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I.‘ JIM.” k‘aoIILW‘I .IJ’I RWJII105\O.\_I III II II - I, . I7. .. .1331Mv’aflfln5 AIWAIIAL. VX Im‘maf .. .V. I. .. ...II 5 ... :0 I,InI . :va V . . III 5.... IVU ’nhn“! I} I ”555'?” .II . .160? . It“. , . I O 0 I I . , _ .00 J ‘00- 60.34.. ‘1'...- IIf I. IIIIO’II. 4 .LI.BRARY 5W m 34 Mfiggggsfgte This is to certify that the dissertation entitled Examining the Caloric Curve in Ar+Sc and Kr+Nb Collisions presented by Richard John Shomin II has been accepted towards fulfillment of the requirements for the Ph.D. degree in Physics Professor Bill Lynch Major Professor’s Signature 7%; 4 4M4 17 December 12, 2003 MSU is an Affirmative Action/Equal Opportunity institution PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 c:/CIRC/DateDue.p65-p.15 Examining the Caloric Curve in Ar+Sc and Kr+Nb systems By Richard John Shomin II 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 2003 ABSTRACT Examining the Caloric Curve in Ar+Sc and Kr+Nb systems By Richard John Shomin II Excited state populations and isotope temperatures have been measured as a function of the energy of the outgoing particle for central Kr+Nb and Ar+Sc collisions at E / A2100 MeV. Temperatures are extracted from these measurements that increase significantly with energy of the outgoing particle. These results indicate the presence of important cooling mechanisms that operate on the time scale of the collective expansion of the system. They raise questions about the widespread interpretation of multi-fragmentation data in terms of equilibrium statistical models and indicate the necessity of using models that take such cooling mechanisms directly into account. In memory of my father Richard John Shomin I June 20, 1933 - February 7, 2003 and to my wife Pearl and my family iii This dissertation was typeset in [MEX 25 and printed on 25% / 20# White Rag Paper - cotton content Zero’th Law or Axiom of Thermodynamics Associated with any simple system there is a function 9 of the variables defining the state such that 6 must have the same value for all substances in thermal equilibrium with one another. As a result some values of the physical parameters of each substance cannot be arbitrary. 6 is known as the emperical temperature and is not unique since any arbitrary single-valued function (MO) will serve equally as an emperical temperature. Once a particular function 4) is chosen for any one substance the emperical temperature is determined for all substances. slight change of ref [126] iv CONTENTS LIST OF TABLES vii LIST OF FIGURES viii 1 Introduction 1 1.0.1 Background and Motivation ................... 1 1.0.2 Thesis Organization ....................... 10 2 The Experiment 11 2.1 Mechanical Layout ............................ 11 2.2 The NSCL 47r detector .......................... 13 2.3 Catania Hodoscope ............................ 21 2.4 IMF Telescopes .............................. 26 2.5 Experimental trigger ........................... 26 2.6 Experimental measurements ....................... 27 3 Calibration and data processing 31 3.1 Data reduction plan ........................... 31 3.2 Impact parameter selection ........................ 33 3.3 Particle Identification ........................... 37 3.3.1 IMF telescope PID ........................ 38 3.3.2 Catania Hodoscope PID ..................... 42 3.4 Calibrations ................................ 42 3.4.1 Silicon calibrations ........................ 42 3.4.2 CsI calibration .......................... 44 4 Single Particle Observables 48 4.1 Moving source Models .......................... 48 4.1.1 Model Description ........................ 48 4.1.2 Single Particle spectra fitting .................. 51 4.2 Improved fits and extraction of the single particle observables . . . . 60 4.2.1 Introduction of source anisotropies ............... 60 4.2.2 Asymmetric source parameterization .............. 60 4.3 Isotopic Temperatures and Isoscaling Parameterizations ........ 82 4.3.1 Extraction of Isotopic Temperatures .............. 82 4.3.2 Comparisons with Equilibrium Multifragmentation Models . . 90 4.3.3 Generalized Isoscaling of isotopic distributions ......... 104 5 Two particle correlations and excited state populations 111 5.1 Pair counting Combinatorics ....................... 111 5.1.1 The Theoretical evaluation .................... 111 5.1.2 Efficiency Calculation ....................... 116 5.1.3 Efficiency Folding and Line shaping(fitting) .......... 122 5.2 Techniques of temperature extraction from correlation analysis. . . . 135 5.3 Dependence of the temperature on incident energy and system size . 140 5.4 Evidence for cooling from excited state population measurements: Breit- Wigner analysis .............................. 147 5.5 Evidence for cooling from excited state population: S-matrix analysis 162 5.6 Summarized results of Breit-Wigner and S-matrix analysis ...... 171 5.7 Evidence of cooling using the Albergo Thermometer .......... 175 5.8 Sequential decay ............................. 179 6 Summary 180 A Derivation of the Relativistic Boltzmann Distribution 184 A.1 Expression for Volume Emission ..................... 184 LIST OF REFERENCES 186 vi LIST OF TABLES 2.1 2.2 3.1 3.2 3.3 4.1 4.2 4.3 4.4 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Mean polar angles for Ball Phoswiches in degrees. ........... Beam and target types ........................... System and Multiplicity Trigger Ball(47rarray) .............. Measured Catania Silicon thicknesses ................... Calibration Reactions. .......................... LCP Single particle fit parameters for Kr+Nb using the 3 source fit with a radially expanding isotropic participant source. ........ Single particle fit parameters using the 3 source, central radial expan- sion model and fourier perturbation. .................. Parameters for emission temperatures. ................. General scaling parameters. ....................... Excited Nuclei studied for standard analysis ............... Excited states Thermometer calculations ................. or“ —->p+t spectroscopic information for Breit-Wigner energy cuts anal- ysis. .................................... Temperature fit information on energy cuts for Breit-Wigner analysis . oi" —+p+t spectroscopic information for S-matrix analysis ........ Temperature fit information on energy cuts for S-matrix analysis . . . Combined table for Breit-Wigner and S-matrix analysis . ....... vii 29 37 45 46 124 138 LIST OF FIGURES 1.1 1.2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 4.1 4.2 4.3 4.4 4.5 Nuclear caloric curve of [91]. ...................... 8 Temperature extractions of Serfling et. al. [100]. ........... 9 Experimental setup. ........................... 12 Experimental geometry. ......................... 14 47r geometry showing position of hodoscope. .............. 15 47r High Rate array ............................ 17 47r Maryland Forward Array . ...................... 18 47r Array electronics. ........................... 20 Side view of the Catania telescope geometry. The front and rear sur- faces are square ............................... 23 Logic diagram. .............................. 24 Catania Hodoscope frame. ........................ 25 Dead time logic ............................... 28 Data Reduction plan ............................ 32 P(NC) and Reduced impact parameter vs charged particle multiplicity for Kr+Nb at 35,70,100 respectively. .................. 35 P(NC) and Reduced impact parameter vs charged particle multiplicity for Ar+Sc at 50,100,150 respectively. .................. 36 IMF telescope dE vs E spectrum for Kr+Nb coliisions at 120 MeV/ A. 39 Imf PID spectrum for the Si-Si telescopes for Kr+Nb scatter plot (E)beam 35 A/U ............................... 40 Imf pid histogram spectrum for Si—Si telescope for Kr+Nb (E )beam 35 A/ U. ................................... 41 Catania PID gates for detector 27. ................... 43 Relativistic Kinematics. ......................... 46 Schematic representation of the participant-spectator model. ..... 50 d,t fits using the 3 source fit with a radially expanding isotropic par- ticipant source. .............................. 57 3H e,4H e fits using the 3 source fit with a radially expanding isotropic participant source. ............................ 58 4H e asymmetric parameterization fit for Kr+N b at 120 AMeV of figure 4.3, the data highlighting at 90M z 90" . ................ 59 Fourier perturbed 3 source fit of 3He for Kr+Nb and Ar+Sc. ..... 62 viii 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33 4.34 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Fourier perturbed 3 source fit of 4He for Kr+Nb and Ar+Sc. ..... 63 Fourier perturbed 3 source fit of 6Li for Kr+Nb and Ar+Sc. ..... 64 Fourier perturbed 3 source fit of 7Li for Kr+Nb and Ar+Sc. ..... 65 Fourier perturbed 3 source fit of 8Li for Kr+Nb and Ar+Sc. ..... 66 Fourier perturbed 3 source fit of 7Be for Kr+Nb and Ar+Sc. ..... 67 Fourier perturbed 3 source fit of 9Be for Kr+Nb and Ar+Sc. ..... 68 Fourier perturbed 3 source fit of 10Be for Kr+Nb and Ar+Sc ...... 69 Rapidity plot of Catania acceptance region of 3 source with expanding central source with Fourier perturbation for 4He for Ar+Sc at 150MeV. 72 Rapidity plot of Catania acceptance region of 3 source with standard expanding central source for 4He for Ar+Sc at 150MeV ......... 73 Rapidity plot of Catania acceptance region of 3 source with expanding central source with Fourier perturbation for 6Li for Ar+Sc at 150MeV. 74 Rapidity plot of Catania acceptance region of 3 source with standard expanding central source for 6Li for Ar+Sc at 150MeV. ........ 75 Rapidity plot of Catania acceptance region of 3 source with expanding central source with Fourier perturbation for 7Be for Ar+Sc at 150MeV. 76 Rapidity plot of Catania acceptance region of 3 source with standard expanding central source for 7Be for Ar+Sc at 150MeV ......... 77 Rapidity plot of Catania acceptance region of 3 source with expanding central source with Fourier perturbation for 4He for Kr+Nb at 120MeV. 78 Rapidity plot of Catania acceptance region of 3 source with standard expanding central source for 4He for Kr+Nb at 120MeV. ....... 79 CM spectra for Kr+Nb (E)beam 35-120 A/U. ............. 81 Apparent temperatures for Kr+Nb (E )beam 35-120 A/ U. ....... 86 Emission temperatures for Kr+Nb (E )beam 35-120 A/ U. ....... 87 Apparent temperatures for Ar+Sc (E )beam 50—150 A/ U. ....... 88 Emission temperatures for Ar+Sc (E)beam 50-150 A/ U. ........ 89 Light Charged Particle Multiplicity .................... 93 Ratio of Light Charged Particle Multiplicity to 4He for Kr+Nb. . . . 94 Best fit of Z distribution and IMF distribution for Kr+Nb 75 MeV/ A. 97 Best fit of Z distribution and IMF distribution for Kr+Nb 95MeV/A.. 98 X—2 vs Fe and Fa. for Kr+Nb 75 MeV/A ................ 99 x‘2 vs Fe and Fa. for Kr+Nb 95 MeV/A ................ 100 BUU calculations ............................. 103 Isoscaling comparison of Kr+Nb for different beam energy ratios. . . 109 Isoscaling comparison of Ar+Sc for different beam energy ratios. . . 110 Detector efficiency and resolution of d—a for Kr+Nb 120MeV/ A . . . 121 p-t Correlation for Kr+N b ......................... 125 d-3H e Correlation for Kr+N b ....................... 126 p—4He Correlation for Kr+N b ....................... 127 p—7Li Correlation for Kr+Nb. ...................... 128 4H e-4H e Correlation for Kr+Nb. .................... 129 p—t Correlation for Ar+Sc. ........................ 130 ix 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.30 5.31 5.32 d-3H e Correlation for Ar+Sc. ...................... 131 p-a Correlation for Ar+Sc ......................... 132 p-7Li Correlation for Ar+Sc ........................ 133 a-a Correlation for Ar+Sc ......................... 134 Temperature dependence on system size ................. 143 Temperature dependence on system Ar+Sc A281 ............ 144 Temperature dependence on system Kr+Nb A2179 ........... 145 Temperature dependence on system Au+Au A=358. ......... 146 Single parameter Breit-Wigner fit of pt correlation on an exponential background . ............................... 150 p-d background used for pt correlation energy cut analysis . ..... 152 Single parameter Breit-Wigner fit of pt correlation on pd background. 154 Two parameter Breit-Wigner fit of pt correlation on pd background. 157 Two parameter Breit-Wigner fit of pt correlation on an exponential background . ............................... 158 Single parameter Breit-Wigner fit of pt correlation for Ar+Sc 100 MeV/A. The upper and lower temperature bounds are the temper- atures determined by two parameter fit. ................ 159 Single parameter Breit-Wigner fit of pt correlation for Kr+Nb 100 MeV/ A. The upper and lower temperature bounds are the tempera- tures determined by two parameter fit. ................. 160 Two parameter S-matrix fit of pt correlation . ............. 164 Single parameter S-matrix fit of pt correlation on pd background. . . 165 Two parameter S-matrix fit of pt correlation versus relative energy(MeV) on an exponential background . ..................... 166 Single parameter S-matrix fit of pt correlation versus relative energy(MeV) on an exponential background . ..................... 167 Single parameter S-matrix fit of pt correlation for Ar+Sc 100 MeV/ A. The upper and lower temperature bounds are the temperatures deter- mined by two parameter fit. ....................... 168 Single parameter S-matrix fit of pt correlation for Kr+Nb 100 MeV/ A. The upper and lower temperature bounds are the temperatures deter- mined by two parameter fit. ....................... 169 Combined temperature limits of Ar100 energy out analysis . ..... 173 Combined temperature limits of Kr100 energy out analysis . ..... 174 Albergo temperatures on energy cuts for Kr+N b systems. ...... 177 Albergo temperatures on energy cuts for Ar+Sc systems. ...... 178 Chapter 1 Introduction 1.0.1 Background and Motivation In general, little can be experimentally learned about nuclei without subjecting them to collisions. Gentle grazing collisions between nuclei can succeed in exciting them to their low lying quantum states. More central collisions at low energies can succeed in merging them together to form excited compound nuclei. Like less excited nuclei formed in grazing collisions or nuclei in their ground states, these compound nuclei are droplets of a ”Fermi liquid” formed of nucleons, i.e. neutrons and protons[8]. This dissertation is primarily concerned with what happens to these hot Fermi liquid drops when one increases the incident energy. Experiments show that in central collisions of equal mass nuclei at incident energies of E/A==35 MeV or more, nuclear systems undergo a multifragment disintegration. This means that the final state of such collisions consists of nucleons, light clusters and intermediate mass fragments with 332330. Models, which assume that statistical equilibrium is achieved during the collision, predict such final multi-fragment final states are the consequence of the nuclear system entering a mixed phase consisting of Fermi liquid drops i.e. fragments, and a gas of nucleons and light clusters [14, 11, 12, 13, 95, 37]. The nucleon-nucleon interaction is attractive at small distances but repulsive at even smaller distances, a characteristic that it shares with the van der Waals inter- action. Like macrocopic fluids where the van der Waals equation of state applies, nuclear matter undergoes a first liquid-gas phase transition from a Fermi liquid [8] into a nucleonic gas. The phase diagram is expected to display critical point predicted to occur at about a temperatures of the order of 10-15 MeV [26, 25]. The experimen- tally observed multifragmentation phenomena, discussed above, are assumed to be a consequence of this underlying bulk phase transition. One key issue concerns how nuclear systems formed in collisions evolve to multi- fragmentation. A commonly proposed scenario for this evolution was described in an early paper by Lopez and Siemens [67]. In this scenario, a hot and compressed composite system is formed after the collision of two nuclei. After initial compression, the system expands nearly isentropically (i.e. at nearly constant entropy) until the compressibility of nuclear matter becomes negative. At this point the system is mechanically unstable, density fluctuations grow exponentially with time, the system breaks up and fragmentation occurs. This scenario is supported by more modern transport theory calculations. [85, 23] The central assumption for many experimental and theoretical studies is that this multi-fragmentation final state is produced at thermal equilibrium [78, 31, 22, 29, 30] The primary support for this comes from the comparison of experimental fragment multiplicities and charge distributions to equilibrium fragmentation model predictions [29, 30]. Even these predictions, however, have difficulty reproducing the data by assuming complete fusion of projectile and target, followed by multi-fragment decay [125, 5, 7]. Instead, one must assume that a significant fraction of the nucleons and the energy is carried away by pre-equilibrium emission prior to multi-fragmentation [125,98,6] Given a scenario of initial contact of projectile and target, followed by preequi— librium emission, expansion and equilibration, analysis have been performed within the equilibrium framework. Such analysis, include the extraction of ”critical expo- nents” from analyses of fragment charge distributions [31], signals of negative heat capacity [22, 28] and extraction of the nuclear caloric curve T(E*) [91, 80, 104, 81]. Fig. 1.1 shows temperatures extracted from Au+Au collisions at E/A=600 MeV by the Aladin group [91]. The horizontal axis is an experimental measure of the thermal excitation energy derived by assuming experimental multiplicities and energy spectra are thermal over a limited angular and rapidity domain and making some corrections for unobserved particles. The vertical axis are temperatures of multifragmentating system obtained by using an isotOpic thermometer first proposed by Albergo [3]. AB T , : _ __ 1.1 ’50 ln(aAY) ( ) where Y(A1, Zl)/Y(A1 + 1,21) AY = , 1.2 Y(A2, Zg)/Y(A2 +1,Z2) ( ) AB = B(A1,Z1) — B(A1+1,Z1) —B(A2, Z2) + B(A2 +1,Z2) (1.3) and a (2.]Z2‘A‘2 +1)(2J21,A1+1 +1) [142(141 + 1)]3/2 (14) (2J21,A1+1)(2J22,A2+1+1) A1(A2 +1) . Here Y(A, Z) is the yield of a given fragment with mass A and charge Z; B(A, Z) is the binding energy of this fragment; and J z, A is the ground state spin of a nucleus with charge Z and mass A. In the data shown in Fig. 1.1, Z1, Z2, A1 and A2 = 2, 3, 3 and 6, respectively, i.e. the relevant isotopes are 3’4 He and 6*7Li and this thermometer is sometimes called the ”HeLi” thermometer. The sensitivity to temperature of the HeLi thermometer is derived from the large difference between the binding energies 3 of the two helium isotope, which minimizes the influence of the secondary decay of heavier nuclei, also emitted during the collision, which decay by emitting the detected Helium and Lithium isotopes, enhancing their yields. Three regions were identified in Fig. 1.1 by the authors of ref. [91]. There is a region at (E0)/(Ao) < 3 MeV, where the isotopic temperature increases rapidly. This trend is reproduced by the curve T He“- 2 \/10(E0) / (A0); the dependence on (E0) / (A0) is consistent with the Fermi gas level density expression for the temperature of an excited nucleus of excitation energy (E0) and mass (nucleon number) (A0). At higher excitation energies, there is a plateau region of nearly constant temperature reminiscent of that for a system undergoing a first order phase transition at constant pressure. At even higher excitation energy, the temperature increases linearly as might be expected for an ideal gas. However, it should be noted that the trends of the data demarcating these regions are not as clear as the lines suggest. Eq.1.1 can be derived from the simple Grand Canonical Ensemble expression for the primary fragment yield for ith fragment in its kth state before secondary decay: A” N ,, Z B,- . N,Z K,k(N.Z.T)=V—jg—(2J.,k+1)exp “ + ”p; ”A l, (1.5) T i where up and un are the proton and neutron chemical potentials, Bi}, and Jik are the binding energy and spin of the fragment in the kth state, and V is the free (unoccupied) volume of the system. The insertion of the ground state yields predicted by Eq. 1.5 into Eq. 1.1 results in the cancellation of the chemical potential terms; the spin and mass number terms contribute to the factor a in Eq. 1.4. The derivation of Eq. 1.1 from the Grand Canonical Ensemble is the most transparent way to obtain these results, but the dependence on temperature predicted by Eq. 1.4 is also observed in microcanonical ensemble calculations [11, 12, 13, 14, 104] and models where the fragments are emitted from the surface of a cooling and expanding thermal source [61]. In all of these cases, however, calculations of the yields of secondary fragments 4 after sequential decay require some accounting for feeding from the particle decay of highly excited heavier nuclei which decay to the measured fragments [14, 104, 140]. Some questions were immediately raised about the interpretation suggested by these curves. These questions become obvious when one considers the projectile fragmentation mechanism that produces the fragments and the helium and lithium isotopes used to construct the temperatures in Fig. 1.1. At 600 MeV, experimental studies show that one is in the domain of limiting fragmentation [98]. The differ- ent values for (Eo)/(A0) correspond to different impact parameters where the impact parameter and the projectile remnant, from which multi-fragmentation is occurring, decreases monotonically with (E0)/(Ao). Taking this dependence into account, this trend of rising THCL, at (E0)/(A0) _>_ 10M eV was alternatively interpreted as indica- tive of a system size dependence of the caloric curve [80]. This proposed system size dependence reflects ideas stemming from calculations of the maximum or limiting temperatures of heavy reaction residues produced in heavy ion collisions , which pre- dicts more highly charged residues produced at larger impact parameters to be more unstable with respect to expansion leading to lower values of the limiting temperature [64]. To disentangle these differing interpretations, it is useful to hold the system size constant while varying the excitation energy. After this dissertation was initiated, the Aladin group attempted to do this by measuring the caloric curve for central Au+Au collisions as a function of the incident energy[100]. Fig. 1.2 shows measured values for the HeLi thermometer (solid squares) as well as temperatures extracted from ratios of the populations of excited states of 4H e, 5Li, and 8Be fragments. According to the grand canonical expression of Eq. 1.4, the ratio of the yields for two states of the same emitted fragment is given by K,1(N? Z, T) :. 2Ji,1+ 1 . e—(Ef—E5)/T, mm, Z, T) 2J,-,2 +1 (1.6) which can be inverted to obtain the temperature T. Clearly, temperatures from the 5 HeLi thermometer increase more rapidly than those from the three excited state thermometers. This led to speculations that the excited state thermometers were somehow incorrect because they did not follow the energy dependence of the HeLi thermometer [100]. Since this measurement, many other caloric curve measurements have been under- taken. Some authors have found significant discrepancies between different tempera- tures obtained by different isotopic thermometers [113]. In principle, all thermometric measurements of a system at equilibrium should yield the same temperature. Some of these differences were reconciled by taking the influence of secondary decay of heavier isotopes into account [50, 140, 113] and by proposing a system size dependence of the caloric curve as discussed above [80] . In fact, temperatures extracted from excited state and isotopic thermometers, measured for central Au+Au collisions at E/A235 MeV and corrected for secondary decay by Huang et al. [50] , are in total agree- ment with each other and yield ”freeze-out” temperature, before secondary decay, of approximately 4.5 MeV. The present dissertation was undertaken to further explore the dependence of the various thermometric measurements on the size of the system. Isotopic and excited state thermometer measurements were made for central Kr+Nb and Ar+Sc collisions as a function of incident energy. When combined with the Au+Au measurements of refs. [100, 50] this provides three systems in which “caloric curve” measurements have been obtained at fixed system size. A second goal is to determine the extent to which global equilibrium can be applied to multi-fragmentation processes. Considering the wide—spread application of equilibrium concepts, it is somewhat surprising that the validity of this underlying assumption is not more rigorously tested. At low energies, where comparatively long-lived metastable excited compound nuclei are formed in central collisions, it is well known that the evaporative emission from the surfaces of such nuclei is well described by a non—equilibrium rate equation. At much higher energies, E/A=200 M eV, there is not enough time in central collisions for equilibration to occur. Indeed, if equilibrium is a valid approximation to multi-fragmentation reactions, it would be the unique instance where equilibrium applies. Thus, the extent to which global or local equilibrium is a valid approximation is of primary importance. One way to probe the validity of global equilibrium is search directly for evidence of radiative (evaporative) cooling. Such tests have been performed at lower ener- gies [137] where isotopic temperatures, extracted from very energetic particles, were compared to those extracted from particles emitted with energies near the Coulomb barrier. Statistical surface emission models [35, 139] generally predict that the most energetic particles originate from the earliest times when the system is at the highest temperature. Thus, clear indications for the importance of non-equilibrium cooling mechanisms can be obtained by comparing temperatures for more energetic and less energetic particles. In addition, tests of the validity of the local equilibrium assumed in statistical surface emission models can be performed by examining whether the relative yields of energetic particles are consistent with a thermal picture. Such tests of global equilibrium and local equilibrium are performed in this dissertation. 12 1 r v TTF'lq—ITH—l—F-r'F-I-F-I— o ‘97Au+ 97Au, 600 AMeV ’ n 12c,‘°o +"“Ag,‘97Au, 30-34 AMeV 10 - A22Ne+‘8‘ra. 8AMeV ,. - 0" i- + 4 41 O I x 2 ’ §( - 2 MeV) 0 0 5 10 1 5 20 I (MeV) Figure 1.1: Nuclear caloric curve of Pochodzalla et a1 1995 [91]. A 5Li' I HeLi 12J5- 0 4He‘ b/bW§O.33 '33 sBe" 10 ~ I 1Z5 — III I l l l l l 1 l O 25 50 75 125 150 175 200 225 EBEASEMeV/A) Figure 1.2: Temperature extractions of Serfling et. al. [100]. The system was 197Au +197 Au for beam energies from E/A=50 to 200 MeV. 1.0.2 Thesis Organization Chapter two describes the setup for the experiment, which involved both the 47r array and the Catania hodoscope. Chapter three describes the calibration of the data and its reduction to physical quantities such as the type, energy and angle of each emitted particle. Chapter four describes single particle observables such as the single particle spectra, multiplicities and quantities, such as the isotopic temperatures, that can be derived from single particle data that have been gated on central collisions. Some simple comparisons to statistical models are performed here that indicate a failure of global equilibrium but offer support for local equilibrium concepts. Chapter five describes the two particle coincidence data and their reduction to obtain excited state populations. Here, additional information about the importance of non-equibrium cooling phenomena are obtained. Chapter six summarizes the findings of this thesis. 10 Chapter 2 The Experiment This thesis experiment involved two major devices: (1) the Catania 96 element ho- doscope and (2) the NSCL 47r detector. The former is a moderate resolution array that was used to measure isotopic yields for 234. The latter covered much of the solid angle not reserved for the Catania array and was used to select the range of im- pact parameters measured in the collision. In addition, four silicon (IMF) telescopes were employed to extend the measurements of isotopic yields to heavier Intermediate Mass Fragments (IMF ’s) with Z=5-8. In the following, the experimental run plan, the mechanical layout and electronic readout of the array are described. 2. 1 Mechanical Layout Figure 2.1 shows a photograph of the experimental setup. In white, one can see various elements of the NSCL 47r detector. At the center, there is the highly segmented forward array of the NSCL 47r detector, a cluster of 45 phoswich detectors centered about the beam axis downstream from the target. The larger white detectors are various hexagonal and pentagonal detector modules of the 47r detector that cover larger scattering angles. During the experiment, two hexagonal modules were removed 11 Figure 2.1: Experimental setup. The beam enters from the right through the hole in the center of the photo of the forward array detector. The Catania array is at the left of the photo. Images in this dissertation are presented in color. to allow the Catania hodoscope to view the target. On the left side of the Figure 2.1, there is the silver colored Catania hodoscope in its aluminum frame. The view of the hodoscope is from the side; however, one can see the front faces of the silicon detectors for many of the 96 telescopes of the Catania hodoscope. The hodoscope was mounted on a rail which allowed the angle of the central detector of the hodoscope to vary over the angular range 40.3°< t9 <67.0° . This angle was varied throughout the experiment to optimize the coverage by the hodoscope for particles emitted at 90 degrees in the center of mass. Figure 2.2 shows a schematic drawing of the layout of the experiment. It is an overhead view that emphasizes the relative placement of the Catania hodoscope and IMF telescopes. The IMF telescopes were also placed at angles where modules of the NSCL 47r detector were removed, but were positioned so as to avoid blocking the passage of particles from the target to the Catania hodoscope. Rough locations of the back surfaces of the NSCL 47r detector modules are shown in outline. Figure 2.3 shows a more detailed three-dimensional view of the 47r detector . The 47r modules are inserted into this hexagonal and pentagonal chamber and occupy most of the 12 inner volume as one can see in Fig 2.1 2.2 The NSCL 47r detector The experiment used the NSCL 47r array as a central collision filter. Impact parame- ters were selected by the multiplicity of identified charged particles. When complete the 47r array detected particles from angles from the beam axis of 70-1570 using 215 plastic AE-E phoswich detectors. This array uses a soccer ball (truncated icosahe- dron) geometry of 32 faces (12 pentagons, 20 hexagons). During the experiment two hexagonal 47r modules were removed to install the higher resolution Catania 96 de— tector hodoscope. The pentagonal sector at zero degrees was covered by 45 phoswich detector elements of the High-rate array plus a 16 element Maryland Forward Array The relative placement of these arrays is schematically shown in figures 2.4 and 2.5. The Maryland Forward array was not used in this experiment and removed. The hexagonal modules of the NSCL 47r array covered a solid angle of 65.96 msr. Each hexagonal module was subdivided into identical 6 triangular plastic phoswich de- tectors. The pentagonal modules of the 47r detector covered a solid angle of 49.92 msr. Each pentagonal module was subdivided into 5 identical triangular plastic phoswich detectors. Table 2.1, gives the polar and azimuthal angles of these hexagonal and pentagonal modules in a rest frame aligned with the beam direction or polar axis and the x axis with the horizontal plane (beam right). The experiment primarily used the phoswich detectors of the 47r array . Each phoswich detector consisted of a thin fast plastic followed by a thick slow plastic, which was coupled to a phototube. In the electronics described below the light from the fast and slow scintillator are digitized. Using the dependence of the energy loss of the fast plastic for the charge and mass of the ion it is possible to determine the particle and the isotope as well if the elements are hydrogen or helium. The various particle types will make different contours on a 13 Hi Rate And MFA f. f | zoo zoo] Brass shield mounted on geared rail 96 element Catania array icosahedron Target beam Figure 2.2: Experimental geometry. Detail drawings of the Hi Rate and MFA Arrays are given in the following figures. 14 4 Figure 2.3: 47r frame showing position of hodosc0pe. Drawing by Craig Snow. 15 plot of the signals from the fast scintillator versus the signal amplitude in the slow scintillator. The different time constants of the fast and slow plastic of the phoswich in the NSCL 47r detector modules allow one to distinguish the signals from each scintillator. In particular, the energy deposited in the fast plastic results in light emission with 1 ns rise time and a 20 ns fall time from the fast plastic with an intensity that is roughly proportional to the energy deposited in the fast plastic. Conversely, the energy deposited in the slow plastic results in light emission with a 20 ns rise time and a 180 ns fall time with an intensity that is roughly proportional to the energy deposited in the slow plastic. The photomultiplier converts the superposition of these two light pulses into an electronic pulse. Figure 2.6 shows an electronics diagram for the processing of signals from the N SCL 47r detector. After the phototube, the signal from each phoswich scintillator is split into three signals by an analogue splitter. One of the these signals is sent to the input of a Phillips leading edge discriminator and the other two are delayed by 100 ns and sent into Fast Encoding and Readout Adc’s (FERA’S). One of these Fera’s is gated by a logic signal of 50 ns length so as to integrate the signal from the fast plastic [119]. The output of this FERA is roughly preportional to the energy loss in the fast plastic. The other FERA is gated by a logic signal of 100 ns[119] length, which is delayed by 10—20 us so as to integrate the signal from the slow plastic [119]. The output of this FERA is roughly proportional to the energy loss in the slow plastic. Each Phillips discriminator handles 16 of the NSCL 47r phoswich scintillators. It generators a linear output which is proportional to the number of signals it receives that are ”hit”, i.e. over the common threshold. All of these linear signals from each of the NSCL 47r Phillips discriminators are combined in a passive linear adder which generates a linear output with an amplitude that is proportional to the total number 16 Figure 2.4: 47r High rate array with exit beam shown. The angle of the detectors closest to the beam axis is about 5.4 degrees. Source: NSCL 47r Group. 17 Figure 2.5: 47r Maryland Forward Array attached underneath the High Rate Array. Was not used during the experiment. Source: NSCL 47r Group. 18 # A, A, B, B, c, C, D, D, E, E, F9 F, 1 23.1 342.0 32.3 5.6 46.0 356.3 51.7 342.0 46.0 324.7 32.3 318.4 2 23.1 270.0 32.3 293.6 46.0 287.3 51.7 270.0 46.0 252.7 32.3 246.4 3 23.1 198.0 32.3 221.6 46.0 215.3 51.7 198.0 46.0 180.7 32.3 174.4 4 23.1 126.0 32.3 149.6 46.0 143.3 51.7 126.0 46.0 108.7 32.3 102.4 5 23.1 54.0 32.3 77.6 46.0 71.3 51.7 54.0 46.0 36.7 32.3 30.4 6 54.7 298.0 54.7 314.0 67.3 317.5 74.6 306.0 67.3 294.5 — - 7 54.7 226.0 54.7 242.0 67.3 245.5 74.6 234.0 67.3 222.5 — — 8 54.7 154.0 54.7 170.0 67.3 173.5 74.6 162.0 67.3 150.5 - — 9 54.7 82.0 54.7 98.0 67.3 101.5 74.6 90.0 67.3 78.5 - — 10 54.7 10.0 54.7 26.0 67.3 29.5 74.6 18.0 67.3 6.5 — - 11 64.9 342.0 72.4 355.0 86.5 354.5 93.5 342.0 86.5 329.6 72.4 329.0 12 64.9 270.0 72.4 283.0 86.5 282.4 93.5 270.0 86.5 257.6 72.4 257.0 13 64.9 198.0 72.4 211.0 86.5 210.4 93.5 198.0 86.5 185.6 72.4 185.0 14 64.9 126.0 72.4 139.0 86.5 138.4 93.5 126.0 86.5 113.6 72.4 113.0 15 64.9 54.0 72.4 67.0 86.5 66.4 93.5 54.0 86.5 41.6 72.4 41.0 16 86.5 306.0 93.5 318.4 107.6 319.0 115.1 306.0 107.6 293.0 93.5 293.6 17 86.5 234.0 93.5 246.4 107.6 247.0 115.1 234.0 107.6 221.0 93.5 221.6 18 86.5 162.0 93.5 174.4 107.6 175.0 115.1 162.0 107.6 149.0 93.5 149.6 19 86.5 90.0 93.5 102.4 107.6 103.0 115.1 90.0 107.6 77.0 93.5 77.6 20 86.5 18.0 93.5 30.4 107.6 31.0 115.1 18.0 107.6 5.0 93.5 5.6 21 105.4 342.0 112.7 353.5 125.3 350.0 125.3 334.0 112.7 330.5 - - 22 105.4 270.0 112.7 281.5 125.3 278.0 125.3 262.0 112.7 258.5 - - 23 105.4 198.0 112.7 209.5 125.3 206.0 125.3 190.0 112.7 186.5 - — 24 105.4 126.0 112.7 137.5 125.3 134.0 125.3 118.0 112.7 114.5 - - 25 105.4 54.0 112.7 65.5 125.3 62.0 125.3 46.0 112.7 42.5 — - 26 128.3 306.0 134.0 323.3 147.7 329.0 156.9 306.0 147.7 282.4 134.0 288.7 27 128.3 234.0 134.0 251.3 147.7 257.6 156.9 234.0 147.7 210.4 134.0 216.7 28 128.3 162.0 134.0 179.3 147.7 185.6 156.9 162.0 147.7 138.4 134.0 144.7 29 128.3 90.0 134.0 107.3 147.7 113.6 156.9 90.0 147.7 66.4 134.0 72.7 30 128.3 18.0 134.0 35.3 147.7 41.6 156.9 18.0 147.7 354.4 134.0 0.7 Table 2.1: Mean polar angles for Ball Phoswiches in degrees. The lettering A,B,... is the azimuthal ordering of each pentagon or hexagon. The first column is the Module(#) number. Source: NSCL 477 Manual 19 _obaoo _ d A 88 iv 83 x95 ~®>®~ mg v :35 Good Sam <._. em 32:; A Bus a 3.34 l 3.3% x], a a as 33584.85 . Q 5:3: can ova—m 32m 3980 8mm + e .Eothscou Emmi“ m m< Rama 055.33 8 pow: 55280 3238:.“ Emma 388.6 «adage 333-3585 "on... 1/47 ma: --.4 ...................... 4 4 .................. 4 ...... 1 _ ...... w M .3585: 353:0 m W “38:: m m m m CH ufi—Ofi mm~hagflflan UNEQU " —0§£O N CH 8t0>goo 0%“ u m u " canon $5250 m m min—mu Congas m m m m 3.094 2.25 ”a 1 4.03 at... m m u. 1. 11111111111111111111111111111111111111111 .. uuuuuuuuuuuuuuuuuuuuuuu . nun-1nnuuuuunnnnunnununnununn...” c. 3:me momznam 98 mom—Em A] 9.2.2? 350 Figure 2.6: 47r Array electronics. reproduced from Dan Magestro’s Thesis and NSCL 477 Users manual. In this experiment, the Bragg curve gas counters was evacuated and were not used. In this mode most energetic particles generate phoswich signals, which were processed by the electronics as described in the text . 20 of NSCL 477 phoswich detectors that are hit anywhere in the NSCL 477 array. This linear output is sent to a discriminator; the threshold of this discriminator selected the minimum desired multiplicity of events and the output of this discriminator is the trigger for the 477 detector. Finally, the Fera ADC’s integrated the signals and converted the charge of the signal within the gate signal to a channel number. The various channel numbers for the detected particles are written to tape by a computer in the vault. The vault computer is controlled via the ethernet from the data room. The vault computer also controls the threshold levels for the various discriminators. 2.3 Catania Hodoscope The Catania hodoscope consisted of 96 identical telescopes. Figure 2.7 shows a schematic drawing of a telescope. Figure 2.9 shows a 3D view of the Catania ho— doscope frame with about three quarters of the detectors missing. Each telescope consists of a 300 um thick silicon detector AE followed by a 6 cm thick CsI(Tl) detec- tor that is read out by a photodiode. The silicon detectors have active areas of 31x31 mm2 and the CsI(Tl) detectors are tapered and large enough to detect all particles that penetrate the silicons when the hodoscope is placed a distance of 70 cm from the target. The silicon AE’s are sufficiently planar so as to resolve isotopes up to 224. Details of the isotopic resolution are discussed in the next chapter. A shield constructed of 20 mg/cm2 brass shim stock was used to prevent electrons produced by interactions of the beam with the target from reaching the detectors. The telescopes of the Catania hodoscope were mounted at a distance of 70 cm from the target in a rectangular array consisting of 14 vertical columns of 7 telescopes. Each telescope subtended a solid angle of 1.83 msr. The angular separation between adjacent telescopes was 0.76 degrees in both vertical and horizontal directions. The 21 horizontal (polar) angle of the center of the array was varied during the experiment. The actual angles are given in table 2.2 along with the corresponding beams and targets for each specific measurement. Operationally, the main practical differences between a photodiode readout of light generated by an ion traversing a CsI(Tl) and a direct readout of ionization generated in a silicon detector is that the CsI(Tl) has a slower time constant and the CsI(Tl) signal is roughly 20X smaller per unit energy loss. In practice, very similar electronics can be used and therefore were used for the readout of both silicon and CsI(Tl) detectors. Figure 2.8 shows an electronics diagram which describes the silicon or CsI(Tl) detectors of the Catania Hodoscope or the Silicon detectors of the IMF(SiLi) telescopes. The linear output of the Pre-amps has a signal that rises to a peak quickly in about 100 us then falls slowly in 40 us. This signal is split in the shaper and filtered to produce a fast and slow signal. The fast signal has a rise time of about 100 ns and a fall time of about 500 ns . The slow signal rises to a peak in about 2 as and falls to zero in about 5-8 as. The fast signal goes to the discriminator and the slow signal goes to a Phillips peak sensing ADC. The output of the discriminator is a standard negative Nim signal. The outputs of both discriminators are OR’d . One output of the OR goes to a stop input of a time to Fera converter(TFC) then to a Fera ADC where a channel number is determined that is proportional to the time of arrival relative to the event trigger, and another output goes to a large OR with signals from all of the telescopes . For each event, there will be an output signal from the OR which has a leading edge defined by the first detector that detects a particle in the event. This output signal is sent to the octal constant fraction(OCF). This OCF is indicated on both the dead-time logic diagram and on the logic diagram for the Catania and IMF telescopes. The output of the OCF is the pre-master signal. 22 ~4mm SI detector 300nm [ 7i 31 on] .1. '5 '27: Z ’ U 5’ E o o E 0 <7 (‘0 v Figure 2.7: Side view of the Catania telescope geometry. The front and rear surfaces are square. 23 PA Si ——-]>—— Shoper ___I ’ Phillips ADC Discriminator PA - - Csl > Phillips . Sho (SiLi ) per ADC Discriminator Del OR oy L, TFC [— stop Si or Csl All other telescopes OR ——> Fast Cleor Circuit Slot 7 PreMoster 0-7 -—> OCF 8000 Figure 2.8: Logic diagram. Applicable for both Catania or IMF telescopes. 24 . x 4 I \ \4 ., f s 4 / ... \\ \ o ]. lilllli . \ 4 / iii] [if \ miliiif Ink. . . lililillll V . . . | - c . nliii _ _ . , ll” ill I ll 4 , \ \ Figure 2.9: Catania Hodoscope frame. Drawing by Craig Snow. 25 2.4 IMF Telescopes Four silicon telescopes were used to measure heavier fragments with 4SZ39. Each telescope consisted of a 75 pm thick silicon AE, and a 5 mm thick Si(Li) E detector. The silicon AE’S are sufficiently uniform as to resolve isotopes up to 2210. Details of the isotopic resolution are discussed in the next chapter. These telescopes were placed at a distance of 27 cm from the target and subtended solid angles of 2.21 msr. The polar angles of the telescopes were 27, 36, 75, 84 degrees; Essentially the same electronics was used to process the signals from the IMF telescopes as for the telescopes of the Catania hodoscope. A description of the electronics, is given in the preceding section of this chapter. 2.5 Experimental trigger The premaster signals on the bottom of Figure 2.8 appears again in the top left corner of Figure 2.10 as inputs to the dead time logic diagram of the experiment . The 477 signal usually arrives first at the OCF because the plastic scintillator signals of the 477 detectors are considerably faster at generating a signal than the signals from the Catania Hodoscope or IMF telescopes. The premaster signal is converted to a master signal if the data acquisition system is not already busy with a previous event. One output of the master signal goes to the input of the 477 Master Logic FanIn/FanOut. Another part goes to the ADC gate circuitry on the upper right part of the diagram that opens the IMF or Catania ADC’s to take the data. If the computer is busy, however, the FaraFaucet will generate a busy signal to the dead time OR and this will veto the master signal. On the bottom of the diagram we have the fast clear cicuits. Because the 477 detector is intrinsically much faster than the other detectors we had to start the data acquisition to read the 477 data even before we knew whether 26 there is data from Catania or IMF telescopes. The purpose of this fast clear circuit logic signals is to clear the 477 ADC’s and TDC’s if there are no Catania hodoscope or IMF telescope signals. The fast clear circuit also provides a signal to the dead-time OR gate to block new data until the clearing process is complete. 2.6 Experimental measurements The objective of measuring the caloric curve in central collisions required measure- ments as a function of incident energy up to the maximum feasible energy of the cyclotron. As the beam intensity generally decreases with incident energy, the target thickness was varied to try to keep the overall counting rate of the experiment more constant. By this means, we kept the experiment counting rate at the maximum value limited by the acquisition system at all but the highest energy. Table 2.2 lists the beams, targets, trigger conditions and hodoscope angles for the various runs. For the main Ar+Sc and Kr+Nb data taking runs, the plan involved measuring coincidence data with the hodoscope and IMF trigger in place and also data with 477 running by itself (477 singles). These latter measurements provide an unbiased multiplicity distribution that can be used to relate the impact parameter to the charged particle multiplicity. Unfortunately, the 477 singles measurements were not accurately performed at 120 MeV/ A. At the other incident energies, these measurements were performed successfully. Looking at the table one can see that experimental runs were peformed using 86Kr beams at energies of 35, 70, 100, and 120 MeV / A using a variety of target thicknesses. Experimental runs were also performed using 36Ar beams, at energies of 50, 100, and 150 MeV/ A on a 45Sc target of 3mg/cm2 areal density. For both experimental runs the targets were mounted at an angle at 45° from the beam axis. As the angle of the center of the detector array was situated at about 40° the detected particles were 27 £0:me Smog 20 40:3 6e 00.»qu EV 4» 00008003 903404.634.“ 8.4 000980 £03800 0.35m "£02m _ _ 0300204. nu): HOMOC—D “O 0%m ,I. l 4.09 All. 0.3 036 l 535 6%..“ MUQ< 90> \ 0000 805 >200 0&3 A GEE 3 GB 00:0 43—09 .806 000.4 3.5 00090483 0:003 do 08:. 000m A , 930 08002034 mom _00 E): a OE.» 03.02 V :80 on: Figure 2.10: Dead time logic. 28 Beam E / A (MeV) Target Hodoscope angle (degrees) 86K r 120 Niobium 40mg/cm2 40.1 86K r 100 Niobium 20mg/cm2 42.6 86K r 70 Niobium 20mg/cm2 48.6 86K r 35 Niobium 20mg/cm2 55.9 86K r 35 Niobium 6mg/cm2 55.9 36Ar 150 Scandium 3mg/cm2 40.6 36Ar 100 Scandium 3mg/cm2 42.6 36Ar 50 Scandium 3mg/cm2 29.3 86Kr 35 Plastic(CH2) Home(40.6) 86m 35 Plastic(CH2) 44.8 86K r 35 Plastic(CH2) 49.8 86m 35 Plastic(CH2) 54.7 86 Kr 35 Plastic(CH2) 59.6 86Kr 35 Plastic(CH2) 64.6 86K r 35 Plastic(CH2) (Limit Switch(68.3) 4H e 22 Plastic(CH2) Home(40.6) 4He 21.79 Plastic(CH2) 50.70 4 He 21.79 Plastic(CH2) 60.64 4H e 40 Plastic(CH2) Home(40.6) a(2IQBi) 6.09/4 Si calibration N/A (228Th parent) 07(212P0) 8.78/4 Si calibration N/A (228Th parent) Table 2.2: Angles measured from beam axis to array center. The difference between the first detector and the center of the array is 18.60 29 emitted at 90° distributed about the normal to the center of the target. Table 2.2 shows the beam type, the beam energy, the target type and the angle of detector #1 of the Catania hodoscope for each measurement in the experiment. In addition to the main experimental Ar+Sc and Kr+Nb runs, there were cal- ibration runs during which hydrogen and helium isotopes of precise energies were measured in inverse kinematics for reactions on a hydrogen (polyethylene) target. Calibrations were also performed with alpha sources. The calibration data from these measurements are discussed in the next chapter. 30 Chapter 3 Calibration and data processing In this chapter we explain how the raw data for the Catania hodoscope that exists in the form of data words from the ADC’s and TDC’s, are converted into energies, charges and masses of specific particles. We also explain how information about the impact parameter for the collisions was extracted from the data for the 477 array. The extractions of temperatures and isoscaling properties from these calibrated data are described in chapters 4 and 5. 3.1 Data reduction plan Figure 3.1 gives an overview of the plan for converting the raw data, which moves as integer data from the ADC’S and TDC’s in the data acquisition setup, into the final results. In the first pass through the analysis, the uncalibrated data on tape is sorted and manipulated to give calibrated data on disk. This stage involves reducing the 477 data to obtain the charged particle multiplicity and identifying the particles in the hodoscope and determining their energies. These principal tasks are described in subsequent sections of this chapter. Below this first pass on the diagram, the analysis splits into two pathways. One 31 Data Reduction Plan Equ1pment I Tape Calibration I T Spectrum Calibration and Inspection 1 Disk Raw spectrum _ ‘ Correlated Mixed 1 Pairs Singles v . v Normalization Correlation Function 1 f v Single Line lshape ‘ 3 Particle Spectrum j Lab ——H Monte ———p Efficiency Cross Check Carlo 0 Yield in 900 itc‘fifisg‘: CMS P Isotope Double Ratio Excited States Ratio Temperatures Temperatures 0 _ B XC = l [(N /N )(2/7E 1)/(2J 1)] _ n 2 1 1+ 2 + ln[A *(Yi/Y2)/(Y3/ Y4)] Figure 3.1: Data reduction plan. 32 involves constructing differential multiplicity spectra, which are needed for the isoscal- ing and isotopic temperature analyses discussed in the subsequent section. The other pathway is used to construct the coincidence and mixed-singles analyses that are es— sential for the correlation function and excited state temperature analyses discussed in Chapter 5. Cross-links between these analyses pathways, indicated in the diagram, are important for the construction of correlation functions. These will be discussed further in Chapter 5. 3.2 Impact parameter selection As a function of impact parameter, many observables for the colliding system change rapidly. To the degree that the change in a specific observable depends strongly and monotonicaly on the impact parameter, the impact parameter can be determined from that observable. Examples of suitable observables that have been so used in- clude the observed total charged particle multiplicity N,, the midrapidity charge [89], the transverse energy [89] and the total charge of beam velocity fragments [91]. In this dissertation, we deduced the impact parameter for total charged particle multi- plicity NC. At incident velocities that exceed the fermi velocity 'Uf = (4%) z 0.3c. N, largely reflects emission from a ”participant” region formed by the merging of approximately equal numbers of nucleons from projectile and target where these two incident nuclei overlap. Both the size of the participant region and the charged par- ticle multiplicity decrease monotonically with impact parameter b ; the remainder of the nucleons are carried away from the collision in projectile- and target-like residues that emit few charged particles. Figure 3.2 shows the distributions for the total multiplicity of charged particles detected in the 477 detector for Kr+Nb collisions. These data were taken in mini- mum bias runs where the ball multiplicity trigger discriminator was set at voltages 33 corresponding to the acceptance of events with NC 2 1. Figure 3.3 shows the corre- sponding distributions for Ar+Sc collisions. Unfortunately, the minimum bias data for Kr+Nb collisions at E / A2120 MeV were not usable and are therefore not shown. If one assumes the charged particle multiplicity decreases monotonically with im- pact parameter, one can use these multiplicity distributions to obtain a relationship between N, and the ”reduced” impact parameter b = b/bMAx, where bMAX is the maximum value of b corresponding to the lowest multiplicities included in the trigger. This relationship is: 13: (11.14;) = \/ N: P(NC)dNC, (3.1) where: Events(NC) P N7 = .2 ( C) 1,322, Events(NC)dNC’ (3 ) Figures 3.2, and 3.3 show the relationship between b and NC given by 3.1 for the Kr+Nb and Ar+Sc collisions, respectively. The relationship for Kr+Nb collisions at E / A2120 MeV, where minimum bias data was unavailable, was extrapolated from the Kr+Nb data at E/A=100 MeV by rescaling the P(NC) of E/A=100 to the E/A=120 MeV P(NC). Careful examination of Figures 3.2 and 3.3 reveals that the majority of events occur at large impact parameters characterized by low charged particle multiplicity. In order to acquire data at small impact parameter more efficiently, the bulk of the data were taken with a hardware threshold on the ball multiplicity trigger discriminator set at voltages corresponding to a higher minimum multiplicity Nam-n. Table 3.1 shows the values for Nam-n for the 477 ball multiplicity trigger setting during the main data taking runs. Even higher thresholds corresponding to higher Nam-n and consequently smaller 6 were set later in software during the analyses. The main analyses in this 34 I :l I I I I I I I I I I I I I I I I I II II t: Tj’ I I I I I I I I I I I I I I I I II I I I L 1 _1~ 4 10 E. ' . - . . ' I “ _25 . Kr+Nb at 35 MeV/A j0.8 10 g” ' . 7 —3E _ A . “0.6 \Z: lO_ E‘ -. : 8 (l 10 g - L04 —5.2 I _ 10 g - _ : —65 —O.2 10 E— . I . : 1 a --.. , ~15 :t 4 10 ts" ar ~ _ 2 g 1:— Kr+Nb at 70 MeV/A i 0.8 10 E. I I I I I I EC. F $10—45:— I‘ll-III.- éi: ‘EO.6 3 —4E . I i]: : B (l 10 E— . I 3" “0.4 -5: . :_r_ : 10 E— 'Et' -1 —6: :h ~02 10 :5— EI- : 1 : I41 I QMAl Ll I 1 114 l l l L l l l FLI l l l l l 1 l l l l 1 1 l4 1;; A r ‘I EI I I I I I I I I I I It IIIj I I I I I _,l—I I I I I I I I I I I I I II I I III I I I L —1 : :i 4 10 g— 3“. _. —25 ii— Kr+Nb at 100 MeV/A —O.8 10 E— ‘3“ :l -3: E; _—]O 6 f3 10 g1 E 4 ' \Z/ “ E ' u I I I I . . . I . . . ' ES]: j 6 Q— 10 E" I . ' I . ‘5— “0.4 ~5: - 57 7 10 g" ' raC 1 —BE 5*— ‘02 10 lEr a : : LJ l L11 1 l 1 L1 1 I l l l l l l_L 1 L Cl Ll lL l l l l 14 L1 l l l l I l l l l 5 10 15 20 25 5 10 15 20 25 NC Nc Figure 3.2: P(NC) (left panels) and Reduced impact parameter (right panels) vs charged particle multiplicity for Kr+Nb at 35,70,100 respectively . 35 1 _I I I I IIIIT frrI I I I I I I I I I I -'I I I I I II I I I rII I I I III IFI I I 1 ~15 _J 10 g— « _2 §_ . - - - - . _ Ar+Sc atSO MeV/A 10.8 IO 5— . I -« —3E - « :3 IO 5* ' :06 6 —4E ' I 8 ‘l 10 E" ' 304 -5E I "‘ I —65 ' ~02 10 E— o ; I '_ =‘ -_ 1 10—1 I: 1 __ 2 E: I; Ar+Sc at 100 MeV/A {0.8 10 I I I I I I I . . El: _4 A 10%;; . ' 2‘: 50.6 g: _ E - “it : s a 10 E- . "a? 50.4 —55 in « 10 g- . EL : —65 ' EF- “0.2 10 g— I - a: j ‘I :1_Tl_%liJcL1 LL1J_1 lLlii l LJ 1 L III-14 1 L1 1 LL Li L44 Ll ALLLf 1 E IIIIIIII I I II I I I I I I I I II HHII III I I I I I III I rI I I ITIIII L —1 : 1- 4 10 +- 4 _2 F I— Ar+Sc at 150 MeV/A -: 0.8 10 E: I I I I I . . . EI I ,3 10‘ g ' ' . 4;;— ~06 \2/ —4E '- it 1 6 a 10 g.- '. 1L 904 "SE I 3— I . 10 g , 2:: : *BE I E_ _O.2 10 E - a; j l 14 lLLl LA l l l l l l l l l l l l l l L FHIJ lJLl LLl l l l I l l LLJ. 5 10 I5 20 25 5 IO 15 20 25 NC NC Figure 3.3: P(NC) (left panels) and Reduced impact parameter (right panels) vs charged particle multiplicity for Ar+Sc at 50,100,150 respectively . 36 Beam and target Beam E/ A (MeV) Multiplicity Trigger Thresholds 86Kr on 93Nb 35 2,5 86Kr on 93N b 70 5 86Kr on 93Nb 100 4 86K?" on 93NI) 120 4 36A?" on 4550 5O 3 36A?" on 45Sc 100 5 36Ar on 45Sc 150 5 Table 3.1: System type and Ball trigger settings. dissertation were performed with software cuts corresponding to I) S 0.45 and I) _<_ 0.21. We note that these minimum settings practically eliminate false triggers on cosmic rays. Even at minimum bias settings, cosmic ray triggers contributed at the level of one percent or less due to the large counting rate difference between the collision and cosmic ray counting rates. 3.3 Particle Identification Both the silicon - CsI(Tl) telescopes of the Catania Hodoscope and the Silicon — Si(Li) heavy ion telescopes used the mass and charge dependence of electronic energy loss to distinguish the masses and charges of the various particles. This energy loss AE, described by the Bethe-Bloch formula, is given approximately by AB = con,9t-Z2-A/E-f(v) (3.3) where Z, A and E are the proton number, mass number and energy of the charged particle, and f (v) is a slow (logarithmic) function of the velocity of the particle. It results in characteristically hyperbolic relationships between the energy loss in the first (silicon) AE detector and second (Si(Li) or CsI(Tl)) stopping detectors in the stack. These correlations are shown and discussed below for both the IMF and Hodoscope 37 telescopes. 3.3.1 IMF telescope PID Figure 3.4 shows the correlation between the energy loss in the silicon AE detector and energy deposited in the Si(Li) stopping detector. Between the two red solid curves in the figure, one can see scatter plot contours with higher counts that correspond to the energy loss relationships expected for the various lithium isotopes 6Lz', 7Lz', 8Li, and 9Li. These red solid curves, drawn using standard energy loss predictions, can be used to define an experimental PID observable that assumes different values for each isotopes. For example, to obtain the PID for the lithium isotopes, one uses the upper PIDB and lower PIDA solid curves to interpolate a PID value as follows: Xp—XA PID .—.———— P XB—XA( PIDB — PIDA) + PIDA. (3.4) This provides a single observable PIDp that assumes a constant value for all particles of a given isotope, regardless of their energies. After similar interpolations are performed for all of the particles, one obtains the results shown in Fig. 3.5 where the PID values are plotted as a function of the energy of the detected particle. The different isotopes are clearly separated, enabling two dimensional gates to be set about each of the relevant isotopes. Fig. 3.6 shows the typical resolution between the different isotopes, obtained when this two dimensional spectrum is projected on the y-axis. Clearly, the resolution on this one dimensional spectrum is quite good. Nevertheless, the use of two dimensional gates, such as the one drawn in Fig. 3.5 about the 10B data, provides a cleaner distinction between neighboring isotopes. 38 2000 1750 1500 1250 dE 1000 750 500 250 llllllJllllllLllllllllllllllllllllLlPll 0 200 400 600 800 1090 1200 1400 1600 1800 2000 Figure 3.4: IMF telescope dE vs E spectrum for Kr+Nb at 120 MeV/ A. Here a point is being interpolated in the 7Li band between the red marked PID curves above and below the Lithium region of the spectrum. 39 P10 ' r v rvv . . . v -‘... . - .‘f ... ' 0.. . ...'.\:;’“ L ‘ “I", .".‘.'."‘H"- \ I3: .,‘ '- K 5”!» \ v .-‘.‘: . - it}? " . "" ’r ta" .0 3:" 2' 3‘. ..Gt'It‘IZf'I’IF I‘I'fi 7,\"- \ ._. ‘. . . L. - . .‘, . I ' . I . . "-.- ‘ I ' ' : - '. . ‘ " 3V“ -'( .1: f‘ '.' II ' . ‘ ' n‘. , _g ‘ . . fix?» 95334.2 {.53.- ,. - ., ... --.3 . .,_ . . .. . . ' . ' ,- 1'. "1:..L‘iur'l l 1"1'1' l'l-I‘} 1“". ' ..L i-‘nr in liLLlllAlllJLl lllllllllJLllJJLl 0 250 500 750 1000 125 150 Energy MeV Figure 3.5: A graphical cut to extract the yield of (E)beam=35 A/U 40 1750 2000 2250 2500 10B for Kr+Nb colisions at 7 . _ LI 6 . _ LI 8 ° 1 103: LI 1. _ 9 _ Be ~ 7 _ Be ‘08 13C e 12 - C — 14 C _ 9 . (1) LI _+_J 11 C1o2~ C 3 E O E Q t 10 _— : 1m l 1 ll ll 1 ll 1 l l l 1L] 1 l l [J #L1 11 l 1 lJ L l l PlD Figure 3.6: IMF pid spectrum for Kr+Nb histogram plot (E)beam = 35 A/U, The yield is obtained by integrating a peak between minima. The yield of 11B is being extracted here. 41 3.3.2 Catania Hodoscope PID PID observables were constructed for telescopes in the Catania Hodoscope similar to those constructed for the heavy ion telescopes. The PID gates on each isotope were de- fined by quartic curves that were fit so as to follow the valleys of minimum counts be- tween neighboring isotopes. Gates for twelve isotopes, p,d,t,3H 8,4 H e,6H e,6Lz',7Lz',8Lz', 7Be,QB 100 MeV and below such energies, to view the emission of Intermediate Mass Fragments I M F (20 > Z > 2) and Light Charged Particles LCP(Z < 3) as originating from three sources after collision as given schematically in figure 4.1. There is a central source commonly termed the participant source. In addition, there is a much slower source and a much faster source from the decay of the target and projectile spectator remnants, respectively [88,16,122,41l In this simplified picture, illustrated in Fig. 4.1, the participant region is com- posed of target nucleons, whose trajectories intersect the projectile and of projectile nucleons, whose trajectories intersect the target. All the other target and projec- tile nucleons comprise the target and projectile spectators, respectively. Naturally, the relative sizes of projectile and target spectators reflect the mass asymmetry of the entrance channel. They have the same masses for symmetric collsions between identical protectiles and targets. The target spectator mass will exceed that of the projectile spectator when the target mass is larger than that of the projectile, etc. Very often, the emission patterns of these three sources are assumed to be isotropic in their center of mass frames. However, this is an idealization because the rapid emission from the participant source, envisioned by this picture, may preclude the complete mixing of the projectile and target nucleons in the participant and the re- sulting participant emission may appear as if it were from a continuum of sources, a concept proposed first in the original ”firestreak” model. [41] Such a continuum of sources is closer to the predictions of dynamical calculations such as those obtained by the Boltzmann-Uehling—Uhlenbeck (BUU) equation. [9] Energy spectra for hydrogen and helium isotopes emitted in Kr+Nb reactions at 13 g 0.45 and E/A = 35, 70 100 and 120 MeV are shown in Figs. 4.2 and 4.3. At 49 projectile -—> Figure 4.1: In the participant-spectator model, it is assumed that the emission pat- tern from the collision can be approximated as originating a participant source formed by the overlap of the projectile and target nuclei during the collision and from pro jec- tile and target spectator sources that contain nucleons whose trajectories avoid the overlap region. 50 most angles, the spectra display maxima at low energies and decrease smoothly with energy. The variation of the cross section with scattering angle and energy is large; care is needed in the analysis of such spectra into order to get accurate differential or total multiplicities for the various particle species. In this dissertation, we first try a simple decomposition of the emission pattern into participant and projectile and target spectator sources in order to accurately parameterize the emission over the angular domain explored by the Catania hodoscope. We modify the form of this decomposition as needed to improve the description of the data. The hodoscope was centered about angles that correspond to 00M z 90" in the rest frame for the collision of a nucleon from the projectile with one from the target. Since this corresponds closely to nucleons emitted to BCM = 90° from the participant source, accurate fitting of the participant source is more important than describing the spectator sources which emit mainly to unmeasured scattering angles. 4.1.2 Single Particle spectra fitting In the following we start our description of the emission from participant, target and projectile spectator sources with the expression of Ref. [52, 53], which was used in that work to describe the spectra of fragments with 23Z§8 produced in Au+Au collisions at E/A:100 MeV. We have used different forms for the fitting expression, which differ in the way that the effects of collective radial expansion of the participant source and for sideward deflection of the two spectator sources are handled. When these effects are included, we use the following expression: d2(M)/deE 2 d2 1 21r R 3 2 i 51 where E denotes the kinetic energy and p, the momentum of the emitted particle; 1),- denotes the velocity and V,, the effective Coulomb barrier of the i-th source; and (DR denotes the azimuthal angle of the reaction plane [53]. Here, 15% (1),, 0, V,) is defined in the rest frame of the source 1),- = 0 by a Relativistic Boltzmann distribution[62, 86]: 47r d? 1 d P ( )‘ < M,- > dEdQ < M.- > dE d E+m02—V- E ’ 2 _ i E 2 _ V2" 2 _ 2 4 — Ti E @(E—Vi)( +ch V)\/((2 +mcr ’) 2mc)e d , (4.2) 2(T—n?)21r1(-"i;—) + (WWW; ) "1366 and %%2 (p,, 1),- ($12)) is obtained from Eq. 4.2 by Lorentz transformation. In Eq. 4.2, T,- is the temperature of the Maxwellian source and 9(E —— V,) is the unit step function and E is the kinetic energy of the particle. The sum in eq 4.1 is over the target and projectile spectator sources, M,- is the particle multiplicity, V,- is the kinetic energy gained by Coulomb repulsion and E,- is kinetic energy. K0 and K1 are MacDonald functions. Note that this equation reduces to the standard classical Maxwell—Boltzmann Distribution for E,- = 0 and V,- = 0 and small T. (PM) In some of the fits, the form of the participant source dEdQ (1),) was chosen in order to model a collective radial flow. For this, a self similar radial expansion WW?) 2 cflerP?/Rs of the spherical participant source (i=1 in Eq. 4.1) was assumed which attains its maximum velocity cfiexp at the surface 7“ : R3. The velocities of individual particles were assumed to be thermally distributed about the local radial expansion velocity with temperature T1. Coulomb expansion after breakup was mod- elled in the limit of large 735W; particles with charge Z f , emitted from a source with charge Z, , gained a kinetic energy of AECW,(7') = Z f(Zs — Z f)e2r2/R3, without changing their direction. In the em. frame one obtains[52]: Rs dam/11> 3 i 2 d2P’ : - . QT, ’ , , I—E E .015 . , 4' dEdQ 417ng ([7 dr/d [dE dEIdQ.(p ”(7') 0)5(E +A C 1(7)) ( 3) 52 where the direction of the particle’s momentum is assumed to be unchanged. The total spectrum is obtained by inserting Eq. 4.3 into Eq. 4.1 as the participant source. In other fits, the collective expansion of the participant source was neglected, the form of the participant source % (131) in its rest frame was given by Eq. 4.2. Regardless of the choice of fitting function, the source spectra must be transformed from the rest frame of the source to the LAB frame, using the formula: d2< Mi )/deEzab = (plab/pso,i)*d2(Mi> / dQ dEsm where E30 and p30,,- are the energy and momentum of the particle in the source frame. In general, Eq. 4.1 has up to 13 free parameters for a three source system, Three multiplicity parameters M,- , three temperatures T,- , three Coulomb parameters Z,- , one expansion parameter 6“,, for the participant, a maximum Coulomb energy for emission from the surface of the participant, one parallel fig and one perpendicular 6x velocity component for the spectators relative to the participant. For symmetric collisions, one can assume certain symmetries in the fitting parameters for the two spectator sources, i.e. (M )2 = (ll/[)3 , Z2 = Z3 and T2 = T3. Even though Kr+Nb and Ar+Sc are not precisely symmetric systems, we invoke this constraint because the experimental data are over an angular domain that avoids much of the spectator sources and is therefore insuflicient to constrain the projectile and target spectator sources independently. Later, the influence of the remaining asymmetry in the emis- sion pattern was addressed by including a multiplicative anisotropy on the overall emission pattern. Fits to the spectra for hydrogen and helium isotopes using Eq. 4.1, with M,, T,- and Z; as fit parameters and non-zero values for the collective expansion velocity 6“,, are shown by the solid lines in Figs. 4.2 and 4.3. The parameters for these fits are given in Table 4.1. In general, the temperature and source velocity fit parameters for the various particles differ significantly at each incident energy. Corresponding 53 fits without collective expansion for 3H8 and 4He are shown in Figs. 4.5 and 4.6 of the next section and the parameters for these fits are given in Table 4.2. The fits with and without collective expansion are comparable. Because the collective part of the energy of the detected fragment Eco” % 1/2Mfmgvfou is proportional to the fragment mass, clear sensitivity to the Beam may only expected for heavier fragments with A 2 12 [53]. In any case, the source parameters for the various particles are not consistant with three defined unique sources. This may indicate either that the sources are not well defined by analysis, or that the three source analysis neglects important effects in the emission or both. Certainly, the angular domain of the data is not sufficient to show the parameters of the spectator accurately enough. The lack of coverage in the experiment at the more forward angles precludes the extraction of the parameters describing the projectile source. Loss of coverage at forward angles also removes lower parts of the center of mass spectra for the participant source and makes it difficult to extract the collective radial velocity from the mean values of the kinetic energy in the center of mass. The lack of coverage in the backward angles makes it difficult to determine the parameters of the target spectator source. In general, the parameters of the fit procedure are not determined uniquely by the present analysis. The values for the parameters are model dependent and less directly connected to the actually measurements than one would like. This is not a seriuos problem for the experiment because the main goal is to determine the excited state population of emitted particle fragments. This is easily achieved with this experimental setup. We therefore use the fits only to assist us in interpolating the data so as to obtain accurate values for the cross section at angles near 00M z 90", and to enable simulation the detection efficiency for the measurements of excited state populations as discussed in chapter 5. For this purpose, the values for the parameters are immaterial; the only relevant issue is whether the fits accurately describe the 54 data. 55 .6808 2082898 :2ku 8.528 68:8 m 2:. mafia QZ+2M :8 232888.82 p@ $05.82 mfiwfim 20A ”:2 @388 8 2.2 8: 882 888 2.2 8.: 828 88.8 2.8 888 O2 3: 8 8.2 228 2.3.: 28.8 8.8 8.: 88.8 888 2.8 88.8 02 an... 8 £8 8.: 88.2. 22:8 8.: 8.2 823 888 28 88.8 2 s 8 £8 8.: 2288 88.: 8.8 8.: 2:3 88.8 2.8 88.8 82 e 2 8.2 88 88.: 8288 8.2 :2 N83 2828 Ed 823 2: 8:2. 8 8.2 88 82:: 82.0 8.2 2.8 22.8 88.8 2.8 28.0 2: m2... 8 £8 8.: 282m 82.: 83: 2.2 22.8 88.0 28 8225 2: s 8 £8 8.: 2:2. 88.8 8.2 8.8 82.8 :88 :8 822: 2: e 8 8.2 2; 223 288.8 2.8 8.2 883 228.8 :8 :83 E ...:.. 8 2.: 88 888 82.8 8.2 8.8 02:8 2:3 :8 :83 E ems 8 8.8 8.: 283 E88 2.3 8.2 883 888 :8 :83 E s 8 2.28 8.: 828 @288 2 88 22.8 88.8 :8 82.0 2. 8 2 >5. >82 EAEV sAEV >22 >22 - - - - >822. - we: as: s: 2822232 2282252 22.2 s2 28an 338 6:538 .58. 28m 2228 56 Kr+Nb B _<_ 0.45 E i _21 E/A=7O MeV — : -2 10 g— { " ‘ E‘ .-,. “E 10 g WW (1 E 2:3’ t 5 -3” 30.0. ’ 30. ‘ -3 10 g 36.5‘ g 36.5' 1; 1o 5 43.2' E 43.2' 3 _4~ 49.8“ - 49.8‘ * -4 10 -— 56.:5' *1, L— 56.3’ —; 10 EL613LOO11141Li14_LLL111#ELq310.111L1 1 i 1 LLl LL1 11: I E/A=1OO MeV C : -2 -2 11.110 :— :— 110 c _ _ a ‘0 : 21.5‘ C 21'5' : \ -:5 28.5’ 28.5’ -3 A10 g— 35.0’ , g 35.0' :10 2 : 41.5‘ : 41.5‘ : “V : 48.5‘ 1 48.5‘ 2 .010—4_4 51435.1 144 1 1 1 L4 1 11 1 1 14 : 1 51415.1 11 ;;1 i 144 4 J 1 1 1 J : 1O _‘ 5 E/A=120 MeV 5 3 -2* ' ‘ —2 10 E— _— :10 E 21.5' E 21.5 E -3; 28.5’ . ~ 28.5’ i -3 10 g 35.0' g 35.0‘ ? 10 E 41.5' i 41.5' ' Data 3 _4~ 48.5’ t 48.5’ — Fit 1 -4 10 E_1 1415.1 1 1 1 1 1 l 1 1 1 1 i 1 1 1 1 15415.1 [4 L 1 L l 1 1 1 1 l 1 1 1 1o 0 5O 1 00 150 0 50 100 1 50 200 E“, (MeV) Figure 4.2: d,t fits using the 3 source fit with a radially expanding isotropic participant source. 57 Kr+Nb B s 0.45 I l LE/A=7O MeV 'M -2 ' .r 3‘0 1° 5 1 ~ 1 -3 330 _310 10 E33 3 ...4 o .1 ...; ' 4 . ' 10 I 1.5' l lllllll (ANN d’1dodE O 3.3.33 O l 0 523.2176 1W UIJ-‘s-F :“ Ill lillll .— .— .1 A .. l' _3 ‘( {’, —l d 0| O # I [111111] / rfi—rIIIHI—I 1111111] ##UNN . .0. § . 21.5’ '28.5’ .212: 5:121. . 1 . 111 , O 200 400 0 Eu, (MeV) Figure 4.3: 3He,4He fits using the 3 source fit with a radially expanding isotropic participant source. 58 —2 10 _- : Kr+Nb at 120 AMeV L. L l —3 L11 10 :— U 2 (:1 _ "O \ l’ NE 1- U r- —4 10 f C r O 100 200 300 400 500 Energy(MeV) Figure 4.4: 4He fit for Kr+Nb at 120 AMeV of figure 4.3, the green curve fit is the fit that includes the asymmetric parameterization of equation 4.4, the data highlighting at 700300149100. 59 4.2 Improved fits and extraction of the single par- ticle observables 4.2.1 Introduction of source anisotropies Most of the fits to the hydrogen and helium isotope spectra in Figs. 4.2 — 4.3 are sufficient to allow extraction of the cross section at angles near 00M 2 90° to within an estimate accuracy of 15 percent. Fits to the spectra for (1 particles at E/ A = 120 do not fit the data at the important angles and energies (indicated as the red points in Figure 4.4) that correspond to 7003001143110”. This problem becomes more severe when fitting the spectra for even heavier particles. 4.2.2 Asymmetric source parameterization In order to obtain a better representation of the data, we have explored alternative fitting functions. It appears that these difficulties can basically be surmounted by making the predicted emission pattern more asymmetric than Eq. 4.2 predicts. The resulting formula is inelegant, but does succeed in representing the data well, enabling the desired extraction of the cross section at angles near 00M z 90°. The inclusion of asymmetric terms in the source fits were needed by problems we observed with the isotropy of the energy spectra in the GM. system. A Fourier cosine series was included as a multiplicative factor to correct for this. This shaping may reflect the small asymmetry in the entrance channel or that the dynamical emission process is poorly approximated by a set of isotropically emitting sources. With this modification, the source function in the CM frame becomes: 60 d2(M) / deE = d2— \ : . 1? .. “(3 l— —4 10 g E -5 10 E O 200 400 O 200 400 Energy(MeV) Energy Figure 4.6: Plot of (ii—1:319 versus Energy. A Fourier perturbed 3 source fit of 4He for 86Kr+93Nb at 100,120 MeV/ A and 36Ar+45Sc at 100,150 MeV/ A respectively. 63 : *- ,o " _ ,1 I ’ »— §§ 9 O N + Kr+Nbot100 MeV - l Kr+Nb ot120A - LL] 4 U _ C: 10 :- _ ‘0 _ + Z \ — l _ NE : ” U _ 1 : _ I . ~5 10 _— LL] 8 —4 .0 10 L— \ i NE l— “ f _5 ' 10 l:— '1 } I 1 .1 l .1 O 200 400 O 200 400 Energy(MeV) Energy(MeV) Figure 4.7: Plot of 2617327346 versus Energy. A Fourier perturbed 3 source fit of 6Li for 86Kr+93Nb at 100,120 MeV/ A and 36Ar+45Sc at 100,150 MeV/ A respectively. 64 163 ._ Kr+Nbot1 :_ Lu - _ 8 ~ . ‘O \ -~4 ”U : : 5 | 10 E— 1 4 4 L L L 1 :— 10—3 _. E r p l- _- LL] F U L .— C 1. U _4 C E 10 _— * N Z P U ~ r -5 1O " 7 L— 4 1 1 l I 7 '11 O 200 400 O 200 400 Energy(MeV) Energy(MeV) Figure 4.8: Plot of 3%"5 versus Energy. A Fourier perturbed 3 source fit of 7Li for 86Kr+93Nb at 100,120 MeV/ A and 36Ar+45Sc at 100,150 MeV/ A respectively. 65 l TlTTll fifiTfi .\ .H I. dZM/deE l A I I T I ITYI l T T TTFTTI I 10 dZM/deE 1 l 7 , 1 l [11 1 l o 200 ' 400 o 200 7 400 * Energy(MeV) Energy(MeV) Figure 4.9: Plot of :2ny versus Energy. A Fourier perturbed 3 source fit of 8Li for 86Kr+93Nb at 100,120 MeV/ A and 36Ar+4SSc at 100,150 MeV/ A respectively. 66 + t t ,1! i 4 + +Nb at 100 AMe +Nb at 120 AMeV 1O - + t :— L1J F : ‘O l— c: _ u f _ l \ __ .- E "O _ .— 4 ” r _5 l ‘ 10 f — 1 , 1 1 4i,, L, l. 1 -4 ch LL110 f D _ c: _ U _ \ ._ E e ‘20 >— —5 l— 10 F“ .1; 1r— 1 1 1, l1l1l [L1, ‘ O 200 400 O 700 400 Energy(MeV) Energy(MeV) Figure 4.10: Plot of 3%.??? versus Energy. A Fourier perturbed 3 source fit of 7Be for 86Kr+93Nb at 100,120 MeV/ A and 36Ar+45Sc at 100,150 MeV/ A respectively. 67 if} u 1 I!" _. +1 l + _ “I 10 _— Kr+Nb of 100 Avlev : +Nb at 120 AMe 1.1 I C a + — ‘O _ _ \ l E a l. F *— 10-5 ~— — i F 1 1 1 l 1H 1 1 1 11 [JD 10_4 2 f F ”a“ f Sc at 100 AMeV f LU h '- 0‘ 8 - l /\ g t 150 AMeV 2 F .. NU J1 _ ll KN» 4 l 165 _ {I .__ *‘ 0 ” 1 11,. .1 1i! lllll. O 200 400 O 200 400 Energy(MeV) Energy(MeV) Figure 4.11: Plot of (iii—(1N5!) versus Energy. A Fourier perturbed 3 source fit of 9Be for 86Kr+93Nb at 100,120 MeV/ A and 36Ar+458c at 100,150 MeV/ A respectively. 68 -.. * + 10 E— . Kr+Nbo .0 .V ' E— “ Kr+Nb+oi . ' ' : E ll L1_J F ”I ' .+ l U +— '- G ‘ 3 -5 | 2 10 E‘ l r NE E E 1 E t ,_ 1 l l J 1 J L l L l C : LcotISIJAMeV LLJ L— 0 Q 0 C6 *5 \ 10 E <1 NE — 0 U i _ lr _ " \0 J} A 4 l I l . \ \ 1 O 200 400 O 200 400 Energy(MeV) Energy Figure 4.12: Plot of jig—3?, versus Energy. A Fourier perturbed 3 source fit of 10Be for 86Kr+93Nb at 100,120 MeV/ A and 36Ar+45Sc at 100,150 MeV/ A respectively. 69 .033 m5 5 53m 98 mass 356580 258 oflumafixmm mahdoflmfisfima “238 was #88598 RES 85:8 ”5053» 60.38 m «5 mafia mumpmfimhwa E Eugen Bwfim ”m6 «BEL 25 8mg 823 «was $8.3 $3.2 $de $23 3 omsm+u< mm: 333 Sad Sid as? 88.3 88.8 km; 88¢ od oflomt< so NEE ago 253 28a 88.3 83me mam; 886 9.0 omsmt< was o as: ammo 35¢ 83% 89:6 325 ES 3 omzztm mm: o 833 m3; 884 88.3 38.: E3 98; ed 8.3th E 0 $26 8m? 2:3 333 :32 £26 $8.0 ed 8371.2 25 - - €sz sAEV >22 >22 - - - >23 a? - as s £23332 £23332 255 as $an 3min 63:3 58% sates 70 Because of its large number of terms, it is difficult to assess what changes in the the emission pattern Eq. 4.4 introduces that makes it better represent the data than did Eqs. 4.1 and 4.2. One way to represent this is to examine the predicted emission patterns as functions of the transverse momentum p 1 = W and the rapidity parallel to the beam. Figs. 4.19 and 4.20 show d2(M)/pdEd§2 fits for 4He for the three source fit using Eq. 4.2 for Fig. 4.20 and the three source fit using Eq. 4.5 for Fig.4.19. In these figures, the colored region indicates the calculated yield within the experimental acceptance bounded by the solid lines and the dashed lines indicate the angular domain 700 g QCM g 110° over which the data were averaged to obtain the final results for data at 6C,” z 90°. Basically, these improvements in the fits resulted in a slight backward peaked anisotropy in the emission pattern reflecting the fact that the 93N b target has a charge Z that is 6 larger than that of the 8°Kr projectile. Figs. 4.13 through 4.18 show a similar trend for the Ar+Sc system for the beam energy of 150AMeV and for °Li, 7Be and 4He respectively. Using these fits, accurate interpolations of the energy spectra of isotopes emitted into the angular interval 700 3 60M 3 110° were obtained. We cross checked this approach by comparing the integration of the fitting function over this angular domain d2 M (1150,,ds20M) where to the the average of the experimental data ( (PM _ _i ‘2’: (PM dE,a,,dn,ab dECMdQC'M>1V 1 dEzabszab dECMdQCM, Here the sum runs over the measured laboratory angles and energies that correspond to the given center of mass energy and over the center of mass angular interval 700 3 00M 3 110°. While the above expression has the advantage of being directly based on experiment, it can potentially lead to errors because the placement of detectors in the lab may not provide a sufficiently uniform coverage of the laboratory center of mass angular domain. Averaging the fit function over 700 3 60M 3 110° avoids this problem. 71 —5 x10 0.5 T i 0.4 I l r 0.45 :— [ .... 035 : l ' l— 0.4 H l i l C 1 ~— 0.3 0.55 T . I i l - 0.25 0.3 — I 2 C ‘ ‘c b . E l— l‘ 0.25 ~— i J 0.2 C 0.2 _— Z 0.15 0.15 l I 0.1 0.1 L E : L : 0.05 0.05 f ..' ; b Ll l_1 14 1 L1 l L1 1 kl 1 1_l 1 l4 1 l 1' l 1 l 1 L1 1 14 J l l l l J_1 l 1 1 l L l l 1 0.05 0.1 0.15 0.2 0. O 5 0.5 0.35 0.4 0.45 0.5 Figure 4.13: Rapidity plot of Catania acceptance region of 3 source with expanding central midsource with Fourier perturbation for 4He for Ar+Sc at 150MeV. The solid lines delineate the experimental acceptance.The dashed lines correspond to the gate 700 3 00M 3 110° 72 0.5 0.45 — . : ... 0.35 r 0.4 l:— C -— 0.3 0.55 L— l- ; -— 0.25 0.3 ~ 2 t \c _ CZ _ 0.25 — 0.2 0.2 l : 0.15 — l 0.15 — r C 0.1 0.1 — . C . .' _ ‘. .' 0.05 0.05 T ‘|‘ '1' l- I‘ : i—ILJJJ.1111ill—Lll_1mllLlll_11‘i'Jlll]lllllllllllllLlJlL O 0.05 0.1 0.15 0.2 0&5 0.3 0.35 0.4 0.45 0.5 Figure 4.14: Rapidity plot of Catania acceptance region of 3 source with standard expanding central source for 4He for Ar+Sc at 150MeV. The solid lines delineate the experimental acceptance.The dashed lines correspond to the gate 700 3 00M 3 110° 73 0.5 0.45 0.4 0.35 NRA/M .0 01 04 0.2 0.15 0.1 0.05 ElleITlrlllllllllleTTlITIIIIIFITIIIIlllelj—[TTTII O 0.05 0.1 _L__1_141_l_l_-l _ l_llllllljllu'lLlJillLLllLllLllJllllLl 0.15 0.2 0.25 Y 0.3 0.35 0.4 0.45 l 1 l l | 1 '01 ' 7' 0.08 0.06 0.04 0.02 Figure 4.15: Rapidity plot of Catania acceptance region of 3 source with expanding central source with Fourier perturbation for 6Li for Ar+Sc at 150MeV. The solid lines delineate the experimental acceptance.The dashed lines correspond to the gate 70° 3 60M 3 110° 74 0.5 0.45 0.4 0.35 NRJM _o 01 L»! 0.2 0.15 0.1 0.05 —6 x10 T 0.2 L 0.18 l _ - 0.16 E e H 0.14 p _ -- 0.12 C l 0.1 C l— 5 - 0.08 f 0.06 i. 0.04 . r 2 0.02 t |‘ '0 _lLlLlldllliLl ll ll lLllll 1“'4L1_1l4111L1111L11L1J1J 1A .0 0 0.05 0.1 0.15 0.2 0&5 0.3 0.35 0.4 0.45 0.5 Figure 4.16: Rapidity plot of Catania acceptance region of 3 source with standard expanding central source for 6Li for Ar+Sc at 150MeV. The solid lines delineate the experimental acceptance.The dashed lines correspond to the gate 70° 3 90M 3 110° 75 0.5 0.45 0.4 0.35 0.2 0.1 0.05 llllIllllllllllTlTlrlTlllilllllleTlTlTlllfTTlTlll O 0.25 Y 0' lllllL114llL_11L1111llL11LlllglLLALlig1LLllLllJLLl 0.35 —6 x 10 0.2 , 0.16 0.14 70.1 ' 0.08 - 0.06 0.04 0.02 Figure 4.17: Rapidity plot of Catania acceptance region of 3 source with expanding central source with Fourier perturbation for 7Be for Ar+Sc at 150MeV. The solid lines delineate the experimental acceptance.The dashed lines correspond to the gate 70° 3 BCM 3 110° 76 0.5 0.45 0.4 0.35 mRa/M .0 01 04 0.2 0.15 0.1 0.05 llllllrlllll[IIIlllTTTerfiTTTIjTTTITTITITTTI‘IIITY O 1411 0.05 0.1 0.15 0.2 0.25 Y 0.3 0.35 0.4 0.45 0.5 —6 x 10 0.2 0.14 0.1 0.08 0.06 0.04 0.02 I, Il11L1111111111l1111‘llllll11111111l4L14iL11LL-O Figure 4.18: Rapidity plot of Catania acceptance region of 3 source with standard expanding central source for 7Be for Ar+Sc at 150MeV. The solid lines delineate the experimental acceptance.The dashed lines correspond to the gate 70° 3 60M 3 110° 77 x10 0.5 _— 0.25 L 045 _ ’ 0.225 0.4 f .... 0.2 C 1.. 0.55 5 - 0.175 0.3 L -— 0.15 2 2 \t _ a L— 025 — 0.125 0.2 L 0.1 l. C . 0.15 F 0.075 C 0.1 _— 0.05 C . 0.05 r 0.025 1: ‘|‘ ." L—1 L1 1_1 1 i411 1141 l 11 1 1 L1 1 1‘..l 1 111 1_1_1 144 1 1411J4L 1111 141,1 - O O 0.05 0.1 0.15 0.2 0.35 0.3 0.35 0.4 0.45 0.5 Figure 4.19: Rapidity plot of Catania acceptance region of 3 source with expanding central source with Fourier perturbation for 4'He for Kr+Nb at 120MeV. The solid lines delineate the experimental acceptance.The dashed lines correspond to the gate 70° 3 90M 3 110° 78 —4 x10 0.5 L“ 0.25 L: n _ r, 0.45 — f“: 0.225 0.4 :— "“1 0.2 0.35 C ...... 0.175 ; 0.3 F 1"“ 0.15 3. E a 0.25 L 0.125 0.2 L 0.1 0.15 _— 0.075 0.1 _— 0.05 0.05 — 0.025 —L1111411411L111_1L1J_1 1412.411111111141__L_141_1_1_J_J_1_1_LL_1_J_1_L_J O O 0.05 0.1 0.15 0.2 0.2y5 0.3 0.35 0.4 0.45 0.5 Figure 4.20: Rapidity plot of Catania acceptance region of 3 source with standard expanding central source for 4He for Kr+Nb at 120MeV. The solid lines delineate the experimental acceptance.The dashed lines correspond to the gate 70° 3 60M 3 110° 79 Figure 4.21 shows a comparison between the average of the experimental data (points) and the average of the fitting function (curve)for 4He. This curve is plotted in red for E/A > 5 MeV and in green for E/A < 5 MeV. Above E/A = 5MeV, where the two procedures can be directly compared, they agree within the statistics of the averaged data. We conclude that the integration of the fit is at least as accurate as the average of the experimental data and therefore use it to extract the average experimental differential multiplicities over 70° S 00M 3 110°. Both are limited by the statistical accuracy of the data and by the accuracy :l:5% of the experimental particle identification gates. We estimate these uncertainties and combine them to obtain the uncertainties in the differential multiplicities we provide below. We note that the experimental acceptance of the device covers ECM/A > 5MeV, but that the yields at lower CM energies have significant systematic uncertainty. In the following, we will apply an energy threshold ECM/A > 5 MeV to many of the quantities we extract from the experimental data. 80 8000 6000 4000 2000 1500 1000 500 300 200 100 400 200 / $111111 4 1\1\\\1"“‘I‘~+--—4L+1 1 1 l 1 L 141 1 41 l J L 1 L 1 0 20 40 60 80 100 120 140 V\\. E 1 1 1 l 1 1 1 l 1 1 1‘ 1k_1_\1fiw1':“1*‘51--~1~—~+~—1~—1 1 L 1 1 1 1 l 1 0 20 40 60 80 100 120 140 5 +42 E \1\\h :— \X\\ ; ~-4\_\_ L N‘Mr-Lx..- 1 1 1 1 1 J 1 l 1 1 1 L 1 1 1 L4 1hr‘l'vi”*‘1“'1“—~i-----t-—+v— -1———1 A O 20 4O 60 80 100 120 140 : 1*KL ; \r\ _— \‘\\‘N 7 1 1 L l 1 L 1 J 1 4 4 J 4 1 1 l 1 “FT‘lfifi—fi—‘rml—‘TT‘fl*'—~l—w+~o~ 0 20 40 6O 80 100 120 140 Figure 4.21: CM spectra of counts versus energy for Kr+Nb, (E)beam=35-120 A/ U. using asymmetric parameterization source model, the yield is obtained integrating from 5A(MeV) to 00 . 81 4.3 Isotopic Temperatures and Isoscaling Param- eterizations 4.3.1 Extraction of Isotopic Temperatures Transforming the fits of the energy spectra to the center of mass and integrating over energy for an angular cut of 70° 5 00M 3 110°, we can obtain integrated multiplicities for the participant source. Dividing by the solid angle corresponding to this angular cut, we obtain the average differential multiplicities (ldM(60M z 90°) / dQICM). From these differential multiplicities, one can form double isotopic ratios and ex— tract isotopic temperatures as discussed in Section 3.1. First, we utilize multiplicities obtiained by integrating over energies of ECM/A 2 5M 61/ where the experimental information is more complete. We have tried to extrapolate these values for ECM = 0, but have concluded that such extrapolation introduces unacceptable contributions to the uncertainties of the integrated yields due to uncertainties in extrapolating the yields from the projectile and target spectators and there are also effects of cooling on the energy spectrum to energies below the experimental thresholds. The isotopic yields of fragments depends directly on their binding energies and, for equilibrated systems, upon the temperature and isotopic composition of the fragment- ing system. Within equilibrium models of nuclear multifragmentation, temperatures can be obtained via Eq. 4.5 [3, 49] from the ratios of yields neighboring isotopes prior to secondary decay : Y(AleI)/Y(A1+1,Z1) _ 1 , . Y(A2, Z2)/Y(A2 +1,Z2) _ EGXMB/Two) (4.5) where Y(A,-,Z,~) is the yield for the isotope with mass A,- and charge Z, , a is a statistical factor determined by spin values and kinematic factors , B = E3(A1,Zl)- EB(A1+1,Z1)- EB(A2,Z2)+EB(A2+1,ZQ), and EB(A,-+1,Z,) is the binding energy of 82 the 2' th nucleus. Temperatures may be obtained from the yields by inverting Eq. 4.5. Feeding of the measured yields from the secondary decay of heavier particle un— bound nuclei alters these ratios and consequently, the extracted temperature values. The influence of feeding can be reduced by focusing upon ratios with a very large binding energy difference B. In practical terms, this requires the incorporation into the double ratio of a single ratio involving either 3He/‘fHe, 11C/12C' or 150/160 as these are the only ratios involving neighboring light isotopes that have binding energy differences significantly greater than 10 MeV. While insufficient carbon and oxygen isotopes penetrate the silicon AE counters of the Catania array to allow analyses, we are able to analyze double isotopic ratios involving the 3Hrs/4He ratio over a large acceptance using the Catania array. In Figs. 4.22 and 4.24, we show values for the isotopic temperature obtained for central Kr+Nb and Ar+Sc collisions from the pd/3'4He, dt/314H e, 6*7Li/3’4He, 7'BLi/3’4H e, and 911°Be/3’4H e isotopic ratios using Eq. 4.5 with the values for a and B given in table 4.3 corresponding to the spin degeneracy factors and binding energies appropri— ate to the fragments in their ground states. All of these ratios increase with incident energy consistent with a monotonic increase in excitation energy per nucleon. At the incident energy of 100A MeV where both systems have measurements, the isotope ratio temperatures obtained in the Ar+Sc are consistently higher than the temperatures obtained in the Kr+Nb system even though the uncertainties are large. We note that a dependence of isotopic temperatures on the system size has been reported previously [80]. There is a big difference between the temperatures obtained for the various double ratios via Eq. 4.5 as shown in figures 4.22 and 4.23. Such differences have been attributed to the secondary decay of heavier particle unstable fragments. Such decays enhance the yields of nuclei that are included in the relevant double ratio. Such decays 83 modify the relationship between the double ratio and the temperature away from that given in Eq. ref 4.5. In ref [111] it was proposed that one can model the correction for secondary decay by assuming that it can be described by Eq. 4.5 but with a different value for a. Because we do not have a good model independent description for the hot primary fragments, we adopt this approach instead of theoretically calculating the secondary decay directly as done by ref. [140, 104, 50]. Specifically, this approach stipulates that the temperature Tem can be obtained from the Tim value provided by Eq. 4.5: 1 1 [7m T... = 2r... ‘ 2’ (4'6) where K. is an empirical correction factor. Figs. 4.23 and 4.25 show values for Tem obtained from the values for Tm shown in Figures 4.22 and 4.24 using the empirical correction factors in Table 4.3 [117]. The resulting values for Tam still display an increase with incident energy, but the variations between the corrected temperatures Tm for different isotope ratios are much smaller than for Tm. The slope of energy dependence of the corrected temperatures, T em, is larger for the Ar+Sc system Fig- ure 4.25 than for the Kr+Nb system Figure 4.23. From the heavy ion telescopes, some analyses would be performed involving the 11C/ 120 ratio. The 150/160 ra- tio would not be analyzed as the statistics were insufficient. Results obtained from the 11112(3/6’7Lz' double isotope ratio are indicated in Figures 423,425 by the square points. Interestingly, these latter points do not increase as strongly with incident en- ergy as do the temperature values obtained from the 1911/ 314H 6, dt/ 314H e, 6'7Lz'/3'4He, 7i°Li/‘°’1"He, and 911°Be/3’4He double isotope ratios. These latter ratios fundamen- tally derive their sensitivity to temperature from the large binding energy difference between 3H6 and 4He. If the carbon and helium isotopes extracted from the same region at the same time, the system is not in equilibrium or there are significant problems with the method of isotopic thermometry. On the other hand, helium and 84 carbon isotopes originate from different regions or they are emitted at different times, one would expect that they would emit at different temperatures. This latter pos- sibility appears likely. In the following two sections, we first examine whether the light particle yields are consistent with them being emitted along with the heavier fragments from a globally equilibrated system. Then we examine whether the yields themselves are consistent with emission from subsystems in local thermal equilibrium, an assumption upon which the validity of isotopic thermometry strongly depends. 85 14 1 1 1 ,2 L 86kr+93Nb L _ O 10 — ’34 d/“He i, - * (fit/“He l . 4:» 6711/3-411e g ’ 78Ll/MHEB 5L 1_< _ . 91oBe/3AHe ' L 8 ‘ 1 l _ 11$ * 1 _ 1 ‘,. 1 1" 6 1; 1* 1 4'1] - .1; ' ‘1' - ’. 1,111 - 1 4 ~— Hi" 44 1 141 1 1 4141 l 1 l l 1 1 41 L 1 L Li 1 1 1 1 0 20 40 60 80 100 120 140 BeomWeV/A) Figure 4.22: Apparent isotope temperatures for Kr+N b (E )beam 35-120 A/ U, obtained by inverting eq 4.7 with the extracted yields from 5A(MeV) to 00 as input. Error bars include PID error on neighboring isotopes. 86 14 12 L. 86Kr’fi1—93Nb 1 I CD \1 1: \ '5 (3 10 ~— CPS 8 \\ 5" A :E m O 12>)!» \JO') 03V 1— \\ (NM #45 :EIECD (DCD 1,..— @— enfisfion l T T m 1’1 J 4231. m1 4L + T T L 1 1 1 1 l l l 1 1 1 1 1 L 1 J 1 l 1 1 1 1 80 100 120 140 60 BeoleeV/A) Figure 4.23: Emission temperatures for Kr+Nb (E )beam 35-120 A/ U. using the asym- metric parameterization source model and yield extracted from 5A(MeV) to 00 Error bars include PID error on neighboring isotopes. 87 14 " 36 45 ,2 _ Ar+ SC TAPP 1.. _ 114 10 r :34“ pd/“He 1 * 91/141418 31? l- , 3i: 7'8L1/3'4He 1 'l‘ 1 ‘4 ' 1 'He 1 O 9'1088/3'41—18 ‘1': 8 1— 1411111141111111111111111111111111111114 0 20 40 60 80 100 120 140 160 180 200 Beom_1J_ x 10 ~10] pd/ffl4e _ 1r 1/“He _ 3414-14 LI/ ' He _ ‘. 91OBEQ/lflde 8 __ Temission T 6" 03 fij f 11.. 21243;: L— 4 L '1 . 1 '1 11111111111111111L1111J14141411111141414 l 0 20 40 60 80 100 120 140 160 180 200 BeamiMeV/A) Figure 4.25: Emission temperatures for Ar+Sc (E)beam 50-150 A/ U. using the asym- metric parameterization source model and yield extracted from 5A(MeV) to 00 Error bars include PID error on neighboring isotopes. 89 4.3.2 Comparisons with Equilibrium Multifragmentation Mod- els Two assumptions underly the caloric curve analyses of ref [91] First one assumes that the emitting system is equilibrated at the time that the intermediate mass fragments are emitted. Second, one assumes that the various yields entering the Albergo ther- mometer expression Eq. 4.5 are emitted at the same time, from the same region and described by the same temperature as describes the fragment observables. Concerning the relevance of a equilibrium description of fragmentation process, we note that comparisons of the Statistical Multifragmentation Model (SMM) calcu- lations to experimental data have widely been performed. This model assumes that equilibrium is achieved and predicts that multifragmentation occurs as the result of a low density phase transition in nuclear matter from a fermionic liquid to a gas. Good agreement between SMM calculations and the measured Intermediate Mass Frag- ment (IMF) charge distributions, multiplicities and energy spectra can be obtained for SMM source parameters that are carefully chosen [97, 30, 113]. On the other hand, comparisons with data reveal that, the optimal choices for the mass, charge and excitation energy of the equilibrated source become progressively less than the mass, charge and available excitation energy of the total system as the incident energy is increased [135, 50, 3]. This suggests that progressively more nucleons are emitted by preequilibrium mechanisms prior to the freezeout stage described by SMM. Sim- ilar problems have been also observed for peripheral collisions [30, 97]. These prior studies suggest that one cannot describe the emission of all particles in a common bulk multifragmentation approach. When one applies equilibrium approaches to the fragmentation process, it is gen- erally assumed that the missing mass, charge and excitation energy is carried away primarily by emitted preequilibrium light particles and not by the IMF’s [30, 97, 113, 90 135, 3]. Such assumptions allow comparisons of IMF observables to SMM calcula- tions but such comparisons become especially suspect when they rely heavily upon the light particle observables as well. Temperature measurements that rely heavily upon ratios of the yields of Helium isotopes may suffer from this problem [3, 140, 75]. Quantitative analyses of such data must account for the influence of preequilbrium light particle emission as well as for other time dependent radiative and expansive cooling mechanisms [43, 91, 98, 142]. In the following, we investigate whether light cluster production in Kr + Nb cen- tral collisions at 70, 100 and 120 AMeV is consistent with an equilibrium picture that assumes simultaneous emission of light clusters and heavy fragments. Previous measurements with this system have revealed that fragment multiplicities are larger in central collisions, where they increase with incident energy [114]. Fragmentation occurs over a short time scale [115, 99]; with many features well described by ”in- stantaneous” bulk multifragmentation models [114, 21]. Nontheless, rapid sequential decay processes cannot be definitively ruled out provided that such processes occur within a time scale of the order 70 fm/c [115, 99]. For such short time scales, however, the distinction between sequential and ”instantaneous” bulk multifragmentation be— comes somewhat semantic. While many features of the multifragment final state are well described by equilibrium at a single freezeout density and temperature, we have shown in the previous subsection that the temperatures extracted from the relative isotopic abundances of helium and lithium isotopes greatly exceed those extracted from the relative isotopic abundances of somewhat heavier 12C' isotopes [75]. Here, we reexamine this discrepancy and make detailed comparisons with statis- tical and dynamical calculations in order to assess whether light particle and IMF emission in energetic central nucleus-nucleus collisions can be self consistently de- scribed within a single freezeout picture. We concentrate on light clusters emitted 91 to center of mass angles of 70° _<_ 60M 3 110° from the ”participant” region, and compare these data to statistical models. The data points in Fig. 4.26 correspond to the observed values for at 60 M = 90° for d, t, 3He and 4He particles and two gates on the impact parameter. Consistent with the trends deduced from inclusive data [40], the extracted (1, t and 3H6 multiplicities increase with beam energy over this energy range while multiplicity of 4He decreases slightly. This difference in the trends for d, t and 3H6 relative to that for 4He can be better displayed by the ratios of the yields for d, t and 3H6 divided by that for 4He. This is shown in Figure 4.27. 92 dM/dn (sr-l) Kr+Nb "l""|""l”"+""lr'rF"'l""|r"'l""|' 0.4:... d + .1; t .3 : ¢ 1:1 :: 1 : 1 -. U1 . 0.21— —— . -— . ‘1 . I 1:1 I 1:1 I - -L -1 0.0 - 3He + £13 E1 —- o1oL ‘ I E . [j¢ 1:1 ¢ : 0.05 L 0 Data 13 g. 0. 131—. t 0 Data s 0.45 - - 1:1 :1 SMM ; O:lllllllllilllllllllllll IllllLlLlllllllllLllllll 60 80 100 120 140 60 80 100 120 140 Elab/A (MeV) Figure 4.26: LCP Multiplicity vs Beam Energy. . 93 0.4 0.2 0.0 0.4 0.2 0.0 (,_Js) UP/NP 2 Kr+Nb :IIIIIIIIIIIITTTTTIWTrl—lq ” de 1 >— D — O 1111111L114111111111111 11'1111l1111|1111l1111l1 — —1 1— th —1 L _ AA 1- —1 N (S vv11'_ —‘ 1- D ~ [— q 01L1111111111111111111 1 _1T[11V1[IIIFI11T1IIIIIII_ 0.4— — 2 _ 1 1 >— —1 1— —1 0.2— — L J _ a ._ 0.0 11114111111111111111111 60 80 100 120 140 Elab/A (MeV) Figure 4.27: Ratio of Light Charged Particle Multiplicity to 4He. . 94 We now consider how well such differential multiplicities can be self consistently described within a statistical model assuming a single freezeout picture. For this illustration, we consider the Statistical Multifragmentation Model (SMM) which has been widely used to describe the emission of intermediate mass fragments (IMF’s) at intermediate energies. In a typical SMM calculation, the emission of light particles and IMF ’s is calculated by assuming an equilibrated breakup configuration at density of about 1/3 to 1 / 6 saturation nuclear density [30, 113, 135, 3]. Some reductions in the size and excitation energy of the equilibrated system be- low the values characteristic of the total system in its center of mass are frequently assumed in such calculations to account for the mass and energy removed by non- equilibrium emission prior to breakup [30, 97, 113, 135, 3, 50]. Following refs. [135, 3], we adjust the excitation energy of the source by a multiplicative factor FE to repro— duce the previously measured IMF charge distribution and adjust the source nu- cleon number by a multiplicative factor F A to reproduce the previously measured IMF multiplicity. We further assumed values for the radial flow energy given by 3.5MeV/A(Kr+Nb 75MeV/A) and 6.5MeV/A(Kr+Nb 95MeV/A) consistent with the systematics of [135]. Fig. 4.28 shows a comparison between the angle integrated experimental charge distribution for 3_<_Z318 measured by ref. [114] at 75 AMeV (solid points) and the charge distribution provided by the SMM calculation after correction for the acceptance of the experimental device (histogram). Fig. 4.29 shows the comparison at E/A=95 MeV. The corresponding values for F E and FA, given in the figure, were obtained by a least squares minimization procedure. Contours of constant x2 as functions of F E and F A are shown in Figures 4.30 and 4.31, where the upper panels are the actual x2 contours and the lower panels are parabolic fits to the contours that are used to determine the sensitivity of the fits to FE and F A and estimate the 95 uncertainties in those quantities. From these sensitivity analyses, we find that the source size is poorly determined with an uncertainty of the order of 20 percent for FA 2 .70(E/A = 75MeV) and FA : .64(E/A = 95MeV), but the excitation energy is more strongly constrained with an uncertainty of about 8 percent for F E = .47(E/A 2 75M eV) F E = .64(E/A 2 95M 61/). Predictions for the statistical contribution to the light particle multiplicities are indicated by the open squares in Figure 4.26. Clearly, there is a problem with these predictions at E/A=100 MeV/ A where the statistical predictions exceed the data for t, and 4He. The origin for this discrepancy may lie in the fact that the parameters F E and FA which determine the excitation energy and size of the source are obtained by comparing the SMM to fragment multiplicities that are integrated over angle, while the experimental light particle differential multiplicities are determined at 60148900. When the emmision pattern for light particles or IMF’s are anisotropic in the CM frame, this comparison may display a nonnormalization problem. Ref. [76] reports that IMF distributions in the CM frame of the Xe+Sn multi- fragmenting system are emitted isotropically and that the light particle yields are considerably more anisotropic[76]. Because the light particle multiplicities reflect more sensitively preequilibrium contributions at forward angles than do the frag- ments, the option of integrating the light particle yield over angle before comparison to SMM would not be very instructive. Instead, we focus on the comparison, shown in Figure 4.27, of measured and calculated values for the relative yields of the light particles. Clearly the measured data displayed as solid points has much higher d/4He, t /4H 6, and 3He/“He yield ratios then the SMM calculation(open squares) that are constrained to reproduce the IMF measurements. From the point of view of statistical physics, this trend is consistent with the actual system being characterized by a larger value for the entropy per nucleon than 96 N(Z) P(Nimf) 100 10—1 10—2 10—3 10—1 10—2 10—3 10—4: 10—5 Kr+Nb ° ” Ebeam/A = 75 MeV J 1 1 1 1 1 1 L 1 1 1 1 1 I 1 1 1 l j 1 1 l 1 1 T 1 I 1 1 1 r I I 1 1 1 FA = .700 :1 O . FE = .470 . .Data 5 E . E ;_ ° —SMM I . 0 j _ l 4 L 1 1 l 4 1 1‘1 1 J 1 l l 4 1 J 1 7 5 10 15 20 Nimf Figure 4.28: Best fit of Z distribution and IMF distribution for Kr+Nb 75 MeV/ A. . 97 100 £1 10“1 —-= \N/ 10—2 z 1: z s 3 10—3 5 ‘2 5 1 5 “E 10_2 . FE = .640 — .... . \Z/ 10_3 Data 114 10_4 -SMM 10—50 5 10 .. 1115' I I 120 Nimf Figure 4.29: Best fit of Z distribution and IMF distribution for Kr+Nb 95MeV/ A. 98 7O 65 60 55 50 45 Fe 40 35 30 70 65 6O 55 50 Fe 45 40 55 50 035 043 025 (12 045 OJ 005 111111111—111111L111111111111111111111111 3O 4O 50 60 70 80 90 100 F0 20 035 (13 025 (12 045 OJ 005 kJJJLLiLi—ilkii111111411111141141L111-{11-1-1 30 40 50 60 70 80 90 100 F0 Figure 4.30: x‘2 vs Fe and Fa. for 75 MeV/ A . The upper panel is the SMM result.The lower panel a parabolic fit, with the dashed curves standard deviations. 99 85 _. , 80 E r-: 75 L 65 L 60 ~+ 55 L (19 (18 OJ7 (l6 (l5 (14 Fe : 033 50 E?— . . . . . . . . . 0.2 45 E ..... .. . . 0‘1 01117 -1--1~--L-1--1--1~-~1--1--1---1---1HL-L‘IL-lui--+--1-~1~-1-4L--1-~u-1---1~1--1---L~-1--1--+--1--1--+--1>--1--i---h4--1--i-~-1>--L--i---1- O 50 55 60 65 70 75 80 85 90 F0 40 4:. ()1 85 80 75 70 65 _‘ L—h _- _— F _5 p; r—. h. 60 ‘+ _— _Q be 1—v— —u _ >— —.— :1.) pa _J h— r.— (19 (18 0f7 (16 (15 0:4 033 (12 OJ 4O .1--L--1-.1--l--1-4--1--1--1~--1--1--1---1.-1--1--+-.1.-1--1--L-1--1---L-1141-1-41“1---1--1--+-1.1-1.41-l--<1---1--L-1--1- O 45 50 55 60 65 70 75 80 85 90 F0 Fe 55 50 45 1 . I o . 1 . o u . . . . n 1 1 n a n . 1 , n I . . . u n c n . 1 . n v o 1 1 u . n u a . o . o n . . u a a n . . . v n u n I I I n . . 1 o o . n . . . n o a n . . . u 1 a c n . . . - ...... .. .. ... . . .... ..... .. ... ...... ...... ...... Figure 4.31: x‘2 vs Fe and Fa. for 95 MeV/A.The upper panel is the SMM result.The lower panel a parabolic fit, with the dashed curves standard deviations. 100 is that given by the calculations [30, 97, 11, 12]. Simply put, the average effective temperature of the system that emits the light particles appears to be higher than is the temperature of the system that emits the fragments. Alternatively, one can propose that there are several sources for light particle emis- sion. One source might be the statistical emission stage during which the fragments are emitted. If one assumes the a’s are also emitted entirely during the statistical emission stage, then once fitted obtains a lower limit on the contributions from pre- equilibrium emission. In this limit the remainder of the deuteron, triton, and 3H6 yields are emitted prior to the freezeout stage. This interpretation is also consistent with the fitted values for the source size obtained in figs 4.30 and 4.31 by comparing central collision data to the SMM calculation. This fitted value stipulate that size of the SMM system is about 65 percent of the size of the total system. The Boltzmann Uehling Uhlenbeck model provides one way to directly to model the fraction of light particles that may be dynamically emitted before the system expands to low density. Such calculation was done by Hong Fei Xi using the numerical code and cluster production mechanics described in ref [108]. Using the approach described there ref. [108], the formation of clusters up to A23 (d, 3He, and 3H) were calculated. The upper four panels of Fig 4.32 shows the predicted differential multiplicity of protons, deuterons, tritons and 3He’s at 6cm 2 90" as a function of time. The solid horizontal lines correspond to the experiment differential multiplicities for protons, deuterons, tritons and 3He’s at 66m = 90°; the horizontal dashed lines correspond to the experimental differential multiplicities minus those predicted by the SMM calculations of Fig. 4.26. The lower panel of fig 4.32 shows the evolution of the central density (density at the center of mass of the colliding system). The predicted multiplicity of light clusters increases very quickly with time. Thus many of the light 101 clusters are produced well before the system attains low density. Indeed, the predicted 3He differential multiplicities approaches the experimental yield as early as 40 fm/c, at which time the system has a central density which is approximately equal to the saturation value. According to this transport model calculation, light clusters indeed are abundantly produced before the system multifragments at low density. This early dynamical emission of light clusters and expansion significantly cools the system. It should be noted that if calculations are followed for much longer times, t>75fm/c, the predicted light cluster multiplicities greatly exceed the measured values. This may be attributed to the fact the present model neglects the competing 4He and IMF emis- sion channels which are not included in the model. As there is no time where the differential multiplicities by the transport model agree with the dotted lines, this sim- ple hybrid calculation consisting of an early BUU emission stage coupled to a later SMM emission stage does not work that well. More work in this area is needed. 102 do/dQ (00m: 90°) J Time (fm/c) Figure 4.32: BUU calculations 103 1 l 1 _L __ L. C5 50 75 100 125 150 1 1/5 200 4.3.3 Generalized Isoscaling of isotopic distributions Dynamical processes appear to contribute to the emission of light particles. Presently, the description of such processes within the BUU model of ref. [110] encompasses only the emission of particles with A S 3. The formation of light clusters, within the BUU involves nucleation processes that, in the long time limit, could produce a form of equilibrium. In this context, it is relevant to compare the isotopic yield to statistical yields in order to assess how well statistical models can describe such fast emissions and cosequently whether local thermal equilibrium is a useful concept. In this section, we compare the isotopic yields to a generalized isoscaling descrip- tion. First we introduce the isoscaling observables and then generalize those observ- ables to enable comparisons of two systems that are not at the same temperature. Finally, we compare this generalized isoscaling to the experimental data. We begin by noting the experimental observation that isotope ratios from two statistical processes with same temperature exhibit isoscaling [142, 114], i.e. the isotOpe ratios depends exponentially on the neutron number, N, and proton number, Z, of the isotope (N, Z) R21(Ni Z) : YZUV, Z)/Y1(Na Z) : C8Xp(QIV + 52)) (47) where a and B be treated as empirical fitting parameters and C is the overall nor- malization factor. Eq. 4.7 can be derived from the simple Grand Canonical model expression for the primary fragment yield for the ith fragment in its 3"" state before secondary decay. .,- 14‘7”2 N,- n + Z,- + B,- Yf} = Vfi-(2J1j+1)ewp “ 1”" , , (4.8) T 2 where 11,, and [1,, are the proton and neutron chemical potentials. Bz'j and Jij are the binding energy and spin of the fragment in the j t" state, and V is the free (unoccupied) volume of the system. The insertion of the ground state yields predicted by Eq. 104 4.8 into Eq. 4.7 results in the cancellations of binding energy terms provided the temperatures of the two reactions are equal. Similarly, the insertion of the ground state yields predicted by Eq. 4.8 into Eq. 4.5 results in the cancellation of the chemical potential terms; the spin and mass number terms that contribute to the factor a in Eq. 4.5. What is measured in an experiment, however, are the secondary yields after sequential decay. Calculations of the yields of secondary fragments after sequential decay require an accurate accounting for feeding from the particle decay of highly excited heavier nuclei [140, 21, 84]. Such calculations are tenable if one can assume thermal population of the unbound levels. Unfortunately, this is subject to uncertainties regarding the levels that can be excited and the structure effects that govern their decay [140, 75, 21, 84, 131, 2]. To construct simple thermal expression, we adOpt instead the thermal expressions in Eqs. 4.5 and 4.7 as rough empirical guides to the possible relationships between the temperature and the charge and mass distributions and explore the extent to which they can be used to describe experimental observations. A similar approach has been taken with Eq. 4.7 in refs. [142, 114] and justified therein by statistical model calculations [115], which suggest that secondary decay corrections cancel when the two systems are at the same temperature. Likewise, this approach has also been taken with the isotope thermometric expression in Eq. 4.5; discussions of the modifications of Eq. 4.5 can be found in refs. [113, 140]. We take this approach in order to see whether the isoscaling relationship can be extended to consider two systems at different temperatures. In general, the binding energy factors in Eq. 4.8 are not cancelled by the ratio in Eq. 4.7 if the two systems have different temperatures. However, one may try to extrapolate the isoscaling be- havior to systems with different temperatures by multiplying R21(N, Z) by a binding 105 energy dependent term: R21(N, Z) exp(k21BE(N, Z)) 2 C" exp(a’N + B'Z) , (4.9) where [1‘21 2 1 / T1 — 1/T2 is a temperature dependent correction factor. Because the two systems are at different temperatures, the scaling relationship of Eq. 4.9 may be more sensitive than that of Eq. 4.7 to the temperature dependent secondary decay corrections of the isotopic yields. In the following, we will use measured isotope ratio temperatures in Eq. 4.5 to test whether empirical isotope temperatures and the generalized isoscaling relationship in Eq. 4.9 can describe the evolution of the isotOpe distributions with excitation energy. We note that it might be possible to invert Eq. 4.9 and obtain a temperature for one system if the temperature of the other is known. To examine whether a generalized isoscaling can be applied to these reactions, we construct the isotope ratios, R21(N, Z) from measurements on the same system at two different incident energies. The top panels of Fig. 4.33 show the isotope ratios measured in Kr+Nb collisions and the top panels in Fig. 4.34 show the isotope ratios measured in Ar+Sc collisions for Z21, (open circles), Z22 (closed circles), Z23 (open squares), and Z24 (closed squares) isotopes and different combinations of incident energies. The different incident energies involved in each ratio are labeled in each panel; the notation ” 70/ 35” in the upper left panel in Fig. 4.33 denotes the ratio of isotopic yields measured at E/ A270 MeV in Kr+Nb collisions to the corresponding yields measured at E / A235 MeV. For simplicity, we adopt the convention that isotope yields from the higher energy collision are placed in the numerator. Clearly, the uncorrected isotopic ratios in the upper panels of these figures don’t show the systematic trend predicted by Eq. 4.7. Instead, the ratios fluctuate from isotope to isotope by a factor of two. To determine whether these fluctuations are consistent with the binding energy term that results from a difference between the temperatures T1 and T2 for the two reactions measured at incident energies of E1 and 106 E2, we compensated approximately for the temperature difference using Eq. 4.8. For kl-z, we used the average value (kapp) where: 1 1 T2,APP TLAPP ’ (11721111111) = ( (4-10) where T1, App and 712,111?!) are the isotopic temperature for a specific isotOpic ther- mometer at the average is over all of the isotopic thermometers in plotted in Figs. 422,424 . These corresponding mean values given in Table 4.4 and used as labels for the lower panels of Fig. 4.33 where the adjusted isotope ratios, R21(N, Z ) exp((k21,app)BE (N, Z )) are shown as the open and closed points. The de- gree to which this procedure removes the fluctuations in isotopic ratios in Fig. 4.33 is remarkable. Alternatively, one can extract the k values by fitting the R21 data in the top panels of Fig. 4.33 with Eq. 4.8. These best fit values, given in the column in Table 4.4 labeled kgl‘m, are statistically consistent with the mean values of (kgmpp). The values for a’ and ,8’ that describe the dependence in Eqs. 4.7 and 4.8 upon neutron and proton number are also given in the table. When one performs the same procedure for Ar+Sc collisions, mean values of are obtained for the pairs of incident energies involved in the left and right panels of Fig. 4.34. The consistency between (kgmpp) and the corresponding best fit values for km is not as good as that obtained for the Kr+Nb system, see Table 4.4 . If the values for (kgmpp) are used to adjust the isotope ratios as shown in the lower panels of Fig. 4.34, the adjusted ratios do not follow parallel lines on a semi-log plot. The same results are obtained if the best fit values for km are used. Thus, the generalized isoscaling relationship is not as well satisfied by the Ar+Sc data as for the Kr+Nb data. Dynamical stochastic mean field calculations suggest that the yields of excited 107 Isotope ratio a B(MeV) ln(k/B)(MeV‘1) 6'7Li3'4He 2.18 13.32 -0.0051 2*3H3’d‘He 1.59 14.29 0.0097 1’2H3’4He 5.60 18.4 0.0496 7'8Li3’4He 1.98 18.54 0.0265 9vaeMHe 0.38 13.76 41084 Table 4.3: Parameters for emission temperatures. System Eg/E1(MeV) < kapp > kfit a B Kr+Nb 70/ 35 -.047i0.005 -.040i0.005 0.3489 0.4034 Kr+Nb 100/ 70 -.025:l:0.005 -.024i0.004 0.0885 0.2073 Kr+Nb 120/70 -.035i0.005 -.028d:0.003 0.1561 0.2340 Ar+Sc 100/50 -.039i0.005 -.028d:0.004 0.2962 0.1461 Ar+Sc 150 / 100 -.028:l:0.005 —.025:l:0.003 0.2303 -0.051 Table 4.4: General scaling parameters. kapp are the averaged data of figures 4.22 and 4.24 a and B are the weighted average from both k’s fragments produced by dynamical models are not as consistent with isoscaling re- lations as are the final yields after secondary decay [18]. Thus, the consistency of the Kr+Nb data with the generalized scaling relationships could be due to a higher degree of equilibration in the heavier system or to a greater abundance of heavier fragments that sequentially decay to the observed ones. In any case it is clear that measurements of other systems would be useful to better establish the validity of generalized isoscaling more. 108 ReleXP(k*BE) R21 .... (\D .0 .O l“ H +—\ (Y) 0" O O C) (I) O N .., ..,....l I . ....,.2.,.. o~ . K ‘1 r O - , . . o . (“ff 3;, "i z... \2 , km S * 24> co 1 01 . /\ v—x "7 8 V II \ l \2 1 O O O . N - 4> . l 03:- OLQ 5—3 8 ._ ;‘\ | m , 163.,“ Q S —=';‘. N7 “9 1% 52 °. . m 4 1 E], El . l. ‘x, ' . *P‘f El... 3 f ‘ ”E a 1 CD 1.1.L.1..|n...|.1111..... . ....L.n..1....‘. Figure 4.33: Isoscaling comparison of Kr+Nb for different beam energy ratios, using the Fourier perturbed source model and yield extracted from 5A(MeV) to 00 . Upper and lower panels show uncorrected and corrected temperature factor respectively 109 Rglexp(k*EE) m R21 (\3 (.0 OT \2 O O ,..1 (\D Tj T I I Y I'I [V'Y‘IV'IIIVIVIIII | I I ' 1 F' Y I. [T I] IIYTUI Y j j WY r O *- /\ —— _l w V .. H r I . m — o _ [ o [ 11>- 00 14> — >2 '7‘ + + J P 01 O -1 O 203 2 2 . 111-.1....1....|....|....1. 1 1 I . . . .1--....1....I....1 I 1 . . . . . . r ml....l...., I o — /\ ——»—* ’X‘ 4- 0‘ v «Q C III " ,_. .. O N - o “o r- 0 ¢ [- m u (II kl}- r —— [. 1 1 O3 ._ __ —1 J_ 4L 1 l l l l l l l l L l l f-lllll1lllllllll l l 1 1 I r—* (\3 11> ... K) Figure 4.34: Isoscaling comparison of Ar+Sc for different beam energy ratios, using the Fourier perturbed source model and yield extracted from 5A(MeV) to 00 . Upper and lower panels show uncorrected and corrected temperature factor respectively 110 Chapter 5 Two particle correlations and excited state populations In this chapter we present experimental results for the two particle correlations and an investigation of time dependence of the temperature extracted from excited state populations is described. Fits to the data are performed using either S-matrix or Breit-Wigner forms and different forms for the background functions are also tried. From these fits the best values for the temperatures and their correlations are ob- tained. The Albergo thermometer is presented here as a consistency check. This analysis also gives the similar conclusion. Comparison of different colliding systems are performed to explore the dependence on system size and on the beam energy . 5.1 Pair counting Combinatorics 5.1.1 The Theoretical evaluation During the breakup of a nuclear system, some of the particles may be directly emitted from the composite system formed by the projectile and target. The emission of 111 other particles may follow a more complicated history during which they may be part of an excited particle unbound nucleus are emitted earlier in the collision and then decay. The excitation spectra of such unbound nuclei will reflect the random ”thermal” excitation of the surrounding nuclear medium. As discussed in Chapter 1, measurements of the relative probabilities that a nucleus, such as 4He,5 Li, etc. is in its ground and excited states provides information about the temperature of that medium. If the unbound nucleus in question decays to two daughter nuclei labelled z' and j in their ground states that move apart with a relative energy Em, we can obtain the differential multiplicity for their excited parent nucleus from a measurement of the total coincidence yield, ch'n(Ere1) as a function of the relative energy of the daughter nuclei. The coincidence yield has two contributions ch'n(Erez) = Yres(Erez) + Yback(Erez) The first term Yres(E,~ez) corresponds to the decay of the unbound resonances of the emitted nucleus and the second term Yback(Erel) corresponds to the yield that does not originate directly from the binary decay of an emitted unbound parent nucleus. Discussion of the data in terms of yields is awkward because yields are not nor- malized quantities and they depend on the efficiency of the detection apparatus. This criticism does not apply to the differential multiplicity. We generally apply gates on the impact parameter in this dissertation using the charged particle multiplicity in the 47r detector and sometimes on the energy of the measured particles as well. Taking the case where an impact parameter gate has been applied, we can define differential multiplicities for single and two particle emission. For single particle emission of par- ticle type 2' at energy E, and angles 9,, one can define the single particle differential multiplicity by W d2M(E,-,S2,-)i _ Y(E1‘1Qi)i dEiin — AEz'AQz'Nevents, (5.1) 112 where Y(E,-, (2,),- is the number of particles measured in energy and angle intervals AE,- and A9,- and Nevent, is the total number of events in the selected impact parameter interval. As defined by Eq. 5.1, the single particle differential multiplicity is the number of particles of type 2' emitted with energy E,- and angles (2,- per collision per unit solid angle per unit energy. Similarly, one can define the two particle differential multiplicity for the coincident emission of particles 2' and j by (14M(Ei1911Ej19j)1j _ Y(EiaQianan)ij dEidQIdEJ-dflj “ AE,-AQIAEJ-AQJ-Nevem,’ (5.2) where the definitions of the terms for the jth particle are analogous to these for the ith particle. Returning to the decomposition of the coincidence yields with resonant and back- ground contributions, we have a corresponding relationship for the multiplicity. Thus, d4M(E,:, (2,, 193,-, 52,-)” | dE,do,dE,-do,- 11.11 ' (5.3) d4A/I(Ei, 9i, Ej’ 9.1-)”- I . _ (14M(Eia mi) Ej? 0])” I + dEidflidEdej cmn— dEiindEdej m . . d4” H,,Q,.E-,n~ ,- . . The decomposmon of lag-dodgfin-LM [com Into resonant and background differ- 4 1 J ' J ential multiplicities, implied by Eq. 5.3, requires one to assume functional forms for each of the two terms. Let’s discuss the background yield first. Explicitly, the background yield corre- sponds to all processes that generate particle i and particle j in coincidence with the exception of those that directly result from the production and subsequent decay of an unbound parent nucleus with Zpar 2 Z,- + Z ,- and AIM, 2 A,- + 11,-. The momenta of the two particles in such processes are more weakly dependent on the relative energy than are the resonant yield described by Yres(Erel) because the two particles must be emitted sufficiently far apart in space and time so that the nuclear final state interaction between the two particles can be neglected. In this case, the Coulomb 113 force is the only mutual interaction of importance and the background coincidence differential multiplicity can be expressed as [56]: d4M(Ez', Qi, Ej, Qj) dEidQIdEJ-dflj dM2(E,-, 52,-),- dM2(Eja 91)]- iJ' lbw: [1 + RCo-ul(Erel)l ' dEdQ. dE-dfl' z z 3 '7 (5.4) Here, 1 + ROWAEM) describes how the background correlation function is modified by the Coulomb final state interaction between the two fragments. As will be seen below, this modification occurs primarily at small relative energies; RCW1(E,.81) gener- ally vanishes at high relative energies. Functional forms for the Coulomb correlation are discussed later in section 5.1.3 of this dissertation. The determination of the exact parameter values of the Coulomb correlation functions are discussed there as well. In some cases there can also be an additional correlation function term from collective motion that influences the background correlation function. This turns out to be of negligible importance for the cases discussed in this dissertation. The single differ- ential multiplicities in Eq. 5.4 can be obtained by fitting the measured differential multiplicities for particles i and j at the same impact parameter gate; however it is easier to obtain the background yield by the mixed event method described in Section 5.1.2. Now, let’s discuss the resonant contribution. The resonant decay occurs in two stages. The first stage results in the emission of the parent unbound nucleus with 2pm. 2 Z,- + Z, and AIM 2 A,- + Aj. It is described by a differential multiplicity ‘12 A’I(Epar19par) dEpardear which must be approximately determined by fitting the experimental data for ground state nuclei with Z 2 Zpar and A 2 AIM, or if such data are not available by fitting the singles data of a neighboring isotope obtained at the same impact parameter as discussed in section 4.1.2. The second stage is the decay of this unbound nucleus and is denoted by %%12° It too, is fitted and constrained along with the Coulomb correlation function by the measured coincident yield. The fitting parameters and detailed procedures are discussed in section 5.2. The resonant. 114 contribution to the coincidence multiplicity is: 1‘83 d4M(E,-, 52,-, E], (2,», I dEidQIdEI-dflj ([2 M(Epa1~1 0pm.) dn(E,.el) dEpardear 47rdErel J(Epa7‘9 Qpara EpaT, Qpar; E1, Qi, Ej’ Qj) (5.5) where, J (Epar, 9pm., Em, QM; E,, (2,, E], (2,) is the Jacobian for transformation from the relative and total to the individual coordinates for the two decay products. Here, it is assumed that the decay of parent nucleus is isotropic in its rest frame. A discussion of the influence of decay anisotropies may be found in ref. [84]. The fits to obtain the parameters of the Coulomb correlation and of the excited state populations that govern the coincidence differential multiplicity are integrated over the parent energies and angles, leaving only the dependence on excitation energy. This integration includes only the energies and angles where the detection system, i.e. the Catania hodoscope, can detect and resolve the particles of interest. In practice, this is implicitly done with the data, where we start by making a change of variables from E,- and Ej to the total energy Etot 2 E,- + Ej and the relative energy Ere, defined by: 1 q2 1 q2 E“, z —— ——, 5.6 1 2771,; + 27713 ( ) where 2 : (SA‘SB/C2 - 5A '53)2 — "1317712304 7773102 + 7712302 + 28,483/62 — 215;; ° {13, (5.7) and 8,4,83 are the total energies, and 111,173 are the momenta measured in the lab frame. Then, we sum over total energy keeping Ere; fixed to obtain ytotal (Ere!) : Z: Yltgtal (E1, E2) coinc 1,2 Erel,fi:ced ’ (5'8) The end result is a one dimensional spectrum of counts as a function of Em, Ycom(Ere,), which contains both the resonant Yres(Erez) and the background Yback(E,.e,) contri- butions discussed previously. When one fits the coincidence data one has to correct 115 for the detection efficiency. Let us first consider how one does an efficiency correc- tion to the theory in order to obtain an efliciency corrected theoretical expression for Yres (Erel) - 5.1.2 Efficiency Calculation The efficiency function €(Ek, Emeas) corrects the theoretic decay spectrum for detector efficiency in order to allow comparisons to the measured one. It basically states how many of the decays that occur at relative E, are observed experimentally at measured relative energy Emm. It is calculated via a computer simulation [79] that assumes for simplicity an isotropic decay in the rest frame of the unstable parent nucleus. This simulation peformed a Monte Carlo integration over the entire detector geometry, which takes into account the position of each detector relative to the target and the beam axis, the solid angle of each detector, the energy resolution of each detector, the beam spot size, and the various energy losses in the target and the brass sheet in front of the Catania telescopes . The simulation assumes the unstable parent nucleus to be emitted according to the fitted single particle spectrum d2M/dEdf2 for the parent nucleus in the ground state. This provides the weighted energy and the angular distribution for the unstable parent nuclei before it decays. In the end, one expresses efficiency function as a summation over the decay spectrum 175—(1% which provides the resonance contribution of the coincidence yield, K83(Emeas) in bin of relative energy Ema, that are of the same size as used in the data analysis. —_cana 59 (IE I1. < > T880". Ymeas(Emeas) : Z AE€(EkaEnlet18) I I: Here the summation is over bins in Ek, the actual excitation energy in the rest frame of the excited particle unbound nucleus. Those bins are typically chosen to be much 0 C d Ere C I I finer than the step 8120 over whlch Ada—7‘2 varies Significantly. ”TC 116 The efficiency function €(Ek, Emeas) is computed following [79] and stored as a matrix. It contains both the information about the probability that both decay products will be detected in the hodoscope and about the resolution with which their excitation energy can be reconstructed. By summing the efliciency function over Emma for fixed Ek €(ET61) : Z 6(E’rela Emeas,l)AEmea.93 (5-10) I one obtains the total detected yield one would expect if the decay spectrum were (1:216: z 6(Ek — Era). An example of this efficiency measure is shown in the upper panel of Fig. 5.1. The normalization is such that it gives the number of exited nuclei you would reconstruct if there were as many excited nuclei produced as there were d2M(Epa~QP°”) and if all these excited ground state nuclel making up the spectrum depardszpar nuclei had an excitation energy of Era. The resolution information is reflected in the degree to which the efficiency matrix is non-diagonal and can be quantified by 2(E'measj — E'rel)2€(Ereli Emeas,l)AEmeas 1/4 0( I) Z €(Erela Emeas,l)AEmeas ( ) l An example of the resolution is shown in the lower panel of Fig. 5.1. Now, let’s turn to the calculation of the background function. In the early analyses of decay spectra, simulations were performed to calculate the non-resonant background via a Monte Carlo integral that is essentially the same as the one use to obtained the decay efficiency [83]. Soon afterward, however, it was realized that the technique of mixed singles could be used to provide the non-resonant background more easily. The expression for the unnormalized mixed single background is: 117 dM2(Ei, 92'» dM2(Ejvflj)j dE,dQ,: dEJ-dn, ’ (5.12) Ymix(Erel) = 607185 ' EY p‘f‘JH 4Hezo_1(Jn:O+) __> p+3H :g ;' Kr+Nb at 120 AMeV Kr+Nb of 100 AMeV :ELil l l l l 1 J 1 L1 l l l l :5 l l l l l I l l l 1 l4 1 l — [1,205 : — A,=o.4 ----- A,:O.8 >_— Ab=0.8 .......... A :03 T. A 20.3 Kr+Nb ot35AMeV 'l I. ..... .0 O “He,,_,(J“:o*) a p+3H l14_1L¥1411L1 '1 l 4 5 EREL< M 6V) 2 4 6 EREL(MeV) The narrow width method(Lorentzian) was used in fitting. 125 E — A,:O.7 1'2 r ----- Ab:1'2 : .......... Ab:O.6 1 1 — /'\ ' __ E : ’ ¢ 9 ¢ LL] 1 V I I 01 t - 0.9 *1 ’ _. + 2 o.’ f. V— 08 : 'If'Ll15_7(Jfl:3/2+) —>d+3He :' 0,7 3. Kr+Nb at 120 AMeV : H ,' 0631"L1111i111111111l111l1111114 . F _ t _ A5:1 I _ ADZI 1.2 _— ----- Ab22 _— ----- Ab=2 : .......... Ab:O'7 : .......... Ab20.8 1 I ~- 1— A (J + + 1 ¢ CK . u... 1. .. 1.. '~ - LL! 1 .................. _ —. . ......... 1- \_/ ..... —+ + ........ +I$ ’.- . O: — ”— ‘ 09 0‘ o" 0" + 'o" _ ...: "v‘ ‘— o.8 511,,,,(,1~:—.3/2+)—>d+5He x}! 511,,,(,1~é3/2+)—>d+3He 0.7 Kr,+"Nb of 70 AMeV ' 1(1be of :55 AMeV 0.6:: ‘L/21111111111'Il'15: 11"L11l111l111 O 2 4 6 O 2 4 6 8 Figure 5.3: The d-3He Correlation for Kr+Nb for 120,100,70,35 MeV/ A respectively . The extremes of the background parameter are indicated. 126 1.5 _ _ 1.4 :— E — A505 1.3 :— """ E— “ ----- Ab:O.7 A ~ _ "‘ $1.2 ;— .......... .__ .......... AbZO 4 LLJ 1 1 :- L— \_/ _ __ . DC 1 E- ....... :_ __________ + >— '. : . 0‘ "'p' a. o.9 , ,- x "' . 5Ligs(d":3/2')—>p+‘He 0-8 Z . 5L195(J":3/2‘)—$p+‘He , 0.7 :5.’ Kr+Nb ot120 AMeV pf? Kr+Nb ot1OO AMeV * : ts‘. 06E iJLiiliillrli_:'1|liiil111_l114 1.4 E — Aszf) E- — Ab=O.5 1.3 E E— ----- Ab=O.6 $1.2 :— E— H .......... AbZO.4 QC ’— r— L1_l L ‘ v11 t € 01 1 I- . + : : ",v'” ‘_. 0.9 :- _— ”O . 0-8 L Sngs(J":3/2_)—‘>P+4He 5;" 5Li..(J"=3/2‘)—>p+‘He 0.7 Kr+Nb ot7O AMeV 3 Kr+Nb ot35 AMeV C — : 0.6—lngllllllllll:[_LllllllllLllL O 2 4 6 O 2 4 6 8 Figure 5.4: The p—4He Correlation for Kr+Nb for 120,100,70,35 MeV/ A respectively . The extremes of the background parameter are indicated. 127 0.6 Figure 5.5: The p-7Lz' Correlation for Kr+Nb for 120,100,70,35 MeV/ A respectively ' ’ "88917.6(Jfi: :‘° . Kr+Nb Ct 120 AMeV 1*)——>p+’Li IIWIIIIIIIIIIII I]!!! I ................ O .. 8Be17_6(J":1")—> p+7Li Kr+Nb at 100 AMeV l l l 1 #Lgl l l J L 1 l 1 III TflTrlrIllTlTTlTlTT i.’.I.UL4444... A508 ----- Ab:1.0 .......... AbZO.6 ..... ...... ' o o Kr+Nb at 70 AMeV .‘o JgingidmlilL1—i IIIIIIIIITfi—[I—ITI—I—lflT'l u"... ...."-n 5" Kr+Nb 0t 35 AMeV :'llllLLllllJ1Ll — Ab=0.5 ----- Ab=0.8 .......... AbZO4 Q ..... c O 8Benfiw 21*) —> p+’L' Oil] 2 4 6 EREL O Trr 8 2 4 6 EREL(Me\/) . The extremes of the background parameter are indicated. 128 . F.— l+R .0 .-‘ ~ CD —‘ M -# Illllllllillllll llellrrTrfirl—lllllllllll[ll t .0 0) Figure 5.6: The 4Hie-4He Correlation for Kr+Nb for 120,100,70,35 MeV/A respec- T T I lTTlllTllll‘IIIIIIIIflI Kr+Nb at 100 AMeV I ' I 6 ERH_(K/leV) :QILl4L411111111lsh’lJlllllllJlll [3.21.2 i — Ab=1.3 ..... Ab:14 i """ Ab:1°5 .......... Ab21.O ... Ab:1 O r- . F'— : .3 u . t— . x . :- .'. ........... ' ...... J é : ... ..-0._.0"'.: ........ :,. ..................... . "‘ 0'. ......... .. Y ....... ' oooo Z Kr+Nb at 70 AMeV C Kr+ Nb 0t 35 AMeV kli.’lJlllllllllllblif’llllllllJQII O 2 O 2 4 6 FREL(MeV) tively . The extremes of the background parameter are indicated. 129 1.4 : .--. — ADZOB ”g : ' ------ Ab:1.1 $1.2 r. ............. Ab:0.3 V E ..... s“ 2...... f]: 1 .— "o ‘w H ...-..-... ‘ + :5 --------- ‘— 0-8 i . """ 4He20.1(J":O+) _> p+3H ~ ,x' Ar+Sc at 150 AMeV 0.6—15L1 ll'iiiiliiirlrLiiliiiilIiiilriiilii11 1-4 r — Ab:0.7 E : """ AbIOQ L512. :- .......... Ab:04 v : ‘~ .4»- Di 1 r - . ................................ + F ......... 4 ,, + 3 r— 08 _: 'o" HEQOJQJ :0 )%p+ H 13 Ar+Sc at 100 AMeV 0.6 1:1u'liiiiliiiiliiiiliiii111—L1l111111111 1-4 r -— A505 Q : ----- Ab:0.9 L13; 12 :- ........... A :03 v : ........ G: 1 *_— ............................................. + : .......... 4 1! + ,3 \——' 0.8 T ' " H020_1(J :O >—_>p+ H ” x" Ar+Sc oi 50 AMeV 06 lh’lllIllllllllgLLlJLnglLIJIlllLJLJIJi O 1 2 3 6 7 8 4 5 FREL(MCV) Figure 5.7: The p-t Correlation for Ar+Sc for 150,100,50 MeV/ A respectively . The extremes of the background parameter are indicated. 130 L. 1.2 A f d l LilC t 1 _______ v ”...uv.."_"_' ---------- . D: E .. '..--;'; """""" q + 08 fl " 5Li16_7(Jfl:3/2+) —9 d+3He ‘— Ar+Sc at 150 AMeV LiSex O 6 '1 J l L l l l l l l l i 1 L l L l l l l l l l l l l l l l l l 1 Li 1.2 — /\ — 32 : ‘ ..... .... LLJ 1 -— ‘. f ......... ‘ ,1,"me m V _ \‘ .o ................................. __- _ Cl: _: “ ;' ................... + w' ‘. ’ ’ ' ....... 5 . K + 1- ___________ 0 8 “2.x" ..... L|,5'7(J :3/2 Z-Tid‘i' H'e ” Ar+Sc_g.t-10’0 AMeV 06 b—l l l l l 14 L1 1 1 L 1 ll l'r'l l l L l l l l L] l l lJ_l_¥LI l l l l A 1'2 f EREL(MeV) _J H ' : 1 LE P T l ’ ’ o 1 p .o ................... V _ ...... s ------------ Q: ” a‘ ¢ 0 """ ’4‘ --------- + 0.8 ; ...:.”: .' ',¢"" 5Ll16.7(dfl:3/2+) _> d +3He : Ar+Sc at 50 AMeV 0.6 ~51 1 13.1 ’l'L #1 L l l l J l J_LL l J l l l 1 l l l l l l l L J l l 1 L4 L 0 1 2 3 4 5 6 7 8 Figure 5.8: The d-3He Correlation for Ar+Sc for 150,100,50 MeV/ A respectively . The extremes of the background parameter are indicated. 131 E — Ab:1 .1 ”E 2 ? ----- Abzi .4 LLCJK 1 5 ”:1 .......... AbZO4 V : .' g 1 ”‘ .......................................... . . + ...................................... . \__ o 5 """""""""" 5Lig,,,(J"=3/2_) —-> p+1He ' E- """" Ar+Sc at 150 AMeV 2 5 _ 1 I 1 1 l 1 l l l l 1 l l l i l l l l l 1 l J [i L 1 1 l L L l l 1 l L l l l /\ 2 E— _ A6:08 5 E """ A 1 .1 LLJ 1 5 r .......... A 05 \_./ : Q: 1 ”— ............ + -.‘ ____________ .-- ._ O 5 E -------- 5LI9,(J":3/2‘) —>p+‘He ' 5 Ar+Sc ot100 AMeV 25 :1 1 1 1 l J 1 l l L 1 A l l ILL 1 1 l 1 l l 1 l 1 1 1 l l l l l l l 1 1 1 l /\ 2 E‘ — Ab:06 d : ----- [3.20.8 CK .. LL] 1 5 T .......... A 04 m 1 :— ..... + : . .............‘.- .......... . ._ O 5 """" 5LIg.,(.J"=3/2’) —-> p+4He Ar+Sc at 50 AMeV llLlllllLlllllllllllellllllll1llllLlll O 1 2 3 4 5 6 7 8 EREL(MeV) Figure 5.9: The p-a Correlation for Ar+Sc for 150,100,50 MeV/ A respectively . The extremes of the background parameter are indicated. 132 q 2 — Ab;O.6 B? """ Ab:l.l Lu 1.5 .......... Ab20.5 V Di 1 oflgff”«..1 I o I, : 88617_6(J"=1+) % p+7Li 0'5 Ar+Sc at 150 AMeV O2’111111111111111111111111111111111111141 f: 2 E —" AbZO'B E‘x’ E """ Ab:0.7 [_LJ 15 j? .......... AbZO4 V ”— D: 1 :— ....... + E .""..:.o' -------- 8 n___ + 7 . ‘— 5 i 9'. ’ B€175 p+ Ll 0' : Ar +Sc at 100 AMeV O:1411L111111111111111111111111111111111L E A 2 f — Ab=?.6 [Tl ..... A : J15 E .......... b_ \/ '- 0: 1 E . 1.7- A _ f 0’ ¢¢ g : O 5 E ........... 88617.6(J":1+)—>p+7L1 ' ‘, " Ar+Sc ot so AMeV O :’I 1 1 J 1 l l 1 l 1 1_1 1 1 L1 L 1 L l 1 114 1_L 1 11 1 1 L1 L1 1 1 l l O 1 2 5 5 6 7 8 4 EREL(MeV) Figure 5.10: The p-7Lz' Correlation for Ar+Sc for 150,100,50 MeV/ A respectively . The extremes of the background parameter are indicated. 133 T l ”E 1.5 :— LL] .— fK ... DJ 2 v 1 ..— [K V ............. l: .............. 8 1T_ + 4 4 + r ........... Begs“ —2 )—> He+ He V. o.5 — ,. ...... Z Ar+Sc at 150 AMeV O .1 14 L1 1 1 1 l 1 1 1 l 1 1 ##1 1 l 14 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 : -— A =12 f3 1 5 l """ A :15 .31, : .......... A 20.9 L1_J .— v 1 L ................................ O: L ..................... 4 4 + E 8Beg,(d"=2+) -$ He+ He _ 0.5 C Ar+Sc at 100 AMeV 1.. O ’— 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1L1 L1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l 1 l 1 1 1 E ' .2 A1 5 :— 3.1.1:: 8 0: t ' M +— V 1 P— . . ..........u-unuunun..-u --.-...;-:;-_ Q: : ................... + Z Y_ 0.5 ~— , I Ar+Sc 0150 AMeV O#4111111_LL1J_111111111411111J_LL11_L11JL111 O 1 2 3 6 7 8 4 5 EREL Figure 5.11: The a-a Correlation for Ar+Sc for 150,100,50 MeV/ A respectively . The extremes of the background parameter are indicated. 134 E — [1.20.8 E — Abzoe 1.8 _ _ : ..... Ab21-O : """ Ab—T-1.O A15 E. .......... Abzo'B E— .......... AbZO.3 ‘63 3 I 1_Ll 14 _ f \-/ ._ .. Bi 1.2 A Jr : : ‘— 1 C- ....... Z ................ : ........... +: .. 0.8 I- f- ”Be.,_6(J":1*)—>p+’Li F _. .' Kr+Nb at 120 AMeV Kr+Nb at 100 AMeV 0.6.:1'1111111111111 llllLlLilllLll 1.8 H _ E ----- 1351.0 ; ----- Abzos d1 6 :- ---------- Ab;O 6 :— .......... AbZO-4 L514 :— . i. V : : % 1.2 _— E— + .— : 1: 1 ......... __ ‘ 0.8 — OX“ 88811501 21+) —‘>p+7L. Kr+Nb at 70 AMeV 0.6,]:‘4’1111111111111 O 2 4 6 EREL(1\/1€\/) ,' Kr+Nb at 35 AMeV 011111111111111 2 4 6 FREL(Me\/) 8 Figure 5.5: The p-7Lz' Correlation for Kr+Nb for 120,100,70,35 MeV/ A respectively . The extremes of the background parameter are indicated. 128 1‘ r- r .. 1.6 _ f: E— _- E _ : DJ 14 — _ \/ : : Q: ~ _ 1.2 1— __ 1' Z : O I 8 1:— 3... o :— 33. a, — g ' Kr+Nb at 120 AMeV . Kr+Nb at 100 AMeV 061—45411111an1111m2131'11gllllLlll.L “ r :+ — Abz1.2 : ___ Abz1.3 18 r- __ C """ A13:1-4 : “““ Ab:1.5 L - A1 6 L— .......... Abz1 O __ .......... A1321 .0 ES 3 : DJ 14 :— ‘ -—< V ’— : Q L. I. __ . CE 12 :— 1. r . '3... \— i . '5 ¢ 1 1 ’— ." .............. ' ........ .1 .,‘ : 9' ........ : .................. - 1— ': ........ 7‘ ......... 08 r E "‘ ~ Kr+Nb 0170 AMeV t ,o' Kr+Nb 0135 AMeV 0.621;-1118111111111111.!‘111111111111 O 2 O 2 8 4 6 4 6 ERELOVleV) EREL(MeV) Figure 5.6: The 4He-“He Correlation for Kr+Nb for 120,100,70,35 MeV/A respec- tively . The extremes of the background parameter are indicated. 129 1.4 ;- — Ab:o.8 ”E : ' ------ Ab: 1 .1 L312 TO -. .......... Ab=O.3 \_/ : ‘ . 9...... m 1 P— "o T‘ . ' ......‘- A + 1:5: ... ........ 4 ‘ n + 3 ‘— 08 i ’,o" Hezo‘1(\J :0 )9 p'l" H t Ar+Sc at 150 AMeV O-6FL2141’11111lI1L1111111lLl_11L111L11111111 ”3 : ----- A 20.9 E 1 2 — .......... A 04 U—l E V _ D: 1 _— .......................... + : """""""""""" 4 n + 3 \—— 08 J "o H620.1(’J :0 )——) p+ H b ' Ar+Sc at 100 AMeV _1'1 h:1 1 1 L 1 1 1 1 I 1 l 1 1 1 1 1 1 1 1 L1 1 l l 1 l l l l 1 l l 1 l 0.6 _ 1-4 L“ — A506 ”3 : ----- Ab:0.9 E12 ~~ .......... A :03 Lu : ....... V _ ..... G: 1 :- ............................................. + : - ......... 4 1! + 3 ‘— 0'8 L— .3. ’¢" Hezo_1(\J :O )__>p+ H - ' Ar+Sc 0150 AMeV 0.6 111,11111;;1L411J1111Lll111_11111|4111111 1 2 3 6 7 8 4 5 ERELWIeV) Figure 5.7: The p-t Correlation for Ar+Sc for 150,100,50 MeV/ A respectively . The extremes of the background parameter are indicated. 130 1.2 - /\ ~ [:1 r— o: _ LL] 1 _ ________ v ~ _...---.'_' ------------ . D: — . .. ',.--;': ;;;;;;; . + 0.8 7 °. " 5L116.7(J"=3/2*)%d+3He ‘— °° Ar+Sc at 150 AMeV LiSex 06 11L111111Llll11111Ll 1.2 ~ /\ ~ d _ m ’— ..... L1_l 1 _ ' v _ --------------------------------- --- _ CE _____________________ + 0.8 ........ 11.5.70“:3/2")_:>.d+-4Fré"‘ Ar+Sc'gt-1’0'O AMeV 0.6 1L111111111L1111.1111L11111LllillllLdlll 1 1— A 2 _ ERELUVle ) LE] 7 . . . ' C C . \J 1 :— -. . ‘ ...... 1..-----"' "....“- a-i- : 7.3.. . .-°' 2., o """ 5‘" 1r + 3 08 — . ,x" l—'115.7(\J :3/2 )9d+ He ‘— ; “ Ar+Sc at 50 AMeV 0'675.1115:Lgm1l11111111J1111114111141111114 O 1 2 3 4 5 6 7 8 Figure 5.8: The d-3He Correlation for Ar+Sc for 150,100,50 MeV/ A respectively . The extremes of the background parameter are indicated. 131 E — Ab:1.1 /"\ 2 f _____ _ n: _ 1_Ll 1 5 :- .......... Ab:O4 v : Q [I 1 ‘ ....... . . + L ....................................... ... 05 : ------------ 5Lig.(.1"=3/2‘)—>p+‘He ' . """ Ar+Sc at 150 AMeV 2'5 1111111111111L114111111111111J1141L1ll11 /’\ 2 E— — Ab:08 g : ----- A :1 .1 1_Ll 15 r .......... A 0.5 V : G: 1 “— ........... _+_ ......,..--~"""':: ............... .— 05 : ------- 5Li9,(J":5/2‘)—>p+‘He ' E Ar+Sc at 100 AMeV 25 1— 1 1 1 1 1J l 1 l 1 1 1 1 1 1 l 1 1 1 1 1 1 1L1 1 ILJ L 1 1 l 1 1 1 1 1 1_Ll i t — Ab:0.6 /\ 2 f d : ----- Abzos c: _ L1_l 1-5 T .......... A 0.4 \_/ : Q: 1 ‘— + : Mao-"2‘. ________ O \— 05 """" 5LIgs(J"=3/2—)-—>p+41-1e - 5' Ar+Sc at 50 AMeV FLILLJLll111111111111111111111111111111111 O 1 2 3 5 6 7 8 4 EREL Figure 5.9: The p-a Correlation for Ar+Sc for 150,100,50 MeV/ A respectively . The extremes of the background parameter are indicated. 132 A 2 :— — Abzoe d : ----- A :11 0’. p b— L1_l 15 1:7 .......... Ab_05 V "' .. CE 1 : ’7’... W + ............... 88617 (Jfl:1+)__> p+7L1 V— 0.5 '6 r ' Ar+Sc at 150 AMeV O 17’1 1 l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l 1 L 1 11 1 1 l l 1 1 l 1 l l /\ 2 E" d : (I 1— |_l_j 15 f- \_/ : C1: 1 L + E 8 1r_ + 7 . ,— . 13817 (\J—1 )—>p+ 1_1 ‘— 05 no '6 : Ar+Sc at 100 AMeV O 1_1 1 11 1 l l l 1 1 l 1 1 1 1 1 l 1 1 1 1 1 1 1 1 1 IL] 1 1 1 11 l 1 1 l 1 1 E A 2 1: _ Ab206 _J H ----- A _— té : b— 1_1_1 15 :— .......... \_/ F DJ: 1 E— __________ : .............. 88917.5(Jn:1+) —> p+7Li V— 0.5 " Ar+Sc 0150 AMeV O ’111111111111L111111111111111L1111_11_1L11L O 1 2 3 6 7 4 5 ERELUVleV) Figure 5.10: The p-7Lz' Correlation for Ar+Sc for 150,100,50 MeV/ A respectively . The extremes of the background parameter are indicated. 133 8 5. ~ — Ab=1 .1 A _ ‘ ----- Abzi .4 L-j 1'5 T .......... A _O 8 (r — 0 b— ' LLJ : .‘ Fr/ 1 j .‘ ' ___..- _____ .. ‘1“ ‘, “ j ...................... 8 ,T_ + 4 4 + 05 _ ............ Beg,(J —2 )—> He+ He V— ' Ar+Sc at 150 AMeV O 1111111111111111L11111111111141111111144 : — A :1 .2 /\ 1 5 f‘ """ A :1.5 E, : .......... A 20.9 LLJ T. v 1 e ________ Q: : _..--""': -------------- 8 fl_ + 4 4 + 05 L; ..... Beg,(J —2 )—> He+ He ‘— ' I Ar+Sc at 100 AMeV O :1 1 1 l 1 1 1 l 1 J J_L 1 1_ L 1 1 1 1 1 1 I 1 1 1 1 1 1_141 1414 l 1 l 14 1 ”E 1.5 l """ ...J _ .......... m ... LU I V 1 r“ ......................................... G: : H a n_ + 4 4 + r s(J —2 )—> He+ He 05 ~ 9 T— . i " "' Ar+Sc 0t 50 AMeV O P- 1 1 l 1 1 1i 1 1 1 1 l 1 1 1 1_14 14g 1 1 L 1 1 1 1 1 1 l #1 J 1 1 1 O 1 2 5 6 7 8 4 5 EREL 0 3H8 I Al’ + Sc a) x” 2 7.5 * £3 Be b/bMAxgoo45 * § .g F— 4) <9 5 I. 4‘15 4 . ........................ 2‘ .............................. , 2115 211%, 215 ~ 1 l j l l l l I l 100 140 160 180 200 EBBESW(M8V1 IR) Figure 5.13: Temperature dependence on system Ar+Sc, Asys : 81. 144 I l l 1 F 1 l 1 12.5 r I HeLi —- G CLi ‘0 r“ A 5LI _ A 4 .. 6 0 8H3 .- - Kr + Nb \{1 7,5 _ '3’ Be b/bw§0.45 _ a. I—-° ‘- 5 " (a! ......... w .......... @413: ............................................. L 2.5 r- _ L l l l L l L l l O 20 4O 60 80 100 70 140 160 180 200 EBWWeV A) Figure 5.14: Temperature dependence on system Kr+Nb, Am =2 179. 145 A 5Li' I HeLi 0 ‘He’ b /b,,,,,§ 0.33 =33 sBe' e o . Q o t 0 o I O § . a Q C Q t ‘. ‘—" Q Q o u c o I 0 Q I e e 0 Q Q Q Q . o o c i Q 5 ‘ 12.5 ~— 2'51— l l l l l l l l o 25 50 75 10 125 150 175 200 225 EeszeV/A) Figure 5.15: Temperature dependence on system Au+Au Asys = 358. Data from Serfling et. al. [100] 146 5.4 Evidence for cooling from excited state popu- lation measurements: Breit-Wigner analysis The difference between the trends shown by the excited state thermometers, the CLi isotope thermometer and the trend shown by the HeLi thermometer is potentially rather disturbing. On the one hand, we have a large number of thermometers that give similar results. We can also partly discount the HeLi temperature because of the theoretical arguments presented in Section 4.3.2 which state that there is a large contribution to the 3H8 from preequilibrium emission with a potentially large dy- namical component. On the other hand, however, the HeLi thermometer combined with the generalized isoscaling results suggest that the early high density stage was considerably more excited than the later breakup stage. This is bolstered by the large excitation energies per nucleon or temperatures one would compute if the early stage could be approximated by a thermalized source. Why is there no indication of this cooling in the excited state population data? Cooling can be most easily examined by looking for a dependence of the excited state temperature on the energy of the detected species. This follows from several considerations. First, the temperature of the system will decrease monotonically with time reflecting cooling by expansion and by radiation. Second, more energetic particles tend to be emitted earlier in the collisions. The simplest reason for this, of course, is that the more energetic particles move away more quickly. If collective motion is important, dynamical BUU calculations predict that the more energetic particles will tend to be on the outside of the system and be emitted earlier. If radiative emission is important, the initial temperature will be higher; the spectrum for emission decrease as exp(-E / T); again the most energetic particles correlate with the highest temperatures. 147 To address this question, we undertook to analyze the 4He thermometer data subject to gates on the energy of the 4He in the center of mass. To select excited 4He nuclei from the participant source, we selected events with f) g 0.45 and 4He nuclei emitted at angles of 70" 3 60M g 110°. Figure 5.16 shows the p-t correlation functions for Ar+Sc (left side) and Kr+Nb (right side) collisions at E / A2100 MeV. The figure is arranged into four rows corresponding from top to bottom to gates on the 4He kinetic energy of ECM S 30MeV, 30MeV < ECM g 50MeV, 50MeV < ECM S 70MeV, and 70M eV < ECM. The solid lines correspond to the exponential background fit shown also in Figure 5.16. The correlation functions show a broad structure at Ere; 3 4M eV which corre- sponds to excitation of four excited states in 4He. For ease of reference, detailed structural information for these states is recapitulated in Table 5.4 and the summary in Table 5.7. The magnitudes of these correlation functions increase significantly as the center of mass energy of the 4He increases. This increase indicates that the spa- tial region from which the p and t are emitted is larger or longer-lived or both for the less energetic particles in the center of mass. If a significant fraction of the protons or tritons were originating from long lived secondary decay of nuclei other than 4He, this could be a reason for the decrease. The extraction of temperature information from these data requires an accurate estimation of the background. The exponential backgrounds shown in Fig. 5.16 represent a reasonable choice for that background, but it is interesting to consider whether one can use data from neighboring nuclei without strong resonances at low energy in order to place constraints on the form of the background correlation and its energy dependence. Thus, correlation functions between the various hydrogen and helium isotopes were analyzed, but many of them have resonances complicating the analysis. Perhaps, the best case is the p-d correlation function, which has been 148 analyzed with the same energy and impact parameter gates as for the p-t correlation function. 149 Ar+Sc 100 A/lvleV Centrol Collisions Kr+Nb 100 A/MeV _ (E, + E,)CM I O — 35 lvieVE (l:p + E,)CM = O — 35 MeV 2 7Oo<®poir<11OOTAPP : 4.61 F7OO<®pok70 MeV; >70 MeV Ii. - - r ‘5; II..: 7.15. 1 :— 0.53 00 Figure 5.16: Single parameter Breit-Wigner fit of pt correlation on an exponential background. The data was obtained using a cut between 70 and 110 degrees in CMS and four energy cuts are indicated. 150 Figure 5.17 shows the gated p-d correlation functions. They are nearly flat except for a minimum at low relative energy that has about the same width for all energy gates. The red solid line in the figure is the function 2 _(L)d BGpd(E)=1—§e A, , (5.32) which lies closer to the data than does the exponential background discussed previously, which is shown here by the dashed line. We designate this as the fitted p-d background. These two functional forms, fitted p-d background and exponential background, were used in the analysis. Surprisingly, the experimental p-d correlation function exceeds unity at some relative energies in the two highest energy gates. This effect cannot be attributed to Coulomb interactions. It could be the result of some repulsive strong interaction effect or due to the influence of collective motion; it is not clear that the p-t background correlation function should exceed unity. 151 2 Ar+Sc 100 A/lvleV Central Collisions Kr+Nb lOO AZMeV (Ep +EI)CM=O-35Mev IGDW— 90° l<20° : (E, + L)“, = o — 35 MeV 1-5 rio,,,,—90°I<2O° IIII WTTI WIL‘IITII . 0 pd correlation l 1 l 1 l 1 1 1 1 1 1 1 1 I 1 35 — 50 MeV -— [Background function 1_ L 1 1 1 1 1 1 l 35 — 50 MeV O ()1 IIII III. IITI IIIIIIIFI III‘ l 1 1 1 1 l 1 1 l 1 i 1 J 1 l ’4 'I "v ... ... >— h E h- ,_ __ , 1 1 1 1 1 1 l 1 1 l 1 1 l l l + Wig.) i'H'P'” “figmi'mi 'IIIII 1 1 l L 1 1 1 1 1 1 1 J J 1 1 I >70 MeV IITIIIIIIIIIII III. I Figure 5.17: p-d background versus relative energy used for pt correlation for both Breit-Wigner and S-matrix analysis. The data was obtained using a cut between 70 and 110 degrees in CMS. 152 Fits assuming the exponential background and a single fixed temperature for all four states are shown in Figure 5.16. Fits assuming the fitted p-d background and a single fixed temperature for all four states are shown in Figure 5.18. The excited state temperatures extracted from the fits are shown in the panels of the figure also in Table 5.4 and the summary table 5.7 . In both cases, the extracted temperatures increase with center of mass energy, confirming the expectation that the excited state thermometers will show evidence for the earlier stages of the reaction being hotter. 153 Ar+Sc lOO A/Mev Central Collisions Kr+Nb IOO A/lvleV 2 :Ep+ECM—O—35lvleV E+EtCM:O—35MeV 15 b (i) o - E 70 .<0W70 lvleV :_ >70 MeV ._ t c 3. _ 15 5 TAPP: 6-935’ 1 :— :— 0.5 ; E Dd DOCkgI’OUfld O b- 1 1 1L 1 1 l 14 l 1 14 1 l ’— 1 J4 1 1 1 1 J 1 1 l 1 1 J 11* O 2 4 6 O 2 4 6 8 Em. (MeV) Ere. (MeV) Figure 5.18: Single parameter Breit-Wigner fit of pt correlation on a pd background. The data was obtained using a cut between 70 and 110 degrees in CMS. 154 In general, the quality of the fits are reasonably good for the lower three energy gates, but the calculation overpredicts the yield at Ere; < 0.35M eV. We have explored three possible explanations for this: 1) the possibility that the background has the wrong shape, 2) The possibility that the states are not populated statistically, and 3) the possibility that the line shapes for the excited states are not correct. Concerning the possibility that the background has the wrong shape, it is clear that one can construct a background that will make the fit go through the points. This can be done by taking the difference between the black solid line and the data and subtracting this difference from the background curve. Keeping the resonance contribution the same, the sum of resonances and background will equal the data. The background necessary to do this is illustrated by the solid green line in the lower right panel of Fig. 5.18. Note, that the green background lies below the fitted p- d background at Ere; < 1M eV, which is opposite to the trend of the actual p-d correlation (see Fig. 5.18). Turning to the possibility that the states are not populated statistically, we have refitted the correlation function, allowing the low spin 0+ and 0" states to have a different population probability and temperature from the higher spin 2“ and 1‘ states. These fits are shown in Figs. 5.19 and 5.20. Clearly, this improves the fit, but at the cost of having a higher temperatures for the higher spin states than for the lower spin states, for the highest Ere, gate. (The differences in the lower energy gates are smaller than the error bars of those fits.) A tendency for high spin state to be overpopulated with respect to the low spin states was observed in case of 10B fragments emitted in inclusive 14N + Ag and peripheral 36Ar + Au collisions at E/A=35 MeV [83, 130]. This was interpreted as an indication that equilibrium was not achieved in those reactions. It could also be the 155 case here that the most energetic particles are emitted before the system thermalizes. 156 2 Ar+Sc lOO A/lvleV Central Collisions Kr+Nb lOO A/MeV E (Ep ‘1‘ E1>CM : O — 35 MeV (Ep + E1>CM : O _ 35 MeV 1'5 §70°<®po,,70 MeV _E_ >70 MeV .5 : 50' background 1 1 A11 1 1 l 1 1 1 1 1 1 l 1 1 1 1 1 1 6 O 2 4 6 8 Ere. (MeV) Figure 5.19: Two parameter Breit—Wigner fit of pt correlation on a pd background. The data was obtained using a cut between 70 and 110 degrees in CMS. 157 Ar+Sc 100 A/MeV Central Collisions Kr+Nb 100 AZMeV _ (15,, + EJCM : 0 — 35 MeV (Ep + EJCM : 0 - 35 MeV 70°<0po,,<110° 70°<0 <110° pair _5 ()1 IIIIIIII ¢ IIWI IIIIITIIIII 1 —— Breit—Wigner fit 0.5 — Background function 0 pt correlation O A 1 L1 1 1 1 1 1 1 1 1 1 1 1 1 l J 1 L1 1 1 L 1111 1 I 1 35 — 50 MeV 35 — 50 MeV .— ._ .— ... ._._ .— .— rIIIIIIII I Q Li - v [1: O T 1 1 I 1 I 1 1 1 I I 1 1 L11 ” I 1 1 1 I 1 1 1 1 I I 1 I 1 1 + : 50 — 70 MeV : 50 — 70 MeV ‘_ 1.5 :— :_ E Z 1 r 0.5 E O 1 1 1 1 1 1 1 1 1 L1 1 1 l .. 1 1 1 1 1 1 1 L 11 l 1 1 1 L 2 ;_ >70 lvleV ;_ >70 lvleV 1.5 M H 1 :— --“ - L- _ f 0.5 r 3, exp background 0 _ 1 L 1 1L11L 1 l 1 1 1 1 1 1 '— 1 1 L 11 l l 1 1 1 1 1 1 1 L 0 2 4 6 0 2 4 6 8 Em. (lvleV) Em. (MeV) Figure 5.20: Two parameter Breit-Wigner fit of pt correlation on an exponential background. The data was obtained using a cut between 70 and 110 degrees in CMS and four energy cuts are indicated. 158 Breit—Wigner fits on energy cuts 10 V 4 . ' . : He ——> p+t Temperatures : 4He —> p+t Temperatures 9 ; Ar+Sc 100 MeV/A .1 Ar+Sc 100 MeV/A : PD background [E EXPONENTIAL background 8 _— t— : / _ 7 r- :_ 6 L T_ L— .— t : i— 5 7— l T C 4 — t e i C 3 _ I. i f 2 F —— Single parameter fit b —— Single parameter fit * t E i * States 21 .84MeV, 23.33MeV ; — States 21 .84MeV, 23.33MeV 1 — —— States 20.21 MeV, 21.01 MeV L — States 20.21MeV, 21.01MeV O P 1 1 1 A 1 11 L1 1 1 1 ’- 1 1 L 1 1 1 L 1 1 1 1 1 1 l 1 O 50 g 100 150 50 100 150 E'CM ECM Figure 5.21: Single parameter Breit-Wigner fit of pt correlation for Ar+Sc 100 MeV/ A. The upper and lower temperature bounds are the temperatures determined by two parameter fit. The two paramater fit gives a measure of systematic error with the previous panels. 159 Breit—Wigner fits on energy cuts 10 : ‘He°——> p+t Temperatures : 4He’—> p+t Temperatures 9 L Kr INb1OO MeV/A L Kr+Nb 100 MeV/A : PD background : EXPONENTIAL background 8 l L 7 :, L C E L L 6 l H T L e— 5 7 e // 4 L L 3 __ ; C l‘ - _ , T 2 l— -——- Single parameter fit r- —- Single parameter fit I- l— i: * States 21 .84MeV, 23.33MeV : — States 21 .84MeV, 23.33MeV 1— >- 1 — —— States 20.21 MeV, 21.01 MeV *- — States 20.21MeV, 21 .O1MeV : C O ’— 1 1 1 L L1 1 L 1 1 1 1 1 1 1 i- 1 1 1 1 1 A 1 1 1 1 1 1 1 O 50 100 150 50 100 150 ECM ECM Figure 5.22: Single parameter Breit-Wigner fit of pt correlation for Kr+Nb 100 MeV/ A. The upper and lower temperature bounds are the temperatures determined by two parameter fit. The two paramater fit gives a measure of systematic error with the previous panels. 160 J7r B(MeV) WidthF(MeV) Branching ratio(%) 0+ 20.21 0.50 100. 0‘ 21.01 0.64 76. 2" 21.84 1.26 63. 1’ 24.25 3.08 51. Table 5.3: of“ analysis.[105] —>p+t spectroscopic information for Breit-Wigner energy cuts System:background CUT (MeV) T(MeV) T0+0— (MeV) T2— 1— (MeV) Kr+Nb100MeV:exp <35 4.23831 3.82095 4.46786 Kr+Nb100MeV2exp 35—50 4.67654 4.34264 4.94783 Kr+Nb100MeV:exp 50-70 5.19825 5.01043 5.36519 Kr+Nb100MeV:exp 70-00 6.53381 5.97037 6.97204 Ar+Sc100MeV:exp <35 4.61394 4.62481 4.63481 Ar+Sc100MeV:exp 35-50 5.65735 5.49040 5.86603 Ar+SclOOMeV:exp 50-70 6.34599 6.22078 6.42946 Ar+Sc100MeV:exp 70-oo 7.15985 6.36686 7.74416 Kr+Nb100MeV:PD <35 3.84182 3.82095 3.88356 Kr+Nb100MeV:PD 35-50 4.34265 4.30092 4.40526 Kr+Nb100MeV:PD 50-70 4.90609 4.82262 4.98957 Kr+Nb100MeV:PD 70-oo 6.30426 5.36519 6.95117 Ar+Sc100MeV:PD <35 4.30000 4.00876 4.63481 Ar+Sc100MeV2PD 35-50 5.34432 5.23998 5.49040 Ar+Sc100MeV:PD 50-70 6.07471 5.65735 6.40800 Ar+Sc100MeVzPD 70-oo 6.93030 5.78255 7.74416 Table 5.4: Temperature fit information on energy cuts for Breit-Wigner analysis. The third column is the temperature from the single parameter fit. The fourth and fifth columns are upper and lower bounds from the two parameter fits. 161 5.5 Evidence for cooling from excited state popu- lation: S-matrix analysis Since the Breit-Wigner analyses yielded poor fits to the decay spectra for the highest ECM gate with ECM > 70M eV, it is natural to inquire whether this is because the resonance line shapes of the Breit-Wigner discussed earlier are inaccurate. For this reason, we contacted Gerry Hale, whose group had performed detailed measurements of the p+t reaction. Unfortunately, that data has never been published in a form that is directly usable for our purposes, but we were able to obtain advice from him as a private communication. Dr. Hale proposed that we use an ”S-matrix” form for the decay spectrum and provided us with a parameterized line shape. As a detailed justification of this form in terms of measured phase shifts was not provided, we take this as an empirical resonance line shape for the relevant states in 4He. This ”S-matrix” expresses the line shape in the following form: 1 cm” E: —E/T_pc E __ 2: ./d e « p+t the channel radius is 4.93fm [45]. Table 5.5 shows the other spectroscopic information for a ——> p+t in this S-matrix analysis. Using the expression for the decay spectrum figures 5.24 and 5.26 show the temperature trend for energy cuts 0-35, 35-50, 50-70, >70 MeV for systems Ar+Sc 100 Mev/ A and Kr+Nb at 100 Mev/ A for the p—d and the exponential background, respectively . Figures 5.25 and 5.23 Show the corresponding examples of two param- eter fitting where as before the 0+, 0‘ state has one parameter and the 211‘ states have the other. We again see that there is a problem fitting the correlation function for the highest energy gates. The temperatures from these fits are also given in Ta- ble 5.6 and the summary table 5.7. Thus, this S-matrix approach suffers the same problem that it fails to fit the relative energy spectrum at Ere; < 0.5M eV for the gate with ECM > 7OMeV when all states are assumed to be populated in a manner that is consistent with a single temperatures. When the high spin states and low spin states are fitted separately, the fit is best when the high spin state are populated at a higher temperature than are the low spin states. Putting this together, we have used the various fits to estimate the systematic error in the extracted temperatures. The cross—shaded area in Fig. 5.22 shows the range of temperatures extracted from the Breit-Wigner fits as a function of the center of mass energy. The cross—shaded area in Fig. 5.28 shows the range of temperatures extracted from the S-Matix fits as a function of the center of mass energy. Both figures show a clear trend of increasing temperature with ECM. Since the Breit-Wigner resonance line shapes are the published values and the S—Matrix parameterization is not published, we take the previous Fig. 5.22 as the most reliable demonstration that cooling influences the excited state thermometer values. 163 2 Ar+Sc lOO A/MeV Centrol Collisions Kr+Nb 100 A/MeV _ (Ep + EJCM : O — 35 MeV ([5,, + EJCM :- O — 35 MeV 1-5 §7O°70 MeVE >70 MeV 2 r F;— ”W . . : An._ . " vavv pd bockground l l L l l l l L l 14* L l l L l l l l l l 4 4 a, (MeV) E... (MeV) l l l O l p l. D'- .0 U1 .4 Imf l\) l 6 Figure 5.23: Two parameter S—matrix fit of pt correlation versus relative en— ergy(MeV). This panel is intended to demonstrate how well the fits are for a two parameter fit to get error limits on extracted temperatures for the single parameter fit. The data was obtained using a cut between 70 and 110 degrees in CMS and rela- tive energy cuts indicated. The apparent temperatures indicated here is the average temperatures of the minimum X2 of the fit. 164 2 Ar+Sc lOO AZMeV Central Collisions Kr+Nb lOO A/MeV " (Ep + EJCM : O — 35 MeV (Ep + EJCM = O — 35 MeV 1. O O O O 5 7O .<®poir70 MeV E_ >70 MeV ‘5 2*" Tm, : 7.38 5 1 E- E— 005 f 1 pd background 0 I L I L I l L I I L I I I I I ’— I L I I I I J I I I I I I I i o 2 4 6 O 2 4 6 8 Em. (MeV) Ere. (MeV) Figure 5.24: Single parameter S-matrix fit of pt correlation versus relative en- ergy(MeV) on a pd background. The data was obtained using a cut between 70 and 110 degrees in CMS and relative energy cuts indicated. 2 Ar+Sc lOO A/MeV Central Collisions Kr+Nb lOO A/MeV (Ep + EJCM = O - 35 MeV (Ep + EJCM : O — 35 MeV 70°<®poh70 MeVE >70 MeV 1-5 W h ' ° E - ““n E . i :— ' -"h-""'~ >— l- 0.5 F L exp background 0 :‘1 1 1 l 1 1 1 L4 1 1 I 1 L 1 1 1 1 l 1 1 1 I L1 1 141 1 O 2 4 6 4 6 E,,,. (MeV) Em. (MeV) Figure 5.25: Two parameter S—matrix fit of pt correlation versus relative en- ergy(MeV)on an exponential background. The data was obtained using a cut between 70 and 110 degrees in CMS and relative energy cuts indicated. 166 Ar+Sc lOO A/MeV Central Collisions Kr+Nb lOO MMeV (B, + EJCM : O — 35 MeV (Ep + E)“, = O — 35 MeV : 4.36 9 05 ,4 — R—matrix fit L E --—1 Background flunction: 0 pt correlation O : 1 1 1 1 1 1 1 1 1 1 1 1 : 1 1 L 1 1 1 1 1 1 L L g : 35 — 50 MeV: 35 -— 50 MeV 15 L L. : ... TApp : 6.01 E TAPP : 4.88 O O ()1 _L Y . I F T. 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 50 — 7e Mev; 5O - 70 MeV TA... : 6.72 3 TA... : 6.46 1 + R(E,e,) O IIIILJLIIILILIJLLI FIIIJLIIJLIIJ+IIII >70 Mevé >70 MeV ._ y— >— )—- p- >- >— h ... U1 In C C \l 07 LN ”cf E' TAPP26.95 1E- 05 E : exp background O¢...1...1...1.1.P.11111111.11..L O 2 4 6 O 2 4 6 8 E,e,(MeV) Ere.(|\/leV) Figure 5.26: Single parameter S-matrix fit of pt correlation versus relative en- ergy(MeV)on an exponential background. The data was obtained using a cut between 70 and 110 degrees in CMS and relative energy cuts indicated. 167 R—matrix fits on energy cuts lO : ‘He‘ —) p+t Temperatures C 4He’-—> p+t Temperatures 9 L Ar+Sc 100 MeV/A L Ar+Sc 100 MeV/A : PD background / ~ EXPONENTIAL backgrou 8 ; L 7 L 5 ~ r 6 :- ~ 1— 5 L f 4 L L 3 L L C E 2 r —— Single parameter fit. L —— Single parameter fit I — States 211‘ ; r States 221” l — —— States OLO‘ :- —— States O’,O' O '— I L I I I l L I l I L I I J I b I I I I I L L I J_ LLL I I I I O 50 100 150 50 100 150 ECM E611 Figure 5.27: Single parameter S-matrix fit of pt correlation for Ar+Sc 100 MeV/ A. The upper and lower temperature bounds are the temperatures determined by two parameter fit. The two paramater fit gives a measure of systematic error with the previous panels. 168 R—matrix fits on energy cuts TO — ‘He°—>p+t Temperatures : ‘He‘—>p+t Temperatures 9 L Kr+Nb 100 MeV/A L Kr+Nb 100 MeV/A ~ r : PD background : EXPONENTIAL background 8 7— L : / ; ~ r 7 f m : t 6 *- a. r— 5 R :4 4 :_ :_ L C ~ r 3 ~— L 2 :- —— Single parameter fit E— Single parameter fit I — States 221" L a —- States 211‘ 1 l —— States 0*,0‘ :— ——— States 0*,0‘ ~ r l— l— O I I I I L I I I I I LI I I I I I I I I I I I I I I; I I I O 50 100 150 50 100 150 ECM ECM Figure 5.28: Single parameter S—matrix fit of pt correlation for Kr+Nb 100 MeV/ A. The upper and lower temperature bounds are the temperatures determined by two parameter fit. The two paramater fit gives a measure of systematic error with the previous panels. 169 J1r CE, E” (MeV) 93p 0+ 1 0113864019627 0.78301 0" 1 1.0309—10.30109 0.56843 2— 5 1.5841-10.73189 0.62836 1— 3 2.8043-1233620 1.0221 Table 5.5: a* —>p+t spectroscopic information for S-matrix analysis.[45] System:background CUT(MeV) T(MeV) T0+0— (MeV) T2_1— (MeV) Kr+Nb100MeV:exp 00-35 4.36352 3.82 4.61 Kr+Nb100MeV:exp 35-50 4.88523 4.46786 5.09391 Kr+Nb100MeV:exp 50-70 5.46953 5.25 5.62 Kr+Nb100MeV:exp 70-00 6.95117 6.50 7.52 Ar+Sc100MeV:exp 00-35 4.82262 4.8 4.9 Ar+Sc100MeV:exp 35-50 6.01210 5.8 6.2 Ar+Sc100MeV:exp 50-70 6.72162 6.59 6.98 Ar+Sc100MeV:exp 70-00 7.63982 6.98 8.5 Kr+Nb100MeV:PD 00-35 3.98790 3.90 4.01 Kr+Nb100MeV:PD 35-50 4.55134 4.50 4.60 Kr+Nb100MeV:PD 50-70 5.17738 5.08 5.31 Kr+Nb100MeV:PD 70-00 6.67988 5.96 7.61 Ar+Sc100MeVzPD 00-35 4.50960 4.34 4.89 Ar+Sc100MeV2PD 35—50 5.62250 5.58 5.85 Ar+Sc100MeVzPD 50-70 6.42946 6.09 7.00 Ar+Sc100MeVIPD 70-00 7.38940 6.51 8.58 Table 5.6: Temperature fit information on energy cuts for S-matrix analysis. The third column is the temperature from the single parameter fit. The fourth and fifth columns are upper and lower bounds from the two parameter fits. 170 5.6 Summarized results of Breit-Wigner and S- matrix analysis We summarize here the previous tables into a single table 5.7 . The extremes in temperature are indicated by the shaded areas in figures 5.29 and 5.30. Each figure picks the highest and lowest temperatures of both backgrounds and both line shapes. We expect that a better choice of background and line shape would fall in these shaded regions. The same trend in a higher temperature at an earlier time is still indicated. 171 .88 mg 8.388 an8-m 8%. 3888 5% 88 5.88685. 2: 8. @8888 8838 £888 88 :888m 833m SE. .855 88mm>>£gm 8888.88 025 8%. 80¢ £883 .832 88 .893 8.8 8838 5% 88 £88 23. .E Hmcmwg-figm .8888qu 8888 85 88.8. 838883 85 mm 8838 838 838 . £8388 xt88-m 88 88wm3-p88m .85 833 883800 ”56 838B mm.w Saw ovmwm .5 335.5 mmmw5.m omommd 865 Dmn>820o8m+8< 8.5 moé ovmmvd cowovd mm5m©.m H5636 c5-om Qmu>82oo8m+8< mwd mm.m ommmod owomvb wammm.m mmvvmfi om-mm Qmu>8EoSom+8< aw.“V wme 8ng4 $358 ©5801.V oooom.“V mm V Qmu>820o8m+8< 8.5 ooh @8506 5:36 330m omvomb 8-o5 Qmu>82832+8¥ 3.1m wo.m wm55fim 5338 mommmfiV @888 O5-om. an>82oozz+8v~ om:V owe 330v mange mmoomnv momvmd om-mm Omu>82©o32+8¥ 3.8 om.m 85345 ommww.m maemmfi $343“ mmv Omn>820o32+8v~ m.w wad $96.5 $35.5 owoomb $34.5 8-O5 axmn>82o08m+8< mozo and 3&56 9436.0 w5omm© 0.33.0 o5-om axou>820o8m+g< m6 Wm Swag-V moomwh 35248 95536 om-mm axon>82008m+8< m8 we momquV $368 5388 63.5.4 mmv axen>82008m+8< 3.5 omd 5:36 vom5m¢ 5mo5m.m memm@ 8-05 Q8H>8Eoozz+8vm mod mm.m mmmovh @Hmwmh mvoSh mmwmfim Q5-om mxmu>82oo32+8v~ Hammoh $594.6 mmmwwn.4 $5388 88va 335:4 om-mm Q8H>8EOOHQZ+8¥ S8 mw.m mmmcmd $5914 mmomw.m Hmwmm.v mmv Q8H>82oozz+8x x588lm xt88lm xt88lm 88wr$lp88m 88w§>Ip88m cocwwglfimgm I I $63: -25 968 -025 9623.5 968 -35 968 5.55 962? 9628.8 88583868 172 4I—Ie'—> p+t Temperatures Ar+Sc 100 MeV/A l—S TTIITIIIIIIIII[I]llllflfirolrllIIIIITTIIIIIIIl limits 0 lJliLl¥JgLPILngiLl+l l l L l L l l I l l l O 20 4O 60 80 CM 100 120 140 160 Figure 5.29: Combined temperature limits of ArlOO energy cut analysis . 173 IO 4He'—>p+t Temperatures Kr+Nb 100 MeV/A IIIlllllllj—rTflrrllITleTTlTFIITIIIIIIIITTT—IIITII O iLllplllligllllillll11111111111 60 80 100 120 I40 160 CM O M O .p. C Figure 5.30: Combined temperature limits of KrlOO energy cut analysis . 174 5.7 Evidence of cooling using the Albergo Ther- mometer It is interesting to ask whether comparable evidence for cooling in these reactions can be provided by isotope thermometer measurements. Evidence for cooling has been observed in Xe+ Cu reactions at E/A=30 MeV [137] where an isotopic thermometer contructed from [Y(3He)/ Y(4He)]/[Y(2H)/ Y(3H)] was analyzed as a function of the energy of the outgoing particle species. Investigation of cooling is potentially more ambiguous with isotopic thermometers than with excited state thermometers because four different fragments enter into the isotopic ratio. Nevertheless, we analyze the isotope ratio with gates on the velocity or equivalently energy/ nucleon in the center of mass system in order to see whether cooling effects can be observed. To calculate the gated double isotope ratios, we integrated the yields of fragments emitted in central collisions at 70" S 00M 3 110" in 4 MeV/ nucleon bins. Yn: 4An 5.35 ./A(n-—d 1) dEdE ( ) and then constructed double isotopic ratios [Y(3He)/Y(4He)]/[Y(2H)/Y(3H)] and [Y(3He)/ Y(4He)]/[Y(6Lz')/ Y(7Li)]. The isotOpic temperatures calculated from these ratios are plotted in Fig. 5.31 for Kr+Nb collisions and in Fig. 5.32. for Ar+Sc collisions. There is a clear increase the resulting isotopic temperatures with the energy/ nucleon in the center of mass. For Kr+Nb collisions, the lowest temperature of T150 z 4 — 5MeV occurs for Kr+Nb collisions at energies that are about 50% above the Coulomb barrier. This is comparable to the temperatures extracted from the excited state populations. The maximum values of T150 as 10 — 12M eV occur at an energy/ nucleon of about 20-25 MeV in the center of mass. Such high temperatures are hard to reconcile to a thermal interpretation. For Ar+Sc collisions at E/A= 50 175 MeV, the temperatures are similar, but there is an increasing discrepancy between the temperature calculated from the [Y(3He)/Y(4He)]/[Y(2H)/Y(3H)] ratio and the [Y(3He)/Y(4He)]/[Y(6L’i)/Y(7Lz')] ratio at the higher incident energies of E/A=100 and 150 MeV. In all of the panels, the values for the isotope temperatures calcuated by integrat- ing over the entire energy spectrum is shown by the solid horizontal lines. Clearly, these integral isotopic temperatures shown here and also earlier in 5.12-5.15 reflect the averaging over faster and earlier emissions characterized by higher isotopic tempera- tures and slower and later emissions characterized by lower isotopic temperatures. It has been noted [61] that phase space consideration favors the emission of more weakly bound 3H8 at earlier stages of the breakup when the temperature is higher and that the emission of more strongly 4He may be rather delayed until the system expands to subnuclear density. Thus the differences between the values extracted from these isotopic temperatures and the 4He excited state temperatures discussed earlier may be a reflection of the fact that these emission sample somewhat different times. 176 18 i kr55 L kr7O 16 :— g L- n— L— .— 14 5 f i t A12 C." r > ~ s £10 :— // F‘ i? V _ / __ / l— ” / — El 8 :— /’/ :_ /// 6 .1 _ /_ __ _q; ——r“g;‘" fa“ “—v v— v“ v r ; 2 e/ 4 :— // _— :// r/ 2 ‘_’— _Z 20 3..1.1..411Uniliullflullimrlmi.lirril 18 E. kr1OO ; [“120 : E o T(“He—dt) 16 E :— E] T(3-‘He—5-7Li) 14 E— :— C : A12 5 2“ ¢ ‘<>’ > f ¢ : # £10 L— , _— _‘Q: v ~ L—Eib — — -— / '— ; :gf/ ; 7.1m, --,/,“,nn- 8 ,_. —:-—-—( '*__,Zr_:-7 7 7 7 —_:4_ I /, : —Cl: / Z —o—‘{:r— 6 L— :8: / ___ Q— h— ’// l- 4 L / :_ E : 2 :4/ L/ O : L l l l l l l l l l l l l l l l l l l l:l l lLl 1 LJ J l LA I l l l l l 41 O 10 2O 30 4(0) 10 20 .50 4O Energy(MeV/A) Energy(MeV/A) Figure 5.31: Albergo temperatures on energy cuts for Kr+Nb systems. This shows a trend in increasing temperature with higher energy cuts. The circles are the 3"‘He-dt thermometer and the squares are the 3"‘He—‘VLi thermometer. The horizontal lines represent the temperatures from the yields of all the data added together without an energy gate. The red line is a guide to indicate the increasing trend with respect to beam energy. The uncertainties are the sum of the statistical uncertainty and a 5% PID uncertainty per isotope in quadrature. 177 20 l- — >— 18 L orSO L orlOO L or15O — — F _ s ~ C) TC”He—ct) _ p F L— .— _— 16 L _ _ U T(" e~°7Li) _ _ ~ y {>{} _ _ - > 14 _ r— (>0 g fl C i I if; 12 — — — T %} a; ~ 4* — — i, 10 +— — — 5 T: ” / “ 4L / b ‘ // ~ / ~ 1 T f - 4%} 8 — — 5} r : 3 / : i t i} 6 _.4747 T 7T; ; _ a), _ / _ _ ._ / F‘ 4 ~ / — /’ e s _ / _ ._ _/ _. 2 L 1 O >- 1 l L l l 1_41_ l_ L l l l 1 l l l_ l l l l l l L O 20 4O 20 4O 20 4O Energy(MeV/A) Energy(MeV/A) Energy(MeV/A) Figure 5.32: Albergo temperatures on energy cuts for Ar+Sc systems. This shows a trend in increasing temperature with higher energy cuts. The circles are the 3"‘He-dt thermometer and the squares are the 3”4He-6'7Li thermometer. The horizontal lines represent the temperatures from the yields of all the data added together without an energy gate. The red line is a guide to indicate the increasing trend with respect to beam energy. The uncertainties are the sum of the statistical uncertainty and a 5% PID uncertainty per isotope in quadrature. 178 5.8 Sequential decay In this thesis, we have not explored the effects of secondary decay on temperatures extracted from the excited state populations and isotopic ratios. Clearly, our obser- vations of significant yields of nuclei in their excited states implies the existence of secondary decay contributions. Correction for secondary decay, however, is difficult because one cannot assume an equilibrium distribution of hot unstable fragments be- fore decay; instead we have limited our corrections to the empirical approach discussed in Section 4.3.3. Because this thesis concentrates largely on the 4He excited state thermometer and the comparison of this to the temperatures extracted from the 3He/“He isotope ratio, the studies of ref. [136] are especially significant. These studies point out that both the 4He excited state thermometer and thermometers based on the 3He/‘iHe isotOpe ratio have very similar secondary decay corrections, because both are primar- ily influenced by secondary decay to comparably well-bound 4He in its ground state. Thus, the differences between the temperatures derived from the two thermometers are unlikely the result of the secondary decay. They are more likely a result of non- equilibrium contributions to the 3H6 yields as discussed in section 4.3.2 or some other effect. 179 Chapter 6 Summary Whether equilibrium can be assumed in modeling the multi-fragment decays of highly excited nuclear systems is an important question. If one can assume equilibrium, then a large number of equilibrium analyses of multi-fragmentation can be attempted. Such analyses are mainly aimed at determining thermal properties of nuclear matter. These include some recent and somewhat more speculative topics, such as determining the caloric curve for nuclear matter [91, 80], whether it displays a region of “negative heat capacit” [22, 29, 30] and extracting critical exponents for the nuclear liquid-gas phase transition [78, 31, 32] Unfortunately for such studies, there are not many cases where the validity of ther- mal equilibrium has been tested. One test, performed for central Au+Au collisions at E/A=35 MeV [50], found excited state and isotopic temperatures in agreement with equilibrium statistical model calculations. In addition a systematic analyses of isotopic temperatures [81] found consistent equilibrium interpretation of isotopic temperatures could be proposed. These analyses also indicated a systematic decrease with system size for the temperature in the “plateau regio”, where the isotopic tem- perature varies gradually with deduced excitation energy. For other systems, the data indicated substantial disagreement with equilibrium 180 predictions [100]. In the work of Serfling et al. [100], isotOpic temperatures extracted from the Helium Lithium isotopic thermometer increased with incident energy. The temperatures extracted from excited state thermometers did not indicate either a problem with thermal equilibrium or a problem with either the HeLi isotopic ther- mometer or the 4He,5 Li and 8Be excited state thermometers or both. In this dis- sertation, we reexamined this issue by studying the incident energy dependence of the same thermometers, plus the CLi isotopic thermometer for central Kr+Nb and Ar+Sc collisions. The trends observed for the Kr+Nb and Ar+Sc systems are in many ways similar to those observed by Serfling. In particular, the HeLi temperatures increase strongly with incident energy, while those for 4He,5 Li and 3B6 excited state thermometers increase only slightly. Interestingly, the CLi thermometer yields values that are similar to those observed for the 4He,5 Li and 8Be excited state thermometers. Calculations were performed with the Statistical Multi-fragmentation model to see whether the observed 3H6 yields, which govern the temperatures for the HeLi thermometer, are consistent with an equilibrium statistical model calculation that reproduces the fragment charge distribution and fragment multiplicities previously observed for the Kr+Nb system. In fact, the statistical calculations that reproduced the fragment yields, grossly underpredicted the yields for deuterons, tritons and 3He’s. This supports the interpretation that early preequilbrium emission of 3H8 is responsi- ble for a major fraction of the observed yield, an interpretation that is also supported by calculations for light cluster production within an extended BUU model. The pos- sibility that secondary decay may be partly responsible for the differences between the HeLi thermometer and the 4He,5 Li and 8Be excited state thermometers cannot be entirely ruled out, but the predicted similarity between the effect of secondary on the HeLi thermometer and on that on the 4He excited state thermometer argues 181 against this possibility. We also examined whether the light cluster yields are inconsistent with local ther- mal equilibrium by comparing their isotopic distributions to an extended isoscaling formalism. Interestingly, we find that thermometers based on the 3H8 —4 He binding energy difference and the isotopic distributions can be analyzed in a consistent ap- proach. This suggests that both are roughly consistent with the assumption of local thermal equilibrium during the early emission stage before the system expands and multi-fragmentation occurs. Despite the existance of a possible explanation for the difference between HeLi temperatures and the 4He,5 Li and 8B6 excited state temperatures in terms of early pre-equilibrium the 3H6 emission, the observation of such low values for the 4He,5 Li and 8Be excited state temperatures is difficult to explain in light of the large available excitation energies of these collisions to explore this issue, the 4He excited state thermometer was analyzed as a function of the kinetic energy of the 4He in the center of mass system. Analyses were performed for ground and excited 4He particles emitted to 70" 3 66m 3 100° and for four gates on Ecm. These analyses revealed that 4He temperatures increase with the 4He kinetic energy in the center of mass. Such behavior is consis- tent with cooling of the system as it expands and radiates particles. Similar trends were observed for the HeLi and HeH isotopic temperatures. In all cases, we found evidence for cooling phenomena that more strongly influence the HeLi temperature values than the 4He temperature values. Thus, there is little support for the validity of overall or global thermal equilibrium at E/A=100 MeV from either thermometer. At other incident energies, the HeLi isotopic thermometer reveals significant cool- ing effects, which are admittedly stronger at higher incident energies than at lower incident energies. 182 Looking beyond this dissertation, we hope more measurements will be performed that explore the degree to which equilibrium is achieved in multifragmentation pro- cesses. Evidence is emerging from analyses of central Sn+Sn [106] that similar cooling effects occur at E/A=50 MeV, in the spectra of heavy carbon fragments. Such obser- vations suggest that fragments with different kinetic energies are emitted at different times from a system that is cooling with time. Thus, it may be more appropriate to consider interpreting the emission in terms of a rate equation that assumes emission from the surface of the expanding system, such as assumed in the EES model. The widespread interpretation of such data in terms of equilibrium approaches cannot be continued in its present form. Model approaches are required that take this time dependence into account. 183 Appendix A Derivation of the Relativistic Boltzmann Distribution A.1 Expression for Volume Emission We start with the integral of ref [42] pp358 fe’BCOthsinh2” :cda: = %(%)”F (V + %) K” (5) , (A.1) This distribution is obtained by evaluating and normalizing the integral [62]: ‘2 . °° _ W jeJ—r” dp (A2) P 0 If we substitute p:mc- sinhx into equation A2 we obtain an equation that is a derivative of A.1 with respect to 6. Setting 1/ = 1: 00 fe‘ficogh‘” sinh2 :r cosh :rda: = — 0 11(1(fi):21{1(fi)+Ko(B) cm 5 a? B ’ here the relation K[ (2) 2 —K1 (2) /z — K0 (.2) was used. (A.3) 184 Now equation A2 after normalization becomes the probability distribution in terms of momentum: _ {m2c4+p262 e T 19%}? PV(P)dP= , 2 2 ,2 * , (A-4) [2 (2%.?) K1 (are) + (...—if) Ko (2%)] W This can be expressed in other kinematic variables using conservation of probability: P(p)(1p = P(E)dE = P(v)dv Perhaps the most useful expression is in terms of the kinetic energy, because it is measured in the lab: _ mc2+E \/(mc2 + E)2 — 772%“ (me2 + E) dE [2 (a)? m (e?) + (.27) K0 ea] m... An asymptotic expansion is used on the MacDonald functions K0(z) and K1(z) in Pv (E) dE = (A.5) the fitting procedure. For large (mcz/T) which is the nonrelativistic case it can be shown eq. A.5 reduces to the classical Maxwell-Boltzmann distribution. Note in the fitting coulomb energy is subtracted from the total energy term so that the kinetic energy at breakup appears in the 3 source fitting function. 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