlc 4O 80 o a . IZO o o 0. I60 Figure 5.3: The time evolution of “Ca (left-hand two columns) and 124Sn (right-hand two columns) ground state nuclei projected in the :r — 2 plane in step of 40 fm/c. The equations of state and the time at which the density is plotted are already indicated in the figure. 96 ‘. MSU-9l-O37 0* Ground State Properties, LH Method A 2 I I I I] I I I I I I I I I II I I I5 I I I I I I I I I I I I I I I I I I O :> - -35- -2 o x Stiff E08 ‘°C 55 ‘2‘ 5 -4 0 Soft nos a ":-" 8n -4 < -6 4;— -6 m -8 'E'm';§$§"i'§£§';§{ " warms-3.: —8 —1o —: -10 0.8 '8 “Ca 35 124Sn 0.8 g. 0.6 :: 0.6 g 0.4 0.4 J.’ 0.2 ‘g 0.2 0,0 : -l--‘R1t-~l"-"'9"" “wail-7373173. ‘, 0,0 3 5 ,0 ‘22—flgoo. ,. ‘ 4a.. -~ 5 a 4 Ca 4;— xx..‘_ xWx- -g.. 4 g 3 ‘ “ .Igp-l-‘Tg'fii-t-‘Q’Ai: 12‘Sn 3 a? 2 ":E 2 1 Mean Fifild Onlly 4;— Mean Field Only 1 O l l l L] 1 l l l L l l l l l 'I l l l l l l l l I 1 l l l l l _ o O 50 100 150 200 50 100 150 200 t (fm/c) Figure 5.4: Stability tests and the conservation of energy for the ground states of 40Ca (left column) and 124Sn (right column) nuclei. the top and the central windows display, respectively, the number of escaped particles Aumped and the root-mean- square radius Rm. (See the text). The bottom windows display the binding energy per nucleon predicted by both the soft equation of state (open circles) and the stiff equation of state (open crosses) calculated from Eq. (5.15). 97 MSU-Sl-OSS 4°Ca+4°Ca, E/A=4O MeV, b=1 fm 0.3 IIIIIfTII‘IIIITIFTIIIIITI AH=H(t=14O fm/c)—H(t=0) ~ Soft EOS, UNN=O * P d D h — .0 N AH/A (MeV) .9 H I 0.0 .41...1IL....I..L.IL1.L 0.0 0.5 1.0 1.5 2.0 2.5' 6t (fm/c) Figure 5.5: The step size dependence of the conservation of the total energy for 40Ca-f-‘oCa collisions with the soft equation of state. AB is the energy difference calculated between t = 140 fm/c and t = 0. Chapter 6 The Disappearance of Fusion-Like Processes and the Nuclear Equation of State Hot nuclei can be readily formed by the incomplete fusion of projectile and target nuclei in a heavy ion reaction. For moderate incident energies, the excitation energies of fusion-like residues increase with incident energy. At incident energies in excess of about E/ A = 35-40 MeV, however, vanishing cross sections for fusion-like residues have been reported, and interpreted as a manifestation of a bulk instability of nuclei at high temperatures. This interpretation may be unwarranted if very hot nuclei decay via unexpected decay modes, or if the reaction dynamics preclude the formation of very highly excited residues. Although this issue is not directly related to the questions raised in the experimen- tal study of this dissertation, early results of the Lattice Hamiltonian code prompted us to direct some efforts to the understanding of the dynamical limits to the residue formation. In this chapter, we will attempt to address 1) what can be learned about the nuclear equation of state and the in-medium nucleon-nucleon cross section from measurements of fusion-like residues; and 2) what are the dynamical limitations to the formation of hot composite nuclei. For such purposes, heavy residue cross sections 98 99 were calculated for ”Ca + 4(’Ca and ”Ar + 27Al collisions using the Boltzmann equa- tion. Qualitatively consistent with experimental observations, the calculated heavy residue cross sections decrease rapidly to zero for E/ A Z 35 — 40 MeV. The decrease in cross section does not appear related to a bulk instability of nuclei at high tem- perature. The calculated cross sections are quite sensitive to the in-medium nucleon- nucleon cross section and the nuclear equation of state (EOS) at sub-nuclear density. I Fusionlike Cross Sections and the Equation of State A 40Ca+4OCa Collisions at E/A :40 MeV The formation and decay of heavy residues is an important process at energies E/ A S 40 MeV. To illustrate our calculations for heavy residue cross section, we consider first the calculations for the “Ca + ”Ca system at E/ A = 40 MeV, performed for an isotropic nucleon-nucleon cross section of 0,... = 41 mb and for both soft and stiff equations of state. Figs. 6.1 and 6.2 show the projections of test particles on the x - 2 plane as functions of time for both stiff and soft EOS. For calculations with the stiff E05 and b=2 fm, one obtains a single fusionlike residue. For the soft EOS, on the other hand, one obtains two residues. The impact parameter dependences of the calculations for the two equations of state are shown in the left column of Fig. 6.3 where we plot the masses (upper left panel) and the component of velocity parallel to the beam axis (lower left panel) of heavy residues produced in the calculations. Two residues with 30 S A S 40 are produced at large impact parameters, b 2 3.3 fm, in calculations with the stiff EOS (open points) and the soft EOS (open squares). A single heavy residue is observed at small impact parameters, b S 3.3 fm, in calculations with the stiff EOS (solid points). For small impact parameters with the soft EOS, however, the projectile and target simply pass through each other, with 100 MSU-9l-03l 4°Qa+4°Ca,E/A=4OMeV,Stiff EOS,b=2 frn Figure 6.1: The time evolution of test particles for 4°Ca+4°Ca collisions at E/A=40 MeV and b=2 fm with the stiff E03 and a’,m = 41 mb projected in the 2: — 2 plane in step of 20 fm/c. 101 MSU-Sl-OZZ 4°Co+4°Co, E/A=4OMeV, Soft EOS, b=2fm : I, O t = O fm/c 20 4O Figure 6.2: The time evolution of test particles for 4oCa+“°Ca collisions at E/A=40 MeV and b=2 fm with the soft E03 and 0,". = 41 mb projected in the 1: — 2 plane in step of 20 fm/c. 102 MSU-90-O7O 4°Cot+4°Ca, E/A=4O M‘ov, oNN=41 mb _ Illllll .l. ...,..-.,... '1'“; 50 " 'ons — 60 “M . Stlf : a: : ‘/Siii§ ‘ 50 — t . A 1 ‘ — 40»: g ; Soft EOS t‘ j v <1 40 :- l i a Z 20 I t , _ 30 {-3 “fiat-53:3 : 20 'l "but " ["1' 4 0 0'3 7v """"""" Stiff nos —. 5 .- beam ’a’o HM : . ’a ’6 ......... _: A O .. ,3 p, .‘ ....... ‘.x E 4 ?o \02_G‘°'&'d’d MJS 2 >N -.__.__._..’ ........... Yen. ; v ifi-G-flfl\ ’/St.iff . Eo/A _g 2 fl 0 1 . \ n Re It (E°-E...)/A _3 1 m 3 Soft E08 ‘15:‘ . o (E'—E,,.,,)/A 5 .1....1....t....t....t....1.“In“? ..l....l....l....l.. - O 0 1 2 3 4 5 6 7 O 1 2 3 4 Figure 6.3: Observables calculated for the 4"Ca+ ‘OCa system at E/A=40 MeV as- suming am, = 41 mb. Upper left: Mean residue masses. Lower left: Component of the mean residue velocity parallel to the beam. Upper right: Mean residue angular mo- mentum for the stiff EOS. Lower right: Mean residue total excitation energy/ nucleon (solid points), after subtracting the rotational energy (crosses), and after subtracting the total collective energy (solid diamonds) for calculations with the stiff EOS. The lines are drawn to guide the eye. 103 their velocities and masses reduced due to insufficient nuclear stopping for the soft EOS at this energy. The right-hand panels of this figure will be discussed in the next section. B Sensitivity of Fusionlike Cross Sections to The Equation of State The energy dependences of the heavy residue cross sections for ‘oCa-t-‘OCa collisions with soft and stiff equations of state are indicated respectively by the solid squares and solid points in the upper half of Fig. 6.4. Each symbol (square or point) is obtained from the largest calculated impact parameter bma, which yields massive fusion-like residues; the upper edge of each vertical bar corresponds to the smallest calculated impact parameter by which yields distinct pro jectile- and target-like residues. These critical parameters, bma, and by , are listed in Table 6.1. As an example, Fig. 6.5 shows the time evolution of the bound test particles for 4oAr +27Al collisions at E/A=30 MeV at the critical parameters, bma, = 4.3 fm (left two columns) and by = 4.5 fm (right two columns). For a constant nucleon-nucleon cross section of 41 mb, the cross sections for fusion-like residues are larger for calculations with the stiff EOS. To see which part of the equation of state is responsible for the varying fusion cross sections, we performed calculations at E/ A = 40 MeV with equations of state having variable low- and high-density behavior. For example, we define a soft-stiff equation of state which follows the soft EOS at low density and the stiff EOS at high density. The parameterizations for this and the analogous stiff-soft equation of state are given in Table 6.2. The 4°Ca+“°Ca heavy residue cross sections obtained with these EOS’s, 330 :f: 30 mb for stiff-soft and 120 :1: 20 mb for soft-stiff, show that the residue cross section depends mainly on the low density EOS. 104 MSU-90-069 20 30 40 50 60 ' Soft EOS P..fi...,. .l....r...r. : 4oCa+"‘°Ca, UNN=41 mb 100° ." - Stiff E08 1 : ' Soft E08 1 A 500 '- 'j .o ; . é . iii 0_:.: P::}:::. .%.r: ri: b3 _ “Ar'i'ZI’rAl I i. 0 Stiff EOS L 500 '- 20 30 4o 150 60 Bub/A (MeV) Figure 6.4: Upper half: Residue cross sections for 40Ca+ 40Ca collisions. The ar— row indicates zero cross section for soft 1308. Lower half: Residue cross sections for 40Art-”Al collisions. The solid points and solid squares describe calculations with the stiff and soft equations of state, respectively. The lines are drawn to guide the eye. 105 MSU-9l-027 4°At+27m, EA/A=30Mev, Stiff eos b=4.3fm 4.3 4.5 4.5 o’ I .’ I t=0 fm/c IZO O l20 o «I . 0' 40 200 40 200 O Q . 0' so 240 so 240 Figure 6.5: The time dependent spatial evolution of 4°Ar+27Al collisions at E/A= 30 MeV at the critical impact parameters, bm, = 4.3 fm (left two columns) and by = 4.5 fm (right two columns). The time at which the density is plotted is indicated in the each panel. The free particles have been suppressed. 106 This sensitivity to the low-density EOS could have been anticipated from the qualitative study of ref. [Bert 78]. There it is shown that compression of nuclear matter in one-dimensional collisions is followed by a rarefaction, and the maximum tensile strength of the nuclear matter in the low-density phase depends on the EOS. Stiffer equations of state have higher tensile strengths, and so the tendency of the system to breakup into two or more fragments is less. A possible determination of the low-density EOS on the basis of the cross sections for residue formation is hindered by the fact that these cross sections are also sensitive to am. To illustrate the possible theoretical ambiguities, calculations were performed for the 4°Ar+ 27A1 system in which the value for am was adjusted separately for calculations with both stiff and soft equations of state to obtain residue cross sections of about 500 mb at Elab/ A = 30 MeV. These choices, (1) an": 25 mb and a stiff E03 and (2) 0,", =50 mb and a soft EOS, provide essentially equal residue cross sections at E/ A S 30 MeV. The critical parameters are listed in Table 6.3. The energy dependence of the residue cross sections predicted by both calculations is qualitatively consistent with experimental observations. Differences between the two sets of calculations at Blob/A Z 30 MeV may not be large enough to discriminate between different equations of state. One must also assess the differences that could arise from variations in the energy dependence in the in-medium nucleon-nucleon cross section. 107 Table 6.1: The critical parameters for fusionlike reactions in ‘OCa+‘°Ca collisions. The nucleon-nucleon cross section are taken to be am, = 41 mb for both calculations. EOS E/ A bm, (in (MeV) (fm) (fm) 20 6.0 6.3 stiff 30 4.5 4.8 40 3.3 3.5 60 1.3 1.5 20 5.8 5.9 soft 25 4.8 5.0 30 3.7 4.0 35 2.0 2.5 108 Table 6.2: Parameters used for the isoscalar nuclear Mean Field Set Label range A(MeV) B(MeV) 7 K(p = po)(MeV) 1 soft 0< p/Po _ 40 fm/c, suggesting much of the collective energy Econ is taken away by particle emission. Finally, the thermal energy E5”, which is of our particular interest, exhibits two maxima: one global maximum at tz 40fm/c and one local maximum at t 1,, = 120 fm/c (t In = 100 fm/c for stiff EOS). The maximum at tz 120 MSU-Sl-OBS 4°Ar+27AL E/A =30 MeV, b=0fm o— L/Etot I I .. /' Enucleon ‘ — 10 _ Efermi '— Eint(T=O'Ares) ‘ -20 Energy/ nucleon (MeV) Soft EOS am=5o mb _30 ._ _ 1111i1111l14‘11l1411 0 50 100 150 200 t (fm/c) Figure 6.8: Decomposition of the various excitation energies as a function of time for 40Ar-f-T’Al collisions with the soft equation of state at E/A=30 MeV, b=0 fm. The bottom line is the nuclear potential energy. From this bottom line up are,respectively, Coulomb energy (difference between the second and the bottom lines), Fermi energy required by the Pauli exclusion principle (difference between the third and second lines), kinetic energy of emitted particles (difference between the fourth and third lines), collective energy of bound nucleons (difference between the fifth and fourth lines) and thermal energy (difference between the top and fifth lines). The freezeout time is indicated by the dotted line. 121 MSU-9l-O4O 4°Ar+27AL E/A =30 MeV, b=0fm I I I I I I I T I I I I l I I I l ‘ 0 /Etot ;;=:% E . th E nucleon ‘ d .1 fl _ ... E . 10 Eint(T=o'Ares) fem ‘ Energy/ nucleon (MeV) Vc Stiff EOS am=25 mb __ -30 _' l [ l l l l l l l 0 50 100 150 200 t (fm/c) Figure 6.9: Decomposition of various excitation energies as a function of time for 40Ar+27Al collisions with the stiff equation of state at E/A=30 MeV, b=0 fm. The bottom line is the nuclear potential energy. From this bottom line up are,respectively, Coulomb energy (difference between the second and the bottom lines), Fermi energy required by the Pauli exclusion principle (difference between the third and second lines), kinetic energy of emitted particles (difference between the fourth and third lines), collective energy of bound nucleons (difference between the fifth and fourth lines) and thermal energy (difference between the top and fifth lines). The freezeout time is indicated by the dotted line. 122 40fm/ c is an artifact of the initial momentum distributions, in which the longitudinal velocities of the projectile and the target nuclei cancel each other, causing a minimum in the computation of the collective energy. At the second maximum, the initial preequilibrium stages have finished and residue has already contracted to a more compact spacial configuration and the thermal energy at its local maximum. After this time, the thermal energy gradually decreases. Due to the evaporative cooling, we take the freezeout time to be the time of the second maximum in the thermal energy. This time is consistent with the time determined by the change in the nucleon emission rate shown in Fig. 6.7. It is interesting to note that the freezeout time is largely determined by the re- laxation time of the surface of the residue. Residues calculated with stiff equation of state, which has a larger restoring force and a larger sound speed, contracts to a compact configuration more rapidly than the residues calculated with soft equation of state. The excitation energies left in the residues are higher for residues characterized by a stiff EOS because they have less time for preequilibrium cooling. Momentum Distributions A third measure for defining the freezeout time may be obtained by the quadrupole moment of momentum distribution [Cass 87, Baue 87]: szu) = (2,1,), / dardspap: - p: — 123mm) (6.19) This criterion is motivated by the belief that a system in thermal equilibrium should satisfy sz = 0. To see how this variable changes with time, we plot Q22 in the bottom panels of Fig. 6.10 as a function of time for 4°Ar+27Al collisions at b=0 fm. For comparison, we show the emission rates in the top panels. The left hand panels show results obtained with the stiff equation of state. The right hand panels show 123 MSU-Sl-O4Z “Ar+27Al. E/A =35 MeV, b=0 frn 2': 6 _ 1—1—1 1 1 I 1 1 1 1 1 1 I 1 I r 1 I 1 B .2. 9 , ‘- ! _‘ § 5 E 1 : Stlff EOS :: I“ :30le E03 1 5 {I 4 :_ p. g am=4lmb If “1 iiar,,,,=50mb_- 4 3:; 5 .' 1 g 35 «,1 f i s *3 3 :— i 1 i :: ' i i -‘ 3 a: : I “ ! .. ,' ' i 3 _ l 1 “L. ' ' 3 2: ' ‘ 30' ‘5: ' l i q _z 2 out I- .I l .I \ «~- + l i I q 31 1 L10 ‘4 ‘ o .‘0 'l'__ 1 - .0 ‘ - g t I Ii " V ‘3" ‘3 II ‘0‘? 1 m 0 3“ l L L I l J l 1 L l l :: l 1 l l ‘I‘L l l l l : :I I—I I I I I I I l I I :: I I I I z I I I r I .4 O C tfreeze-out 3: tfreeze-out. j 2 — i '— i '1 2 «I: : i :: ‘ { - ! .. I > 1 .— ; 1.— “: 1 0 r- - "' '1 8 : 3 i: ‘ .... O :_ ........... :. .............. 4:. ....................... J 0 f3 - ' «- 1 : °' : i 2: i : —1 :- Stifl: EOS “j:- Soft :EOS j —1 : 1 1 1 1 L 1 1 l l l l l :- l I 1 1 I f 1 1 L I L 1 j 0 100 200 O 100 200 t (fm/c) Figure 6.10: The emission rates of nucleons (top panels) and the quadrupole mo- mentum distributions sz (bottom panels), defined by Eq. (6.19), for 40Ar+27Al collisions at E/A=30 MeV, b=0 fm. The left panels (right panels) show the results for the stiff EOS (soft EOS). The vertical dot-dash lines indicate the freezeout time discussed in the text. The dashed lines in the bottom panels include the calculations for all nucleons, while the solid lines include only nucleons in the bound residues. 124 those from the soft equation of state. The dashed lines include all nucleons while the solid lines include the nucleons bound in the residual nuclei. Clearly at the t Ire defined previously, Q 22 is significantly reduced from its initial value at t=0. However, the values of Q 22 continue to oscillate about zero for a long time after thermal freezeout, reflecting the existence of macroscopic quadrupole vibrations. Such long term collective vibrations render Q 22 less useful in defining the thermal freezeout time. In summary, a consistent freezeout time was obtained by checking three different variables. In the next few subsections, we will study the excitation energies and the angular momenta of residual nucleus at freezeout. C Collisions at E/A=30 MeV In Fig. 6.11, we show different contributions to the excitation energies of residues at freezeout as functions of the impact parameter for 4°Ar+ 27 Al collisions at E/A=30 MeV assuming alternatively the stiff (lower panel) or the soft (upper panel) equations of state. ( A similar analysis was shown in lower right hand panel of Fig. 6.3 for 40Ca+“°Ca system at E/A=4O MeV assuming stiff EOS.) The solid symbols in the figure represent the calculations in which a single heavy residue is observed in the final state. The open symbols represent calculations at larger impact parameters in which the system breaks up into projectile-like and target-like residues at a later time. The total excitation energy E" (solid circles), calculated from Eq. (6.10) increases slightly with impact parameter. This increase can be partly attributed to the collective rotation. The crosses in Fig. 6.11 depict the excitation energy after the rotational energy E‘ see Eq. (6.15), has been subtracted. The remaining part, rat 1 E‘—E‘ m, becomes constant for the central collisions where heavy composite residues were formed. We note here that the rotational energy E:o,, indicated by the difference 125 MSU-SO-ZIO “Ar-+2711, E/A=30MeV fl IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 4' E- ’ 0 E. _i .. G . : :: . 1 3 :- .,..XE -E,5;: A 2 ENNo-“muEI-o—é I: 2 RN ”Econ: 2 1 :- ~ . '2 v : Soft EOS, am=50mb ‘3 E , 1 <1 .. \ 0 - . 1'1 5 *3 E' -: 4 ..XE'—E,;;-; 3 NOE.-Emé 2 -Em;g B E. ._:, 1 sun nos. am=25mb u:- 5 olllllllllLllllllLllLll[ILLILIIIII " 0 1 2 3 4. 5 6 b(fm) Figure 6.11: Decomposition of the predicted excitation energy at freezeout for differ- ent impact parameters in 40Ar+27Al collisions at E/ A =30 MeV, assuming the soft EOS, (top panel) or the stiff EOS (lower panel). The solid and open symbols are explained in the text. The solid lines are drawn to guide the eye. 126 between the points and crosses, increases with impact parameter until reaching the maximum impact parameters for fusion, it then becomes constant at larger impact parameters. The constant behavior of :0, at large impact parameters can be expected since the moment of inertia I scales as I ~ I)”, and the angular momentum J scales as J ~ I) at large impact parameters. Thus from Eqs. (6.15)-(6.17), one would expect constant values of Ejo, at larger impact parameters. The compressional energy, which corresponds to the difference between the crosses and the diamonds shown in Fig. 6.11, exhibits little dependence on the impact param- eter. The collective energy E;.,_, indicated by the difference between the diamonds and the squares, remains roughly constant for central fusionlike reactions at impact parameters at b 5 4 fm. This constant value for fusionlike residues reflects the energy stored in macroscopic vibrations which may have a significant monopole components [Rema 88]. At larger impact parameters, b 2 4 fm, the collective energy EL. in- creases with impact parameter suggesting an incomplete dissipation of the incident collective motion of projectile and target nucleons. It is this incomplete stopping, not the thermal instability, which cause a decrease in the residue cross section as the incident energy is raised. The thermal excitation energy E5“, designated by the squares in Fig. 6.11, de— creases slightly with impact parameter. Thus in these dynamical calculations, the formation of heavy residues for “Ar + 27Al at E/ A S 30 MeV does not appear to be limited by the stability of the residual nucleus at high temperature. Indeed, in larger impact parameter collisions, where the residue formation is less likely, the intrinsic thermal energies are somewhat smaller. Experimental investigation of excitation energy have been based on the mas- sive transfer models [Lera 86, Auge 85, N ife85, Goni 89, Wada 89, Deco 90, Grif 90, Fahl 86, Bour 85, Gali 88] in which the measured residual velocities were used to es- 127 timate the excitation energies. In Fig. 6.12, we show the comparisons of the massive transfer models with the BUU calculations. The open points depict the results of massive transfer models (for details, see Appendix B) using the residue velocity from the BUU calculations. The massive transfer models significantly over-estimate the total excitation energy, suggesting the present BUU calculations are inconsistent with expectations of massive transfer models. D Limiting Angular Momenta Fig. 6.13 shows the total angular momenta for residues, obtained for both the stiff equation of state (solid circles) and the soft equation of state (squares), as a function of impact parameter for 40Ar +27Al collisions at E/A=30 MeV. The angular momentum increases linearly with impact parameter to a value of Jm, z58 h at b=4.3 fm for stiff EOS (Jm, =44 h at b = 4 fm for soft EOS), comparable to the maximum orbital angular momentum predicted by the liquid-drop model [Cohe 74, Ring 80] for mass A=56 (A=52 for soft EOS). Similar results were shown in the upper right panel of Fig. 6.3 for “Ca +4°Ca collisions with a stiff EOS. This suggests that the formation of a residue at E/ A = 30 MeV may be partially limited by the maximum angular momentum that a nucleus can sustain. To examine how the maximum angular momentum evolves with incident energy, we display, in Fig. 6.14, the energy-dependent residue masses (top windows) and the maximum angular momenta (bottom windows) for both the soft equation of state (right-hand side) and the stiff equation of state (left-hand side). At each energy, the solid symbol corresponds the maximum angular momentum 1m” which occurred at the largest impact parameter bma, for which a fused residue is observed in the final state. The Open symbol corresponds to the minimum angular momentum J11 calculated at slightly higher impact parameter b” for which the system breaks up 128 MSU-91-044 40.A.r+z".111,El2/A=3O MeV -1’1111l1111 1111 1111 111- ““A---Al"" ““-AE' ur ME. Soft EOS, am=50mb E'/A(MeV) Opmwammqmpmwammqm . ---A ----- ‘A""AE'm. ME. Stiff EOS, UNN=25mb LlllLLLLLLlILlllIllllllll 0 1 2 3 4 b (fm) Figure 6.12: Comparisons of the total energy from BUU with that from massive transfer models in 4°Ar+27Al collisions at E/ A =30 MeV, see the text for the details 129 MSU-SI-OSZ wAr+27AL E/A=30MeV. t=1201m/c 80 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII U 550 60 ’ I l l l l I l l l l I l l l l l l l llllllllllllllllllllllllllllll 0 1 2 3 4 5 6 b(fm) Figure 6.13: Angular momenta of residues as a function of impact parameter for calculations with the soft EOS (squares) and the stiff EOS (circles). The solid and open symbols are discussed in the text 130 MSU-90-ll2 40Ar+27Al _"'l""l"”l""l"':_"'l'”'l”"1""l'“ 603- :— . —§ 60 : I :- . g : m 50_— ' u {E— o —' 50 § 5 Soft EOS . :: Stiff E08 5 4o 70m=50 mb {—oNN=25 mb -{ 40 A 2..:(tmlttnlnnlmsgmitttqlunlunlv'j a so"— —;5—°\2\ -3 so 1 ._ D 55 3 ..8 4°; - ‘3? O. ‘3 4° 20;. 2:;— . : 5 Soft EOS :: Stiff E08 9 7 20 O-..1....1....I....1...:5...1....11...1....1...‘ 25 30 35 4o 25 30 35 4o Elub/A (MeV) Figure 6.14: Residual masses (top) and angular momenta (bottom) at the freezeout as functions of the incident energy for calculations with the soft (right side) and the stiff EOS (left side). The curves and the open and close symbols are explained in the text. 131 eventually into two pieces. These boundary values, 6",”, Am”, In.” and Jm“, and, by, In and J11, were listed in Table 6.3. For both sets of equations of states, the fusion-like residues are comparable to the maximum angular momenta (solid curves in Fig. 6.14) predicted by liquid-drop models [Cohe 74] at E/ A S 30 MeV. At higher energies, E/AZ35 MeV, however, the maximum angular momenta from BUU calculations decrease more rapidly than that expected from liquid-drop model calculations, suggesting that the liquid- drop model calculations provide little theoretical guidance at high energy collisions. E Limiting Excitation Energy Much effort has been devoted to the determination of the maximum excitation energy that a metastable composite nucleus can sustain. To learn about the possible dynam- ical limitations to the residue excitation energy, it is interesting to see how the calcu- lated total excitation energy evolves with the incident energy in fusionlike collisions. On the left hand side of Fig. 6.15, we show the decomposition of the excitation energy for central collisions as a function of incident energy for the soft EOS (top panel) and the stiff EOS (bottom panel). On the right hand side, we provide the corresponding calculations for the maximum impact parameters bm, that lead to residue forma- tion. With both equations of state, the calculated total excitation energy and the thermal excitation energy increase slightly with incident energy, a phenomenon also predicted in other simulations of light systems. [Snep 88, Boal 88a, Boal 88b]. The calculated excitation energies are generally larger for calculations with the stiff EOS, a trend also predicted by static models, [Levi 84] even though 0,". was adjusted to make equal residue cross sections for the two sets of parameters. The total excitation energy for the stiff EOS increases gradually from E‘/A= 3.8 MeV at E/A=25 MeV to E‘/A=5.5 MeV at E/A=40 MeV; the thermal energy increases correspondingly 132 MSU-Sl-O45 Ener Dependence of excitation ener y forwh-t-"Al III'IIrIlrITIl’II I IIIII II 1 l - b-O tn 1 E. b‘bmu IIIIIII' llLJlll M .11. 1F \>\ IIIII'IIII II II 1III I l «r- I :1- .1- F q d q- 1. I. .1- b «1- M ‘ . ‘b I «11- 1-1 - llljlllllllll’ll Ill 0 as... p E ”I I i ii“ [1...]!111111 N (D Soft E08. cal-50ml) Soft mos, ”unsanb 1 111111 I'ILLlllllll I'IIII1TIIII' II‘IIIII IIII b 1. 111111 1P - q E°/A (MeV) 10 (1) 1% 0| 05° 0' 1" M o IIIIIITII'IITIIIIII IIII IIII II 1 I I F b p n I b b F b CI I- I I f” r .l l I ll llLl I‘LL II to ti 3 «1111 lllllllll ll jllllllllll llllllllll _ l. NUIFGGO Stiff EOS. au-zsmb Stiff EOS. Un'ZGmb llllllIlllllLllllll 111 '11 I llllllll 25 30 35 4o 45 25 30 ‘35 4o 45 H IIII'IIIIIII 1J1 C Figure 6.15: Decomposition of the excitation energy at freezeout for b=0 fm (left side), b=bma, (right side), soft EOS (top panel) and stiff EOS (lower panel) as functions of incident energies in 40Ari-”Al collisions. Here, the solid circles and squares denote the total and thermal excitation energies, respectively. For other symbols, see the text and the figure caption of Fig. 6.11. The values of BM, were listed in Table 6.3. 133 from Ban/A: 2.4 to 2.8 MeV. The maximum predicted thermal energy, Ea, / A =2.8 MeV, is not small compared to predictions for the maximum excitation energy that a non-rotating nucleus can sustain. Thus it is possible that, besides limited by dynamic effects in large impact collisions, additional reductions in the calculated residue cross sections may occur for central collisions at the highest energies due to thermal insta- bilities [Boal 88a, Boal 88b, Levi 84, Saga 85, Gros 88] of the hot residues which are not considered by our calculations. Similar excitation energies have been estimated from the experimentally measured residue velocity distributions using massive transfer models. An analogous procedure using the calculated residue velocity provides the open points in Fig. 6.16. As also shown in Fig. 6.12, application of the massive transfer to BUU calculations greatly model overestimates the residue excitation energy of this light symmetric system at all energies because it underestimates the cooling due to preequilibrium emission. Part of this discrepancy may also be due to the fact that the present calculations seem to lead to residue velocities which are smaller (for reverse kinematics) than the measured ones. This discrepancy could be reduced by choosing smaller values of nucleon-nucleon cross section that lead to residue velocities which are more similar to the measured ones. A detailed study of this issue would require a large amount of computer CPU time and it is beyond the scope of the present study. III Conclusions In summary, calculations have been performed with the Boltzmann equation to as- sess the sensitivity of heavy residue cross sections to the E08 and the in—medium nucleon-nucleon cross section. For specific choices of 03m and the nuclear EOS, the calculated residue cross sections decrease and eventually vanish for incident energies 134 MSU-9l-O46 I I I T rrr I I I I I I fr I I I I L; I I I I I l' I WV I I I L 1° b b=0 frn [91:3, 3: babmax , or... ; 1° 1- / <1- , q 8 :- , , ”Y J:- , ”3’ -€ 8 I- B j!- e’ u: c , ’ 1: , ’ 1 6 r 9’ ‘.:" °' 1 6 I 1: 1 D ‘- . d 4 r /3. 1;— /t 1 4 r d. 1 S‘ 2 L J:- ‘ 2 § ; Soft nos. o.-50mb 3; Soft nos. o.-50mb ; V P ' '1 .lLl A l l l l 1 L1 1 l4 1 1 l [144 L 1. L 1 LL] LL 1 1 L1 1 l 1 LJ 1 ‘ < 10 r I l r rI I l I I I I 1' fl I I I :I I I {L I I % I I I I lrr fI I }I I er 10 III 1. b=0 1m ’08 n j: babmu ’03.“ : .. / <1- ; '1 n 41- 0' q s r , , I“ "— , , ’ -; a 1- B «1- E It 6 :- 0“ E' ‘0- G E' 1 6 t f: ./’—_'/. : 4 r 1.— -: 4 2 E- 31111 nos. urn-25ml: {- Sttrt’ nos. «...-sz1: -: 2 P db 1 1- ulb '1 O 1 Lil l l l A L L] I l 1 LI 1 l l l l l L l LLI 1.1 l l I 1L1 LL]. 1 l l l l I o 1% Ol 25 so 35 4o 45 25 so 35 4o Eu/A (MeV) Figure 6.16: Comparisons of the total energy from BUU with that from massive transfer models in 4"Ar+27Al collisions at freezeout for b=0 fm and b=bm, with the soft E08 and the stiff EOS as a function of incident energy. see the text for the details 135 above E/ A _>_ 35 MeV, consistent with experimental observations. This decrease in cross section does not seem to be related to a bulk instability of nuclei at high tem- perature. The calculated residue cross sections are sensitive to both the nuclear E05 and the nucleon-nucleon cross section. This dual sensitivity constitutes an ambiguity which may be reduced or eliminated by measurements of observables like the in/outo of-plane ratio and the mean transverse momentum that are related to the isotropy of the emission patterns of coincident light particles. By using a decomposition technique for the excitation energy, we have investigated in detail the thermal and dynamical limitations for the formation of heavy residues formed in the 40Ar-l-T’Al collisions. For a given incident energy of E/ A S 30 MeV, the calculated excitation energy is slightly lower than the maximum values extracted from experiments and the cross sections are mainly limited by dynamical considerations in large impact parameter collisions. At higher energies, however, the calculated maximum angular momentum decreases much faster than that predicted by static model calculations, indicating large dynamical effects. Moreover, the thermal energy increases from Ban/A: 2.4 to 2.8 MeV as the incident energy is raised from E/A=25 to 40 MeV. The maximum predicted thermal energy, ED" / A =2.8 MeV, is comparable to predictions for the maximum excitation energy that a non-rotating nucleus can sustain. Thus it is possible that, besides limited by dynamic effects in large impact collisions, additional reductions in the calculated residue cross sections may occur for central collisions at the highest energies due to thermal instabilities [Boal 89, Levi 84, Saga 85, Gros 88] of the hot residues which are not considered by our calculations. The present calculation has several limitations. Because the theory has insuffi- cient fluctuations, it can not predict, for example, under what conditions and how the hot residues will disassemble. Even within the model itself, there are considerable uncertainties concerning the nuclear E03 and in-medium nucleon-nucleon cross sec- 136 tion. Further investigations are also required to assess the sensitivity of the calculated observawa to the detailed algorithm for Pauli- blocking and to the surface energies of the computational nuclei. Chapter 7 Nuclear Temperature and Nuclear Equation of State In the preceeding chapters, we presented experimental measurements of emission tem- perature obtained from the relative population of excited states of intermediate mass fragments. In chapter 5 and 6, we discussed BUU calculations which were under- taken to determine whether excitation energies of residues are sensitive to theoretical quantities such as the nuclear equation of state and in-medium nucleon-nucleon cross section. Some sensitivity of the excitation energies to these quantities was obtained for light systems and they were presented in chapter 6. Such light systems may not be ideally suited to address our original questions about limiting temperatures since the residues formed in such light systems do not survive collisions for incident ener- gies in excess of E/ A z 40 MeV. Emission temperatures, on the other hand, have been obtained for heavy asymmetric systems at incident energies up to E/A=94 MeV [Chen 88a]. Here we present Boltzmann-Uehling-Uhlenbeck (BUU) calculations for the asymmetric 40Ar+27Al and 4°Ar+1“Sn system. We show that the calculated thermal temperature for the residual nucleus is sensitive to the nuclear equation of state, as well as the impact parameter, and surprisingly, in-sensitive to the in—medium nucleon-nucleon cross section. 137 138 This chapter is organized as follows. In section I, we check whether the criteria used to define the freezeout time are also satisfied for this heavy asymmetric system. We discuss the dependence of the total excitation energy on incident energy and compare the results with the predictions of massive transfer models in Section II. In Section III, we consider a simple model for extracting the emission temperature from the thermal excitation energy. Some conclusions are drawn in Section IV. I Freezeout Conditions Before we present the calculated excitation energies and temperatures, we would like to check whether the freezeout conditions discussed in the previous chapter give consistent freezeout times for 4°Ar+1243n collisions. Figs. 7.1-7.4 show the decomposition of the excitation energy using Eqs. (6.1)- (6.6) for 4°Ar+1245n collisions at b=0 fm. In Figs. 7.5-7.6, we display the emission rate of nucleons as functions of time for 40Ar-l-lMSn collisions at E/A=35 MeV and 65 MeV, respectively. The solid (open) circles depicted the calculations with the stiff (soft) equation of state. Both calculations are performed assuming an isotropic in-medium nucleon-nucleon cross section of 41 mb. Similar to 4oAr +27Al collisions shown in the last chapter, one sees a prompt non-equilibrium peak at t 2 60-80 fm/c followed by a lower emission rate characteristic of slow statistical evaporation from a equilibrated system. The solid arrows in the figures indicate the freezeout times we choose for the stiff EOS while the open arrows indicate the freezeout time for the soft EOS. By examining the thermal excitation energies shown Figs. 7.1—7.4 and the emission rates shown in Figs. 7.5-7.6, consistent freezeout times were achieved. The freezeout time t In for 40Ar+124Sn collisions at b=0 fm are, respectively, tfre z 140 (the a: 120) at E/A =35 MeV; and t," z 160 (t1,e z 120) at E/A=65 MeV; for 139 MSU-9l-O47 4°Ar+12‘3n, E/A =35 MeV, b=0fm I I I I l I I I I l I I I I I I T I I O *- /Etot -— f; 0 E f g ‘10." E...(T=0.A...) Bum. T o _ . . a - tl'free;ze-out « E —2o M H Q) I: [a] Soft EOS _ " UNN =41 mb ‘ . llllJLLllLkLLlllllL 0 50 100 150 200 t (fm/c) -30 Figure 7.1: Decomposition of various excitation energies as a function of time for 4oAr+1243n collisions with the soft equation of state at E/A=35 MeV, b=0 fm. The bottom line is the nuclear potential energy. From this bottom line up are,respectively, Coulomb energy (difference between the second and the bottom lines), Fermi energy required by the Pauli exclusion principle (difference between the third and second lines), kinetic energy of emitted particles (difference between the fourth and third lines), collective energy of bound nucleons (difference between the fifth and fourth lines) and thermal energy (difference between the top and fifth lines). The freezeout time is indicated by the dotted line. 140 MSU- 91-048 wAr+12‘Sn, E/A =35 MeV, b=0fm I I I I -( o /Etot . : A ' E 3 Enucleon % - coll V 1 Z a - g _ g ‘10 T E E...(T=0.A...). 2 .- fem - g - tfreeze—out. - i ' ' ‘ >~. -20 — Q0 . S .. :5 - l Stiff EOS V « 2 n..— ‘30_ UNN=§41 mb . Ill 0 50 100 150 200 t(fm/c) Figure 7.2: Same as Fig. 7.1, but for 4oAr+124Sn collisions with the stiff EOS at E/A=35 MeV, b=0 fm. 141 MSU-9l-049 40.es.r+‘2"Sn, E/A =65 MeV, b=0fm ""l""l""l§'r' P /Etot 5 E. A O the nucleon I > . § j G I Efermi ‘ 8 —1o — a '8 - Eint(T=OtAres) g: . fi - . >s ' = 4 2," -20 — Vn ‘3 o - t ; LE] - freese—out ' v Soft EOS = . -30 C O’NN =41 mb j m. L L l . . . . l . . . . l i . . ‘ O 50 100 150 200 t (fm/c) Figure 7.3: Same as Fig. 7.1, but for 4oAr+12“Sn collisions with the soft EOS at E/A=65 MeV, b=0 fm. 142 MSU-Sl-OSO “AH“‘Sn. E/A =65 MeV, b=0fm I ' g ' ' l ' r ' r j I I I I I I I I I ' /Etot Enucleon Efermi - Energy/ nucleon (MeV) tfreeze-out —so L Stiff EOS am, =41 mb- '....l...L|.;Ln|.ff 0 so 100 150 200 t (fm/c) Figure 7.4: Same as Fig. 7.1, but for 4°A1'+12“Sn collisions with the stiff EOS at E/A=65 MeV, b=0 fm. 143 MSU-SI-OSI 10 4°a£+T12418n,'153/A =35 MeV, oNN=41 mb : I 8 ft E10; I 1:! I I I l I r r I I y : 10 : 0 o “ - M _ ‘ 1:— _‘ 3 8 : gag, oStiff E08 '3: 33"... b=2 fm : 8 : : \O I: : b ...: 8 6 :- t'-' b. 1.— ’-' "'0 '1 6 “5 : 0' ‘\'-. 1: f I '-. : 9.4%? PM“ ::—,° -—: 4 \ I i .‘i (R _ :: 3)., ° : "" '— 95 \ \° 61- 4:. cf 3 _- 2: 2 .3? , I! b O ‘8“) Eliot \. 0.0.3./ \8 3 2 4.) '- 8 61 1 I 1.1 '1’- 1 1 1 1 l 1 1 1 1 l 1 1 I g 0 : F f I I I :C I I l I 1 I f I T I r f: O 8 8 5" 9,6,9 b=3 fm _:-:-— AG b=4 fun "i 8 3’: 6953113., €I— ,6"; ' +26 B E 9' to EE *5 *2 3 El 4 r I, \1 —-..._ 3 \\' 1 -:. 4 I I t l. 1: g \ 1°; 2 I- . \ -Q .1.- ." . o. .1 2 :‘g .0103“ 0:? 0-0.39.1 2 O y l l I l l l l l L L .1 l l I I l I l l l I l l : O of 100 200 O 100 200 t (fm/c) Figure 7.5: The emission rates of ‘oAr+”‘Sn collisions at E/A=35 MeV for impact parameters b=1 - 4 fm. The solid circles are results with stiff E03 and the open circles are results with soft EOS. The corresponding freezeout times are indicated by the arrows (solid for stiff ROS and open for soft EOS). The respective lines are used to guide the eyes. 144 MSU-Sl-OSZ ‘°Ar+“"‘Sn, E/A =65 MeV, aNN=41 mb : I I éqI I OISI ;tIEIOé I .1: I T6: I I I I I I I I o : 912, 08111: EOS :: 2". b=2 fm 1 \ I .-’ ‘9 :: i ‘- : 5 10 .— I ‘4 ‘1:- ; I '1 10 8 : I \‘g b=1fm :: g ‘9; 1 < 5 LI! ‘2 ALI lQ .: 5 :1 : ' Q~ .;‘\Q. I l ‘46.!” I ' ‘09, .o’ 0 ‘P’ '0. >90. ‘ B 1 1 1 l 1 {0'1 1 1oL-m‘l'h 1 1 1 I 1 101 [_161 . g 0 : I roIoI I I I I I l I I ll. I' I I I I I I I r I I I : O 15 — 5 '- - 9 -+ 15 5 : fl; b=3 fm 1: JQ‘ b=4 fm : “Id '- § o 1: '\ d VJ - . . -- O . .."3. 10 - F 0; —— 9 § - 10 g : t \‘e :t ! \2 : m _ c ‘1. + 6 g . 5 Lf’ ‘1 .i .3... 1°} . _‘ 5 I E \ O I: ' -°’ "3)! j ' g ' .5... "' é . 1a). 1 0 '1 l 1 I I. l 101 L I 1‘!“ l l I l l 1 L l 1.01 O O 100 200 O 100 200 t (fm/c) Figure 7.6: The emission rates of 40Ar+1248n collisions at E/A=65 MeV for impact parameters b=1 — 4 fm. The solid circles are results with stiff E08 and the open circles are results with soft EOS. The corresponding freezeout times are indicated by the arrows (solid for stiff E08 and open for soft E03). The respective lines are used to guide the eyes. 145 soft (stiff) EOS. In general, the freezeout time reached with our criteria are slightly shorter for stiff EOS compared to that obtained for soft EOS. More detailed studies also indicates that the freezeout time does not depend on the impact parameter at E/A=35 MeV for either stiff or soft equation of state. However, at higher energies, E/A=65 MEV, the freezeout time depends very sensitively on the impact parameter, particularly for the soft EOS. With O’NN = 41 mb, the freezeout time decreases from the z 160 fm/c at b=0 fm to the z 120 fm/c at b=6 frn for the soft E03; and the freezeout time decreases from the z 120 fm/c at b=0 fm to the z 100 fm/c at b=6 fm. The bottom panels of figs. 7.7-7.8 show the quadrupole momentum distributions as a function of time for 40Ar+12“Sn collisions at b=0 fm. For comparison, the emission rates at b = 0 are presented in the top panels. Similar to 40Ar +27Al collisions discussed in the previous chapter, sz does not provide accurate freezeout time due to complications from quadrupole vibrations. The final spatial configurations at freezeout depend very much on the incident energy. Figs. 7.9-7.13 show the final spatial distributions for 40Ar+”‘Sn collisions at E/A=35 MeV and 65 MeV with soft or stiff EOS. At E/A=35 MeV, one always see a single well defined residue at the freezeout time, even for the larger impact parameter collisions in which the bound residue decays into distinctive pro jectile-like and target-like residues at a later time. In contrast, at E/A=65 MeV and impact parameters greater than b m 2 - 3 fm, the projectile-like and target-like residues (sometimes more than two residues) at freezeout appear to be more distinct. This indicates that the time scale for breakup in high energy collisions becomes shorter than the time scale for relaxation and equilibration of the extended residues. We also note here that at E/A=35 MeV, the residues appear to distribute over a larger volume for the calculations with CNN = 20 mb (Fig. 7.10) than the corresponding 146 usual-053 “AH“‘Sn. E/A =35 MeV. am=41 mb, b=0 fm % 10 '- I I I I I I Ifi I I I I. I I I I I I I I I I I I d 10 \ : ,I‘ ,swr nos 2: Soft 9 E03 2 g. 8 L— ’ \ ! _‘L 90 9 '1 8 o b I ‘ I d!- I \ I 4 9’. I r \ i i: I ‘\ y 4 - \ -- l I - > 6— ' o i —— a 0‘ i -: 6 ... 2 fi \ i 2: o \ 3 « fl . ' \ - .. l - . *’ . l \ 5 .. , b ! .. c2 4 " ' g! '1:- 1 “ ! ‘1 4 ' l | ' .. 1 ' . " - Q \ ’ e g I .' 1: ’x 2:,0 .3 ,‘e . 5‘. 27° fifth-2:“: H! ‘7 2 . _ I ‘4 C _m , h. \ .- g ‘Q.’ l l d“ i b : m I. L l l l l l L L l l d. l l l l l l l l l O 1- I I I I I r I I I I I ‘h I I I I I I I I I I : O : . :: g .. 4 '— ‘ -—" ' -E 4 : tfreeze-out 1. t-freeze-out 3 a?" 3 :— 9 75" '1 3 ’6‘ : 5 :: : \ ' I " _.‘: 3f, 2 E— i :: : 2 o : i :: _3 :2: 1 :- i .. : 1 - I 2: < Ga 0 E.- .......... \.‘...'.’ ‘I. 2; ---------- \I : 0 1 1151111 nos; 1: Soft mos; .3 _1 ”1 l 1 1 I I l l l I l I: l 1 LJLLI l l I l : 0 100 200 0 100 200 t (fm/c) Figure 7.7: The emission rates of nucleons (top panels) and the quadrupole mo- mentum distributions sz (bottom panels), defined by Eq. (6.19), for “AH-”‘Sn collisions at E/A=35 MeV, b=0 fm. The left panels (right panels) show the results for the stiff EOS (soft EOS). The vertical dot-dash lines indicate the freezeout time discussed in the text. The dashed lines in the bottom panels include the calculations for all nucleons, while the solid lines include only nucleons in the bound residues. 147 MSU-Sl-054 ‘°Ar+‘2‘Sn, E/A =65 MeV, oNN=41 mb, b=0 frn : I I I I I I I I I I I I rj I I I I I I o I— l ' I «I- I l l \ I . fl ' 1 E - R ; Stiff EOS ‘r ,\ 50ft i E03 * ._ 15 :- I e . -— o ‘\ i — 15 o ¢ \ . “ ’ i ‘ 95 : ' \ . ‘: T ‘. . ‘ W — l . 9 ' -——— ' i 3 ~— 3 10 . I \ ' ..- l “ l - 10 _ \ I _,_ l I _ a? - + \ I .. O k i 4 . I I a... l I a . . " .9. 5 _ I, *1 | "— I, ‘ 9 "I 5 U) - I ~- I - .59. b I. ‘1! f.\ 1' J k ' l.‘.\ f " I / .A '41- | \ q s 1‘ 1.1. 1 . 4" kx V‘ . ‘ O+f§1rfh§§ ‘r% #%+#L%%‘|—¢—-|—l—¢— O : I i I “t l i ‘ - , -- i 6 :— t i _‘L i j 6 .... : freeze-out :: tfreeze-out ; %; 4: ' I ' I 9 : I ‘ : 4 (g 1- -l- . 4 f: 2 :- ‘tt’ - -: 2 g )- dh- ‘ .. C? P " ‘ ‘ - -_~- L ‘ O __.. ............ _ ........ .- ......... i . 4 O :Stiff EOS! ;: Soft EOS ! 3 _ y . ! _2 1 1 1 I 1 1 1 1 l 1 1 1 1 1 1 l 1 1 1 1 l l _2 O 100 200 O 100 200 t (fm/c) Figure 7.8: The emission rates of nucleons (top panels) and the quadrupole mo- mentum distributions Q 22 (bottom panels), defined by Eq. (6.19), for 40Ar+1245n collisions at E/A=65 MeV, b=0 fm. The left panels (right panels) show the results for the stiff EOS (soft EOS). The vertical dot-dash lines indicate the freezeout time discussed in the text. The dashed lines in the bottom panels include the calculations for all nucleons, while the solid lines include only nucleons in the bound residues. 148 calculations with (INN = 41 mb (Fig. 7.9). We will come back to this point when we evaluate the density dependent level density parameter. II The Excitation Energy at fieezeout A Excitation Energies In Fig. 7.14, we display the decomposition of the excitation energy at freezeout using Eq. (6.18) for ”At +‘2‘Sn collisions at E/A=35 MeV. The freezeout configuration at this energy consists of a single bound system, similar to those obtained for 4oAr +27Al system at E/A=3O MeV and the qualitative behavior of various contributions to the excitation energy is also similar. At higher incident energies, E/ A =65 MeV, the freezeout configurations shown in Figs. 7.12-7.13 are more complex. The left hand panels of Fig. 7.15 display the decomposition of the excitation energy if all the residues are included in the calculation. The right hand panels show the corresponding decomposition if only the largest residue (target-like residue) is analyzed. In addition to the density requirement for the bound residues, a sphere of adjustable radius with its origin at the center of the target-like residue is used to separate the target-like residue from other bound clusters. Applying our analysis at large impact parameters, b 2 3 fm, to the entire system yields a total excitation energy per nucleon which is significantly larger than that for the target-like residue alone. These large values for the excitation energies are a consequence of the large relative velocities between the various residues. These large relative velocities could be responsible for the significant increase in the non- rotational collective energies, E“ denoted by the difference between the diamonds nDr. , and squares of the left hand panels of Fig. 7.15, at larger impact parameters. The thermal energy per nucleon (squares), however, is practically the same if one considers 149 MSU-91-023 b=0 fm 3 Q Figure 7.9: The spatial distributions at the freezeout time for 4oAr-I-““Sn collisions at E/A=35 MeV with the soft E03 and amv = 41 mb. The values of impact parameters are indicated in each corresponding panel. The beam directions is in the vertical direction (projectile moves from top to bottom). 150 MSU-9|-026 b=0fm 3 | 4 2 5 9 g} .9 Figure 7.10: The spatial distributions at the freezeout time for 40Ar-I-u‘Sn collisions at E/A=35 MeV with the soft E08 and a'NN = 20 mb. The values of impact parameters are indicated in each corresponding panel. The beam directions is in the vertical direction (projectile moves from top to bottom). 151 MSU-Sl-OZS b=0fm 3 O C O 9 Figure 7.11: The spatial distributions at the freezeout time for 4°Ar+1243n collisions at E/A=35 MeV with the stiff E08 and 0N” = 41 mb. The values of impact parameters are indicated in each corresponding panel. The beam directions is in the vertical direction (projectile moves from top to bottom). 152 MSU-9l—021 b=0fm 3 g. 4 2 5 Jr Figure 7.12: The spatial distributions at the freezeout time for 40Ar+1“Sn collisions at E/A=65 MeV with the soft E08 and 0'sz = 41 mb. The values of impact parameters are indicated in each corresponding panel. The beam directions is in the vertical direction (projectile moves from top to bottom). 153 MSU-91-024 b=0fm 3 Q Figure 7.13: The spatial distributions at the freezeout time for 40Ar+n4Sn collisions at E/A=65 MeV with the stiff E03 and (Inn; = 41 mb. The values of impact parameters are indicated in each corresponding panel. The beam directions is in the vertical direction (projectile moves from top to bottom). 154 MSU-9| -055 “Ar+‘“$n. E/A=35MeV, am=41mb 3 ITIIIIIIIIIIIII[TIIIIIIIIFWIlIfiIIII " J 2 Soft 1208 E' j 2 I- . ”var-Em: I i i “It”. I ' Mfr-E . : ‘13” ‘B“B~\B 'Ecomp: . \ j E‘ / A (MeV) - )1 , :ETN . I .- ‘B’ \U'~a‘ E ..Ero‘fi 1 _ ‘ El- —Ecom;: l: m _, -III1 lllllllLLllllllllllllllllllllllllllllll O 1 2 34567 b(fm) Figure 7.14: Decomposition of the excitation energy at freezeout for ”Ar-klz‘Sn collisions at E/ A =35 MeV, assuming the soft EOS, (top panel) or the stiff EOS (lower panel). The respective symbols indicated in the figure are the different components by using Eq. (6.18). The lines are drawn to guide the eye. 155 MSU-91-056 ‘°Ar+12‘8n, E/A=65MeV, UNN=41mb IIIIIIIIIIIIIII IIII IIII ‘3‘ EL ‘13-- Soft nos '3~ llllllllllllllllllllllllllllll IIIII E'/A(MeV) HmwpmqummIIr-moaxi Whole Bounded System “f 13‘" fl "- tifl EOS " ~ 3 I: QB \ Stiff EOS IIIIIIII IIII IIII'IIIIIII‘I Illllllllllllllllllllllll ‘13-- HMO-FUOQHNCD-FUIONI .5? 01234560123456 b(fm) Figure 7.15: Decomposition of the excitation energy at freezeout for ‘OAr-I-u‘Sn col- lisions at E/ A =65 MeV, assuming the soft EOS, (t0p panels) or the stiff EOS (lower panels). The left hand panels display the decomposition of the excitation energy when all the residues are included. The right hand panels show the corresponding results when one includes only the largest residue (target-like residue). The respective symbols indicated in the figure are the different components by using Eq. (6.18). The lines are drawn to guide the eye. 156 the whole bounded system (left hand panels) or if one simply considers a single target- like residue (right hand panels). This result is consistent with the assumption that the thermal excitation energies per nucleon are the same for all clusters. On the left panels of Fig. 7.16, we show the decomposition of excitation energies with soft E08 and mm = 20 mb at E/A=35 MeV (top panel) and 65 MeV (bottom panel) at freezeout time. For comparison, the corresponding calculations for amv = 41 mb are re—plotted on the right panels. The qualitative behavior with both values of UN” is very similar, indicating that no significant dependence of excitation energies on nucleon-nucleon cross sections. Indeed, one might expect that a larger nucleon- nucleon cross section could generate more excitation and therefore a larger excitation energy. However, this effect is partly compensated by the fact that the calculations with smaller aNN tend to freezeout earlier. In fact, the thermal excitation energy and the non-rotational collective excitation energy (the difference between the diamonds and the squares) at E/ A =35 MeV calculated with O’NN = 20 mb are slightly higher than those calculated with (INN = 41, since the systems freezeout at t In a: 120 fm/c, earlier than t fr, 3 140 fm/c obtained with (INN = 41 mb. we will come back to this point when we evaluate the temperatures. In the bottom window of Fig. 7.17, we show the energy dependences of the total excitation energy (circles) and thermal energy (squares) per nucleon for 4oAr +12‘Sn collisions at b=0 fm. The corresponding residue mass is shown in the top window. For both the stiff EOS (solid symbols) and the soft EOS (open symbols), the excitation energy increases with incident energy. However, this increase with energy becomes more gradual at energies E/ A 2 65 MeV, indicating that the excitation energy per nucleon may be reaching a saturation value. The difference between the excitation energies for different equations of state is of the order of z 1 MeV per nucleon at all energies. For comparison, the solid diamonds depict the excitation energies deduced 157 4°A1~+12"'Sn, Soft EOS 4: ‘l"”l""l"”l""l':'l""l'"'l""l""|””l‘1 4 .. 43-93112. is I; it I d11111 Ell a; IIIIII I E‘/A (MeV) P3 p ll O) 01 z (‘D i l_:‘ I l l l O s_ —.:— a. 3 : :3 ’ 1 -B’ an- _: 2 .- —.:' _ :2 3 11. ,aii " - — fl’ -- .- 1 — .— 1 ONN=41 mb \‘a UNN=20 mb . .. .1 - _ .- .- q I lllllllllllllllllllllllllllll llllllllllllllllllllIllll 012345012345 b(fm) Figure 7.16: The dependences of the excitation energies on UNN for 4°Ar+124Sn col- lisions at E/A=35 (top panels) and 65 MeV (bottom panels) with soft EOS. The left hand panels are results with UNN = 20 mb and the right hand panels are results with UNIV 2 41 mb. the excitation energies indicated by respective symbols are the same as those shown in Fig. 7.15. The lines are used to guide the eyes 158 USU-Ol-OSS “Ax-+‘3‘Sn. amaumb. b=0fm ""l""I""I"‘ 150- I- .. 1 I- 8‘3... s ..‘u ' °‘o\"c.._‘ r “ON"... 100.. ‘n. e - I ‘~ ”0 a ' “o I < b .1 ~ 0 Stiff EOS * 50— d E 0 Soft EOS L 5 f . '. 4 C- i F; . . O E. I I 3 —q s t < 0 1' ‘ '23 " ‘ 5 2 L J .\ I- ‘ a '- ’ I O : . 93'! P ° E.“ (stiff nos . " a o E A N O ab 0 G O (D O H O 0 Figure 7.17: The dependence of residual mass (top window), the total energy and the thermal energy (bottom window) on the incident energy for ”Ar+”‘Sn collisions at =0 fm. The solid symbols represent the results with the stiff E08 and the open ones represent the results with the soft EOS. The total energy and the thermal energy are indicated by the circles and squares, respectively. The diamonds are extracted from ref [Jian 89] for 4°Ar+232Th collisions. The lines are used to guide the eyes 159 from neutron multiplicity measurements for the “Ar +232Th system [Jian 89]. In ref. [Jian 89], only the total excitation energies were given. For simplicity, we have assumed the residue mass Am = 272 for the ”Ar +232Th system. If the residue mass for this heavier system decreases with the incident energy as dramatically as we have calculated for 40Ar + 124Sn system, the excitation energy per nucleon of the residues for the “Ar +232Th system would actually be increasing with incident energy. With the present calculations, we have not attempted to determine the EOS from the available data though the calculated excitation energies per nucleon are closer to the predictions for soft EOS than for stiff EOS. To further illustrate the difficulties in making these comparisons more quantitative at present, we show the total excitation energies of these systems. The total calcu- lated excitation energies predicted for 40Ar+ l“Sn are lower than those determined experimentally for 4oAr+ 232Th system. Clearly, to extrapolate our calculations to the 40Ar +232Th system, we need to know more about how the residue excitation energies depend on the target mass. The calculations do show that the total exci- tation energies predicted by both equations of state appear to be in-sensitive to the incident energy at E/ A Z 40 MeV, similar to the insensitivities demonstrated by the experimental data. This result indicates that, because the residue mass is decreasing with incident energy, the total excitation energy may even saturate at energies where the excitation energy per nucleon shown in Fig. 7.17 is still increasing. B Massive Transfer Models Experimental studies of residue excitation energies are frequently based on analysis of residue velocities using massive transfer models. Such models assume that part of the projectile ‘fuses’ with the target, while the remaining part of the projectile escapes with the beam velocity. Using this simple assumption, it is easy to deduce the 160 MSU-QI-OSO 4°Ar+12‘Sn, ONN=41mb, b=0fm 800 __ l l I _ I “Ar+232'rh, j E . ““““““ 1 600 _ “f’ hang et a1 _ I L E : . ....... . ....... '. °°°°°°° . j :J 400 _— ....,.. — lid - o" ..I ----- , ..7-7'.'""5 ‘ h . .-- .”e’ I .. ...-’9’ ,’B~~ 1 200— 95813 ,’E3’ ‘B~--B — ' I" ,E” ‘ I 3’3 43 f O r- I I I L l I J L I J_ I I I I I I I I I q 20 40 60 80 100 Emu/A (MeV) Figure 7.18: The dependence of the total energy and the thermal energy on the incident energy for 40Ar+”“Sn collisions at b=0 fm and and for mAr+232Th extracted experimentally. The solid symbols represent the results with the stiff EOS and the open ones represent the results with the soft EOS. The total energy and the thermal energy are indicated by the circles and squares, respectively. The lines are used to guide the eyes. 161 excitation energy from the measured residue velocity (see Appendix B for details). To examine whether our BUU calculations are consistent with massive transfer models, we show in Fig. 7.19 the residue velocities (top panels) and the total excitation energies predicted by the BUU and the excitation energies extrapolated from the residue velocities (bottom panels) according to the massive transfer assumption. The left hand panels show the results for “Ar + 1“Sn collisions at E/ A =35 MeV and the right hand panels show the results for E/A=65 MeV. The circles in the bottom panels depict the excitation energy extracted from the calculated residue velocity using the massive transfer models. The squares depict the total excitation energies obtained directly from the BUU calculations. Both the residual velocities and the excitation energies are determined for the targetlike residues which have survived the collisions. At E/A=35 MeV, the predicted residue velocity by BUU is slightly less than the velocity of the center of mass and shows little dependence on impact parameter. In contrast, the velocity at E / A=65 MeV depends significantly on the impact parameter. At both incident energies and at all impact parameters, the massive transfer model significantly overestimates the excitation energy for the largest residue. The discrep- ancy is largest for the central collisions. In our simulations, the massive transfer model fails because it significantly underestimates the cooling due to preequilibrium emission and because the pre-equilibrium particles have velocities which are signifi- cantly less than the beam velocity, inconsistent with massive transfer models where the pre-equilibrium particles have the beam velocity. Caution must be taken when interpreting the calculated residual velocity because we have not adjusted any to reproduce the experimental data. Smaller discrepancies will occur for smaller values of O’NN. Additional uncertainties may arise because the BUU models only predicts the average trajectory and can not accurately predict the multifragment breakup, due to the suppression of fluctuations by the ensemble averaging in the calculations. 162 MSU-SI-OSS “Ar+‘2‘Sn, om=41mb, E/A=35 MeV 0.10 0.10 0.08 Vc [ii—320;... V“ 0.08 ' :E ‘05:- Q 0.06 -°-'='O=+4': :E' 3;»! 0.06 N i. "' .. > 0'04 OStiff EOS 1E 031111 EOS 8%. 0'04 0.02 0Soft E08 4; OSoft EOS 0 02 A 12 .. m' we. ::.. m: m: :3: film: an :::: m: m: ::::: 12 % . I :58'59;'., a 10 OjMassive Transfer-3:- 03. 10 g 8 —::— 8 g, 6 --o--o~o--o--o~.- jL- 6 :3 :: \ .tB .. 22.41.1344;— 2 0 ' “ 0 01234560123456 b(fm) Figure 7.19: Comparison of BUU results with that from massive transfer models for 4oAr+”‘Sn collisions at E/A=35 MeV (left hand panels) and 65 MeV (right hand panels). The top panels show the residual velocities and the bottom panels show the comparison of the excitation energy. The details are discussed in the text. The lines are used to guide the eyes 163 Pre-equilibrium emission of complex fragments via multifragment breakup could re- duce the velocity of the residue, and therefore might reduce the excitation energy estimated by the massive transfer model. III Nuclear temperatures of the Residues A Formalism To allow a comparison with experimental emission temperatures, one must relate the excitation energies of the residues to the corresponding temperatures. For our computational residues, the level densities are sensitive to the density distributions and thermal energies of the residues and not solely to the total excitation energy, because there are sizable non-thermal collective energies and because the density distributions of the residues may retain some memory of the collision dynamics. We estimate the temperatures of the residues by integrating the Fermi-gas expression 5‘(T,£p(p(r”))) for the excitation energy per nucleon over the nuclear density and equating this value to the thermal energy provided by the numerical simulations, as follows. a... = 1; «mp. ...-(T, w.» + p. «12:12.00» (7.1) Here, p, and pn are the matter densities for neutrons and protons, respectively. For simplicity, we approximate €‘(T,€p(p)) by its low temperature limit, and thus Eq. (7.1) becomes :he = GT2 (7'2) with a, the level density parameter, given by m 37r2 a = WIT)”: deSI—pl/Bm (7.3) 164 In expressions 7.2 and 7.3, we have assumed equal Fermi energies for protons and neutrons, and have used the local density approximation, II.2 31r2 er(p)=—--( ,” 2m )2“ ~ (7.4) In our calculation, the level density parameter is evaluated from the density distri- bution produced from BUU calculations at freezeout. Since we already calculated the thermal energy, we can calculate the temperature from Eq. 7.2. The results are discussed in the following subsections. B Results 4"Ar+""’Al collisions Fig. 7.20 shows the predicted temperature at freezeout as a function of impact pa- rameter for ”AH-”Al collisions at E/A=30 MeV. The solid symbols indicates the calculations where a single fused residue is produced while the open ones correspond to the calculations in which two residues are observed in the final states. For both the stiff (circles) and soft (squares) equations of state, the predicted temperature decreases slightly with impact parameter. At larger impact parameters where no fu- sions occur, the predicted temperatures are smaller, thus indicating that, at E/A=30 MeV, the fusion cross sections are not limited by thermal instability at high tem- peratures. For all impact parameters, the temperatures predicted by the stiff EOS are much higher than that predicted by the soft EOS, even though the nucleon cross sections with the stiff EOS has been adjusted to produce similar fusion cross section as that of the soft EOS. The predicted values, T2: 3 — 5, are comparable to inclusive experimental observations. To indicate the range of the level density parameter, the different curves in Fig. 7.20 are the results obtained by Eq. (7.2), with respective values of a indicated in 165 MSU-SI-OGO 4°Ar+27AL E/A=30 MeV :I I I I l fI I T I I I I I I I I I I I I r I I l I I: 5 "-3 .................... m .- . ..................... : 4 - - g 3 _'--—........-.-.-.-._,~.-.-:: ..... .3 a '- -' ‘. ..... - E-' z ........... a=A/9 - ‘K ‘ l. _ a=A/8 i Efthe=aT2 .1 2 _ ....... a=A/7 q 1 ;_ 3; Stiff EOS. am=25mb _; 313m EOS, am=50mb O P I I I L I I I L I l I I L L I I I I Ll I L l I l I I 1 2 3 4 5 b (fm) 0 Figure 7.20: Dependence of the temperature on the impact parameter for wAr+27Al at E/A=35 MeV. The circles are results with the stiff EOS and CNN = 25 mb. The squares are results with the soft EOS and O’NN = 50 mb. The solid and open symbols are discussed in the text. The respective lines are calculations by E" = aT2 with corresponding values of a indicated in the figure. 166 the figure. The predicted values of the level density parameter show little sensitivity to the nuclear equation of state. These in-sensitivities of the level density parameter to the nuclear EOS may indicate that rather similar freezeout density distributions are produced by both equation of state. We note here that although the values of the level density parameter, a z A/ 7 - A/9, are similar to the empirical values of A/ 8 commonly used to relate the total excitation energy to the temperature via the relation E“ = 0T2, it is important to recall that we are using the level density parameter to describe the thermal excitation energy and the temperature. The dependence of temperature on incident energy is shown in Fig. 7.21. The left hand side is the calculations at b=0 and the right hand side is at b = 0",”, the maximum impact parameter for fusion. The predicted temperature at b = 0 for both equations of state increases slightly with energy, a trend also seen experimentally [Chen 87c]. In all energies presented here, the calculations with the stiff EOS are ~ 1 MeV higher than that with the soft EOS, though the corresponding amv is half as large. At lower energies, E / A z 25 -30 MeV, the values of T at b...“ are smaller than that at b = 0. At higher energies, E /A z 30 — 40 MeV, the values of temperature at 0m, approach those at b = 0 fm, reflecting the fact that bm, —I with increasing incident energy. Based on the calculations on 4oAr+ 27A1 collisions, it is possible to extract the information concerning both the equation of state and O’NN, if the observables for ex- periments are properly selected. At low energies, for example, at E/A=30 MeV, the temperature at most central collisions, b S 3 fm, are relatively insensitive to impact parameter, but very sensitive to the nuclear equations of state (Fig. 7.20), even if the corresponding nucleon-nucleon cross sections are constrained to yield equal fusion cross sections. Thus it may be possible that, by measuring the fusion cross section, and by measuring the excitation energy or temperature for central collisions, one 167 MSU- 91-06l [IIIIleIIlIIIIIIr “Ar+271u, b= I-leIIIlIIIIrTTIIlII-I .- 6 “Ar+2"A1, b=0fm T (MeV) TITIIIII 00 I: an =A/ 9 a ——a=A/8 - ..... a=A/7 0 Stiff EOS 0 Soft EOS 0 Soft EOS ILIIIIIIIIIII IJILIIIILIIIILI II 0 (003430103 IIrIITIIIlIIIIlIIjIlI N ' Stiff EOS IILJILIIIIIILIIIIIILIIIIIIIllI IIIIIIIIIIIIIIIIIII'IIIIIIIII'II IIIIIIIIILIIIIIIIIIIIIIIILIIII In— I- I- I- I- O l 25 30 35 40 25 30 35 40 Elab/A (MBV) Figure 7.21: Dependence of the temperature on the incident energy for 4°Ar+27AL The left hand window is calculations at b=0 and the right hand window is the cal- culations at bmu, the maximum impact parameter for fusion. The solid circles are results with the stiff EOS and aNN = 25 mb. The open circles are results with the soft EOS and CNN = 50 mb. The respective lines are calculations by E‘ = 0T2 with corresponding values of a indicated in the figure. 168 may obtain the minimum two constraints needed to separate the dual dependences of the observables on the nuclear EOS at low density and an”. The emission pat- tern of coincident light particles, discussed in Chapter 6, can also provide additional information about O’NN. ‘°Ar+124Sn collisions Fig. 7.22 shows the predicted temperature as a function of impact parameter for 40Ar+1uSn collisions at E/A=35 MeV (top window) and E/A=65 MeV (bottom window), respectively. The dependence of the temperature on nuclear equation of state is shown in the left-hand panels. The solid circles in the left hand panels depict the results obtained for the stiff EOS, while the open ones depict the results for the soft EOS. For comparison, the respective curves are the results obtained with Eq. (7.2), and a constant level density parameter a with its value indicated in the figure. At E/A=35 MeV and central collisions, b5 4 fm, the predicted temperature depends weakly on impact parameter, but depends sensitively on nuclear equation of state. In contrast, the temperature depends rather strongly upon the impact parameter at higher energies, E/ A =65 MeV. The sensitivities to nucleon-nucleon cross sections are shown on the right hand panels of Fig. 7.22. The predicted temperatures show surprisingly little sensitivity to the nucleon-nucleon cross section. This result is due to the fact that the calculations with smaller nucleon cross sections yield earlier freezeout times by using the criteria of nucleon emission rates and the thermal energies. For example, at E/A=35 MeV, the freezeout times with soft EOS and 0'va = 20 mb are about the z 120 fm/c in contrast to the z 140 fm/c for the corresponding calculations with amv = 41 mb. We also note here that the predicted level density parameters for the soft E08 and UN” = 20 mb at E/A=35 MeV, are higher than that for other calculations shown 169 MSU-Sl-OGZ 40Ar+124Sn 4 : x "L; om=41mb17 Soft E08 '5 4 3- -Z: """"" ; ----------- -Z 3 : I: 0 ‘0mm“- 3 2 :- —..'." °°°°° a=A/9 0 -E 2 E 3: "" a=A/8 : 917;° -'-"-‘."“‘—‘A/" ‘21" a) : 0 of 0 ‘3 00:01:20318 O. i: a 6 = ..... , = ‘ 6 5 e 5 ....f’uu=41mbff ..... ”ms... nos 5 .. 4 £35.11 -22- ' ----- 4 3 -\ \ i— ..... a=A/11 ‘6‘ x =A/11 O 3 g -- t'a:A/10 :E— "" a=A/10 o 2 .... 0 IE ;" a=2A 9 1 fit f.1110; '1? 0331::4 g 1 O l ” I 0 01234560123456 b(fm) Figure 7.22: Dependence of the temperature on the impact parameter for 40Ar+1248n collisions at E/A=35 MeV (left hand window) and 65 MeV (right hand window). The solid circles are results with the stiff EOS and the open circles are results with the soft EOS. The respective lines are calculations by E" = aT2 with corresponding values of a indicated in the figure. 169 4’OAr+124Sn 5 . - 4 : x I“ ,ym=41mb—:: Soft EOS 3 : .\ ‘T. _E: .......................... : ' :: ° “o"O‘x 2 :— _:_ a=A/9 o : :: — a=A/8 g 1 _-_-_ S __:E_ ;" a=A/7 : :_ Com-:20 mg ID .. O of E S -- a 6 . iii-:-if:Flilfiilfimiiiml;o :— 5 am=41mb : ..... ..msm nos 4 " --:;— ........ 3 —:— ----- a=A/11 ‘ \ :: =A/11 o 2 -- a=A/10 :— -- a=A/10 . -t'fa=A09 55 ;- a=A 9 1 0331’ Ea? ‘3;— omii s O MSU-Sl-OGZ 10wa ...a IOIIIIII‘IIIIIIIIIII _ T (MeV) OHNCIDAOIO) 01234560123456 b(fm) Figure 7.22: Dependence of the temperature on the impact parameter for ‘oAr-l-u‘Sn collisions at E/A=35 MeV (left hand window) and 65 MeV (right hand window). The solid circles are results with the stiff EOS and the open circles are results with the soft EOS. The respective lines are calculations by E“ = aT2 with corresponding values of a indicated in the figure. 170 in the figure. This occurs because the densities at freezeout for this calculation are distributed over a more extended volume than those for the other calculations. Fig. 7.23 shows the energy dependence of the temperature for the 4OAr+1248n system at b=0 fm. The experimental values of the emission temperature obtained in this dissertation study, along with other experimental results [Poch 85a, Poch 87, Chen 88a, N aya 90], are depicted by the squares. Both equations of state predict a gradual increase in temperature as the incident energy was raised from E/A=30 MeV to 55 MeV. The stiff EOS predicts consistent larger values of temperature at all inci- dent energies. At higher incident energies, E/ A Z 65 MeV, however, the temperature increases very little, suggesting a possible saturation in the temperature with inci- dent energy. At all energies, our calculated temperatures are similar to experimental ones. Due to large uncertainties in the choice of the impact parameter averaging and the uncertainties in the Pauli-blocking and nucleon-nucleon cross section, we can not make more quantitative conclusions concerning the compressibility of the EOS at low density from our present comparisons. IV Summary In summary, guided by numerical solution of an improved BUU equation, we have studied the global features of the reaction dynamics for 40Ar+12‘Sn collisions. Using various criteria, we found that consistent freezeout times could be defined. The predicted thermal temperature for the heavy residues at freezeout is comparable with experimental measurements. Our studies indicates that the extraction of information concerning the equation of state and the in—medium nucleon-nucleon cross sections is not trivial. Although the predicted excitation energies and emission temperatures display a significant sen- 171 MSU-Sl-063 wAr+12‘Sn, om=41mb, b=0fm I I r I l I I l T r I I r T ‘l I I I I III — IIIUIIIjII‘IIIIIIIIII H—I T (MeV) as ............ a=A;13 ; E‘ T2 __ a=A 1 =3 ----- a=A/9 the. 0 Stiff EOS 0 Soft EOS N [IIIIrIrTI LLJlllJllllIljlllllLllllllllllllllll IIII O r F p r- F' 20 4O 60 80 100 Ebb/A (MeV) Figure 7.23: Dependence of the temperature on the incident energy for 40Ar+1243n collisions at b=0 fm. The open circles are results with the stiff EOS and the open squares are results with the soft EOS. The respective lines are calculations by {M = aT’ with corresponding values of 0 indicated in the figure. The sold diamond is the experimental result of this dissertation. The solid square is taken from ref. [Naya 90] which is derived from a large number of particle unstable states. The solid circles and solid crosses are results of excited states of 5Li and 6Li, respectively [Chen 88a]. 172 sitivity to the nuclear equation of state, a comparable sensitivity to impact parameter is also observed, particularly at higher energies. Fortunately, our calculations show that the predicated thermal excitation energies and the temperatures are relatively in-sensitive to in-medium nucleon-nucleon cross section, reducing the possibility of ambiguities in the interpretation of emission temperatures. Chapter 8 Conclusion In this dissertation, we have measured the average emission temperature for a large number of particle stable states of intermediate mass fragments for 323 induced reac- tions on ““‘Ag at the incident energy of E/A=22.3 MeV. To assess if measurements of nuclear temperature can provide information concerning the nuclear equation of state and the in-medium nucleon-nucleon cross section, we have performed dynamical calculations based on the Boltzmann—Uehling-Uhlenbeck' (BUU) equation. To test the statistical assumptions for the fragment emission and to check the degrees of thermalization and the internal consistency of thermal assumptions, 28 independent 7-ray transition intensities were measured using the Spin Spectrometer [J aas 83]. The measured relative populations of these states were compared to those calculated from a thermal model which include sequential feeding from higher lying states. This comparison indicated an average emission temperature of Tz 3 -4 MeV. This result is consistent with the trends established by measurements of the particle unstable states of 4He, 5Li, 6Li, and, 8B nuclei [Poch 85a, Poch 87, Chen 88a]. It is also consistent with the results of a recent investigation of large number of particle unstable states of intermediate mass fragments in MN induced reactions on Ag at E/A=35 MeV [Naya 90]. Putting together these results suggests that the emission temperature increases gradually with incident energy from values of T m 3 - 4 MeV 173 174 at E/A z 23 MeV to T w 5 — 6 MeV at E/A=94 MeV. To study whether the emission temperature can provide any information about the nuclear equation of state and the in-medium nucleon-nucleon cross section, we. have performed dynamical calculations based on the Boltzmann-Uehling-Uhlenbeck (BUU) equation. Since the BUU equation is an one-body theory, it does not have the many- body fluctuations required to produce the intermediate mass fragments. We therefore attempted to study the excitation energy and the emission temperature of the heavy residues. We believe this approach may be justified because molecular dynamics calculations [Lenk 86, Schl 87] indicate that all reaction products, regardless of their masses, could be characterized by a common temperature. This result indicates that the information concerning the emission of intermediate mass fragments may be obtained from the emission temperatures of heavy residues predicted by BUU calculations. To improve the stability of the ground-state nucleus and the conservation of energy during nucleus-nucleus collisions, we have used a Lattice Hamiltonian method to solve the BUU equation. With this improved code, consistent thermal freezeout times are obtained from the emission rates of nucleons and the thermal excitation energies of the heavy residue produced in ”Ar +27Al and 4"Ar +12‘Sn collisions. The predicted total excitation energies and emission temperatures at freezeout are comparable with those obtained from experiments. These predicted values for the excitation energies and temperatures are quite sensitive to the equation of state. Surprisingly, little sensitivity of the emission temperature to the in-medium nucleon-nucleon cross section is observed. Unfortunately, the predicted emission temperatures are also sensitive to the impact parameter, particularly at high incident energies, making it difficult to determine the EOS and O’NN from inclusive measurements. Calculations for 4oCa-{-"°Ca and 40Ar-l-T’Al collisions also indicate that the residue 175 cross section is rather sensitive to both the equation of state at sub-nuclear density and the in-medium nucleon-nucleon cross section. This result provides another observable which may be used to obtain information concerning the equation of state at the low densities. We also demonstrate that measurements of the emission pattern of the coincident light particles may provide information concerning the in-medium nucleon- nucleon cross section. The dynamical limitations to the formation of fusionlike residue are investigated by calculating the excitation energies and the total angular momenta for the residues formed in 40Ar + 2"Al collisions. These calculations indicate that the dynamics, not the Coulomb or thermal instability [Levi 84, Besp 89, Gros 82, Ban 85, Gros 86, Gros 88, Bond 85, Mekj 90], plays a decisive role in limiting the production of fusion- like residues at energies E / A S 30 MeV. At higher energies, E/AZ 35 MeV, dynamics are still important, but we can not rule out additional reductions of the residue cross sections due to instabilities of hot nuclei at high temperature. Based on our theoretical study, it is possible to extract information concerning the nuclear equation of state at sub-nuclear density and the in—medium nucleon-nucleon cross section from exclusive investigations of emission temperatures, cross sections of fusionlike residues, and the emission pattern of light particles [Xu 90, Tsan 89]. Addi- tional theoretical work is needed to check the sensitivity of experimental observables to momentum dependent interactions as well as to the details of the Pauli-blocking algorithm. Appendix A Correction of Finite Statistics to the Collective Excitation Energy In this appendix, we discuss two methods used to correct for spurious contributions to the collective energy due to finite number of test particles. I The Goldhaber’s Problem To illustrate the finite particle number effects, we first consider the problem of pro jec— tile fragmentation. In order to understand the projectile fragmentation in relativistic heavy ion collisions, Goldhaber [Gold 74] considered the following question: Suppose that A nucleons are assembled with a zero net momentum, 5,4 = 0. If K of these nucleons are chosen at random and are emitted as a single fragment, what would be the mean square total momentum p} of this fragment? Goldhaber [Gold 74] solved this problem by the following arguments. By assump- tion, the total momentum of A nucleons has < pi >=< (:13?)2 >= 0, (M) and, in consequence, A+2<13§-133>=0, (A.2) #1 176 177 01‘ <>=—/(A—1), (A.3) where < p2 > is the mean square nucleon momentum and the double bracket denotes an average over all 1' ;£ j. A similar exercise applied to the momentum of K nucleons yields K <19}>=<<(Zi2’.~)2>>=K+K(K—1)<> (AA) i=1 Here, the double bracket indicates an average over all possible choices of the K nu- cleons from given A nucleons.'Substituting Eq. (A.3) into Eq. (A.4), one obtains =K—K(K—1)/(A—1) =K(A—K)/(A—l) (A.5) Goldhaber used a Fermi gas value for < p2 > and was able to interpret quite success- fully the mass dependent fragmentation spectra with Eq. (A.5). In the limit A -* 00, (A — K)/(A — 1) z 1, thus Eq. (A.5) becomes < p} >z K < p2 > (A.6) As we will see later, this expression is very close to the spurious collective momentum of K test particles located at a given lattice point in the hot residue. II Correction of Finite Statistics to the Collective Excitation Energy In our calculations, the collective excitation energy is obtained by summing up con- tributions from individual lattice cells in which the collective current are evaluated. Let’s look at a given lattice cell with K test particles. Suppose this cell has a true collective momentum of 131.0" per nucleon in the continuum limit (an infinite number 178 of test particles). The true collective excitation energy of K test particles can be represented by 2 E0 _ Kpcoll. (A7) coll _ 2m Here m is the nucleon mass. In practice, this true collective energy is not known. Instead, we calculate the apparent collective energy E30,, (235‘ 131')2 a =< w" 2K m > . (A.8) Rewriting 1?,- = 15',- ’ + 15201: (here , 13',- ’ is zero on the average), a = < [25(E’+ficoll)]2 > coll 2K"! 1 K - K .. K - = mm... + (2,...)- < D.» > + < (21».- '>2 >1, (A9) i=1 i=1 i=1 In our simulations, the second term can be ignored, since the vector sum satisfies 2?; p;- ’ z 0 , when summin over all possible ensembles or over all lattice points. 3 1 g The third term, however, is non-zero and has a value corresponding to Eq. (A.6). Using this expression, Eq. (A.9) becomes . NKPZOu coll 2m + 2m ' < p’2 > = E30" + 2m - (A~10) Thus the apparent collective energy is larger than the true collective energy by a value of 5%,}. Rewriting Eq. (A.10) in terms of the true collective excitation energy per nucleon, one gets 0 ’2 Econ E20“ 1 < P > = coll _ —K 5.0", (A.11) 179 with 1 5corr K 2m - (A.12) This correction to the collective excitation energy is due to the finite number of test particles. We note here that p’2 is viewed in the frame of the true local velocity and is not known. In the following two subsections, we describes two different ways to estimate < p’2 >. A Thomas-Fermi Approximation Since [1" is evaluated in a frame moving with local current, one may estimate 5m, in the local Thomas Fermi approximation. Using this assumption, Eq. (A.12) becomes 1~13 co" = — ~ —- , A.l3 5 K 2m K 56F ( ) here the local Fermi energy 617 is given by 2 ... LF 6}? — 2m’ (A.14) with 31r2 p, = h(—, ”)“3. (A.15) In these expressions, we have assumed that the proton and neutron have the same local Fermi energy. For a lattice cell of size 1 fm‘3, with local density of pa = 0.17 fm‘3 and Nm. = 80, one would obtains K z 14, thus the correction to collective energy would be 5m, z 1.7 MeV/ nucleon. This is clearly a non-negligible correction. As we discussed in Chapter 6, we have calculated the collective energy for a nucleus in its ground state. After correcting the local energy by a term given by Eq. (A.13), one indeed obtains a zero collective energy ( for details, see the discussions in Chapter 6). 180 B Local Momentum Analysis Another way of calculating < p’2 > in Eq. (A.12) is to try to relate it in terms of momentum 13',- which is known. For this purpose, we use the identity 2652-133) =20)?" -p‘,-') - (A-16) {#j '3‘] Averaging over ensembles, the right hand of this equation becomes < gang-13')? >= 2K(K — 1) < p'2 > —2K(K — 1) << p‘.’-p‘,-’ >>, (A.17) #1" Using Eq. (A.3), this equation becomes -u «:2 ’2 (Pn> =2K(K—1)

+2K(K—1)o i¢j A—l =2KK— - - '2 ( 1) A_1

z 2K(K — 1) < p’2 >; (A.18) in the limit of large A. Similarly, the left hand side of Eq. (A.16) has the form K <20) (56-51) >=2(K—1)2—2, (A-19) #1 i=1 #1 Using the identity K K <;fi-m>=<(;fi)2>—<2p? >, (A20) I J I: i=1 we can rewrite (A.19) into -o = 2K[Z —K<(— '—=—K‘p') >] i¢j i=1 __ —2K < Eb)?“ ZP_K__1')2]1'=1 >; (A21) Here the term Tm —(z:-'f(’-E'-)2 is the apparent collective energy. Equating Eq. (A. 18) and Eq. (A21), one obtains 6601’? = 5:71" K(K < 2b?_Z:K=I(1—)2]pj > ' (A'22) i=1 181 which can be evaluated at each lattice point. This expression corrects for all spurious collective motion coming from the Fermi motion and the thermal motion. In our calculations, We have evaluated numerically the corrections given by Eq. (A.13) and by Eq. (A22), and they essentially give the same results, indicating the dominant contribution from the Fermi motion. Further discussions are presented in Chapter 6. Appendix B Massive Transfer Model In this appendix, we present the basic formula of massive transfer models which are commonly used to estimate the excitation energy in incomplete fusion reactions. For convenience, we start by considering reactions leading to complete fusion. I Complete Fusion Suppose a projectile of mass mp with beam velocity 17,, is fused completely with a target of mass m,, the residue velocity 13', of the fused composite system satisfies .. mpvp v,= (13.1) mp+mt Neglecting the ground state Q-values, the total excitation energy E" has the form 1 1 E’ - -2-m,,vID — -2—(mp + m¢)v, = l . "W"? + mt) 2 (B 2) 2 mp ' ' Eqs. (B.1)-(B.2) are constantly used in estimation of the excitation energies for reactions leading to complete fusion. 182 183 II Incomplete Fusion The massive transfer models are referred to the following assumption used in in- complete fusion reactions. For a normal kinematics in which m, < m,, the massive transfer models assume that a fraction, f, of the projectile mass fuses completely with the target, with the remaining fraction, 1 — f, escaped with beam velocity. The excitation of the composite system can therefore be evaluated from the measured residue velocity. Under this assumption, one has (fmp + mtlvr = fmpvpi (3'3) or mgv, f = , B.4 mp(vr - ”r) ( ) and similar as Eq. (8.2), the excitation energy E‘ are given by ._l mt(fmp+mt) 2 E — 2 fm, 12,. (3.5) From Eqs. (B.4)-(B.5), one may obtain the fraction f and the excitation energy E‘ once the residue velocity is measured. Similar exercise can be performed for reverse kinematics (m, > m,). In this case, the projectile fuses with part of the target f. The expressions for f and E‘ are given by f = mp(vp - Ur), (3.6) mt”,- l. fmt(mp+fmt) ~v2 2 m, " E‘ = (3.7) Using these equations, We have calculated the excitation energies from the residue velocities predicted by BUU calculations. 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