A :Yutl?.b.. H .. .. la .hil ..A§-i! 1': f» ‘1 to}. ‘6.» 1.3-... 38?”;- . t: a if?! .5! .7. 5.1:!!! .ESlfrvblpf’J?! 7.4.43.2}! !. f‘ .01»). .V: 1:... It! 2:35.. {if Kilt!!! 6 7. .1... It!!! 2‘. l.€.ia 3.3!!!!) ~ 1! {lizz‘n‘ 1' .1. lob-2.x ' all...- c.).ru|...1..ol .. I MICHIGAN SITY LIBRARIES \H \\\\\\\\\\\\\\\\\\\\\\\1\1\\\\\\ \\\\\\\\\\I I This is to certify that the dissertation entitled Emission Temperatures From The Decay of Particle Unstable Complex Nuclei presented by Tapan Kumar Nayak has been accepted towards fulfillment of the requirements for PhD degree in Phys iC 5 Major professor Date July 18, 1990 MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY Michigan State University 1_ PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE MSU Is An Affirmative Action/Equal Opportunity Institution ottoman-pd EMISSION TEMPERATURES FROM THE DECAY OF PARTICLE UN STABLE COMPLEX NUCLEI By Tapan Kumar Nayak A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1990 ABSTRACT EMISSION TEMPERATURES FROM THE DECAY OF PARTICLE UNSTABLE COMPLEX NUCLEI BY Tapan Kumar Nayak Q Relative populations of particle-unstable states were measured for complex frag- ments emitted in the reaction 14N+Ag at E / A = 35 MeV by using a position sensitive high resolution hodoscope. The hodoscope consisted of 13 telescopes, four of these telescopes were designed to isotopically resolve fragments with 3 S Z _<_ 10 and the other nine to resolve hydrogen and helium isotopes. In order to optimize the excita- tion energy resolution of the hodoscope, each telescope contained an :1: — 3/ position sensitive gas proportional counter. A position resolution better than 0.5 mm was obtained for 5.8 MeV a-particles. For the a-decay channels of the particle unstable loB nucleus produced in the reaction, an excitation energy resolution of about 50 keV (FWHM) was achieved. Experimental population probabilities of particle-unstable states were extracted by fitting the coincidence spectra of the decay products by an appropriate R-matrix or Breit-Wigner formalism. According to thermal models, the populations of excited states at freezeout are expected to follow a Boltzmann distribution weighted by the emission temperature of the system. Tests of this freezeout assumption were made by comparing the experimental population probabilities to the predictions of statistical calculations. Extensive statistical calculations which include the effect of sequential feeding from heavier particle unstable nuclei were performed to estimate the population proba- bilities of the states starting with a thermal distribution of primary fragments at an initial temperature, Tem. A global comparison of the measured and calculated population probabilities and the ratios of population probabilities indicate emission temperatures of about 3—4 MeV. But a detailed comparison for individual fragments for a calculation with Tam = 4 MeV reveals that about half of the measured popu- lation probabilities and one third of the ratios of the population probabilities differ significantly from the predictions of statistical calculations. Calculations which in- clude rotational effects could not satisfactorily account for this discrepancy. These results suggest a possible breakdown of the assumption of local thermal equilibrium at freezeout. To my Parents Madhab and Renu iii ACKNOWLEDGMENTS I would like to acknowledge the excellent support of the staff, students, and faculty of the National Superconducting Cyclotron Laboratory and the Physics department of Michigan State University. I wish to express my deepest appreciation to my thesis adviser, Prof. Bill Lynch, for his guidanCe, support and encouragement throughout the course of this work. Sincere thanks go to Prof. Konrad Gelbke for his invaluable advice, suggestions, and continued interest in my work. I am grateful to Dr. Tetsuya Murakami, whose numerous criticisms, assistance, and moreover his friendship had a profound impact on my graduate career. I would like to thank Dr. Betty Tsang for her help during the course of the work. Thanks to Dr. David Fields, Dr. Josef Pochodzalla and Dr. Kris Kwiatkowski for their help. Thanks to Ken Swartz who had a head start in the gas detector project before I got involved in it. Special thanks go to my fellow graduate students Ziping Chen, Hong- ming Xu, Fan Zhu and Yeong Duk Kim for their assistance during the experiment and their friendship. I would like to thank Dr. V. K. B. Kota of Physical Research Laboratory, Ahemedabad for providing me with the statistical level density program, which I regret I couldn’t implement to our code. I appreciate the help he had pro- vided to me. Thanks to Prof. Aron Galonsky for the interest he has shown in my iv work. I wish to thank Dr. N. Anantaraman for his helpful suggestions. Very special thanks go to Prof. S. D. Mahanti who brought me to MSU. His en- couragement, interest in my research, and assistance both in my research and outside is highly appreciated. Last but not least, I would like to acknowledge the understanding and love of my parents who valued my education very highly and supported me even during the hardest times. Special thanks also go to my brothers and sisters for their love. Contents LIST OF TABLES ix LIST OF FIGURES xi 1 Introduction 1 I Motivation ................................. 1 II Organization ............................... 7 2 Experimental Setup 9 I Description of the Detection Array ................... 10 II Position Sensitive Gas Proportional Counters ............. 17 III Position Calibrations of the Gas Counters ............... 19 IV Energy Calibrations ............................ 20 V Particle Identification ........................... 23 3 Data Analysis and Reduction 29 I Single Particle Inclusive Cross Sections ................. 30 II Two Particle Coincidence Cross Sections ................ 36 A Detection Efficiency and Resolution ............... 36 vi B Fits to the Resonances : R-matrix theory ............ 42 III Excitation Energy Spectra for Particle Unstable Nuclei ........ 52 4 Sequential Feeding from Higher-lying States 84 I Levels and Level Densities ........................ 85 II Primary Populations ........................... 87 III Details of the Decay Calculations .................... 90 IV Results ................................... 92 5 Nonstatistical Excited-State Populations 96 I Non-statistical Populations of States in 10B ............... 97 II Angular Momentum Effects on Populations of States ......... 99 A Rotational Effects : Statistical Theory of Compound Nucleus Decay ............................... 99 B Rotational Effects : Sequential Feeding Calculations ...... 103 III Decay Angular Distributions ....................... 106 A Experimental Angular Correlations for 6Li and 10B Decays . . 106 B Comparison with Statistical Calculations ............ 110 IV Discussion ................................. 117 6 Emission Temperatures 119 7 Summary and Conclusion 132 APPENDICES 136 vii A Electronics 136 B Details of the Efficiency Calculations 139 LIST OF REFERENCES 145 viii List of Tables 3.1 3.2 3.3 3.4 Source parameters of three moving-source fits. The Coulomb repulsion energies Uc and the temperature parameters T.- are given in units of MeV, and the normalization constants N,- are given in units of pb/(sr MeV3/2) ................................... Spectroscopic information for Lithium and Beryllium isotopes which was used to extract excited state populations. Branching ratios I‘c/I‘ are given in percentage. Except for 5Li, relative populations in are defined relative to the particle stable yields for the same nucleus. The group structure is explained in the text. ................ Spectroscopic information for 8B, 10B, and 11C isotopes which was used to extract excited state populations. The branching ratios are given in percent, and m are defined relative to the particle stable yields for the same nucleus. The group structure is explained in the text ....... Spectroscopic information for 13N and 14N isotopes which was used to extract excited state populations. Branching ratios I‘c/I‘ are given in percent, and n), are defined relative to the particle stable yields for the same nucleus. The group structure is explained in the text. ..... ix 33 67 3.5 Spectroscopic information for 16O and 180 isotopes which was used to extract excited state populations. Branching ratios I‘c/I‘ are given in percent, and n ,\ are defined relative to the particle stable yields for the same nucleus. The group structure is explained in the text ....... 81 List of Figures 1.1 1.2 2.1 2.2 2.3 2.4 2.5 Inclusive differential cross section for 10B at the laboratory angles shown in the figure. The solid curves are described in the text. The dashed curves correspond to fits with a “three moving source” param- eterization which will be discussed in chapter 3. ............ Apparent emission temperatures extracted from the relative popula- tions of states of 4He,5Li,6Li, and 8Be nuclei extracted for the three reactions indicated in the figure [Chen 88a]. .............. Front view of the hodoscope showing all the nine light particle(LP) and four heavy fragment (HF) telescopes. The actual dimensions of heavy fragment detectors are displayed. Since they are closer to the target, however, they cover larger solid angles than suggested by this projection .................................. Schematic cross sectional view of the hodoscope ............. Expanded drawing showing the LP (top) and HF (bottom) telescopes. Photograph of the assembled hodoscope. The four HF telescopes are in the foreground. Eight out of nine LP telescopes can be seen at a larger distance from the target. ..................... Photograph the full experimental set up in the scattering chamber . . xi 11 12 13 15 16 2.6 Photograph showing individual components of the HF position sensi- tive gas detector. From left to right one sees the entrance window, front cylinder, middle foil, back cylinder, and exit window which also serves as a mount for AE and E silicon surface barrier detectors. A ruler provides the scale in inches. .................... 2.7 Two dimensional position spectrum of the calibration mask for one of the heavy fragment detectors. The missing points were used to identify and establish the orientation of the different detectors. ........ 2.8 Image of the full calibration mask for all the telescopes. The missing points seen in the spectra correspond to holes that were blocked in the mask in order to identify and establish the orientation of different telescopes .................................. 2.9 Sample particle identification spectrum for a heavy fragment telescope. 2.10 Particle identification in the central region of a light particle telescope as a function of p2, where p is the radial distance from the center of the detector ................................. 2.11 Upper part: particle identification spectrum without correction. Lower part: particle identification spectrum for a light particle telescope after correcting for non-uniformity of AE detector. ............. 3.1 Inclusive differential cross sections for H and He isotopes as shown for laboratory angles listed in the figure. The solid lines represent “moving source fits”. ................................ xii 18 21 22 25 27 28 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Inclusive differential cross section for selected isotopes of Lithium, Beryllium, Boron, Carbon, Nitrogen and Oxygen are shown for lab- oratory angles listed in the figure. The solid lines represent “moving source fits” . ................................ Inclusive differential cross section for 10B fragments. The solid curves describe the full “three moving source” fits and the dashed curves show the emission from a slow moving “target like” source. ......... Calculated total efficiency (upper part) and rms resolution (lower part) for the detection of p-13C pairs resulting from the decay of particle unstable 14N. The efficiency has been normalized to 1 at Em = 0.42 MeV (E‘ = 7.97 MeV) ........................... p-13C correlation function. The excitation energy in the 1“N nucleus is indicated on the top. The dashed curve indicates an estimated back- ground and solid curve is a fit described in the text. The dotted curve shows an alternate description of the background. .......... Correlation function as a function of relative energy for a-p. The solid curve is the fit to the data assuming the background designated by the dashed line. The dotted line shows an alternate background. Correlation function as a function of relative energy for 3He—d. The solid curve is the fit to the data assuming the background designated by the dashed line. The dotted line shows an alternate background. . The d-a correlation as a function of the relative energy. The fits to the resonances is shown by solid lines assuming the background shown by dashed line. The dotted curve shows as an alternative form of the background. ............................... 32 35 38 41 53 56 3.9 3.10 3.11 3.12 3.13 3.14 t-a correlation function as a function of relative energy. Location and spins of particle-unstable states in 7Li are indicated. The insert gives an expanded view showing the second maximum. The solid curves are the fits to the data assuming the background designated by the dashed line. The dotted line shows an alternate background. ......... p-6He correlation function as a function of relative energy. The exci- tation energy in 7Li is indicated on the top. Location and spin of a particle-unstable state in 7Li is shown. ................. Correlation function as a function of relative energy for 3He-a. The solid curves give a fit to the data with the background shown by the dashed lines. The dotted line shows an alternate background. p-6Li correlation function as a function of relative energy. Location and spins of a particle-unstable state in -7Be is indicated. The solid curve shows a fit to the data with the background designated by the dashed line. The dotted line shows an alternate background. ..... 8B —+7Be+p correlation function. The excitation energy in 8B is in- dicated on the top. The solid curve shows a fit to the data assuming the background depicted by the dashed line. The dotted line shows an alternate background ............................ 6Li-l-or (upper part) and 9Be+p (lower part) excitation energy spectra. Location and spins of particle-unstable states in 10B are indicated. The solid curves show the fits to the data assuming the background depicted by the dashed line. The dotted lines indicate an alternate choice for the background ............................... xiv 59 61 63 64 66 69 3.15 3.16 3.17 3.18 3.19 4.1 Excitation energy spectrum of 11C obtained from the coincidence cross section of 7Be+a. The excitation energy in 11C is indicated on the top. The solid line is a fit to the data assuming the background depicted by the the dashed line. The dotted line shows an alternate background. . Excitation energy spectrum of 13N obtained from the coincidence cross section of 12C-p. The solid line is a fit to the data assuming the back- ground depicted by the dashed line. The dotted curve shows an alter- nate background. ............................. Energy spectrum resulting from the decay of particle unstable l4N. Solid curve is a fit described in the text assuming the background shown by dashed curve. The dotted curve shows an alternate description of the background. ............................. Excitation energy spectrum of 160 obtained from the coincidence cross section of 12C+a. The solid curve describes a fit obtained by assuming the dashed line as one possible background. The dotted curve shows an alternate background .......................... Excitation energy spectrum of 180 obtained from the 14C-oz coincidence cross section. The solid curve describes a fit obtained by assuming the dashed curve as one possible background ................. The level density of 2°Ne as a function of excitation energy [Chen 88a]. The histogram gives the number of known levels whereas the solid curve shows results of level density predicted by eq (4.3). ....... XV 72 74 77 79 83 88 4.2 4.3 4.4 5.1 5.2 5.3 5.4 Element yields at 0 = 38° summed over measured energies. The dashed and solid histograms show the primary and final yields of particle stable fragments produced by the feeding calculations. Results for T¢m=2, 3, 4, 5, 6, and 8 MeV with the corresponding parameters f are given in the figure. ................................. Comparisons of measured and calculated isotopic yields at 0 = 38°. The solid histograms show final fragment distributions for feeding cal- culations at Tem=2, 3, and 4 MeV. ................... Comparisons of measured and calculated isotopic yields at 0 = 38°. The solid histograms show final fragment distributions for feeding cal- culations at Tem=5, 6, and 8 MeV. ................... Relative populations, 12,-, of different groups of particle-unstable states in 10B are plotted as a function of excitation energy. The vertical scale is normalized so that 2,,(2J;c + 1)erc = l, where the summation is restricted to the particle-stable states of 10B ............... Calculations for n,- in the limit of full spin coupling are shown as dotted, dashed, dot-dashed and solid lines for parent nuclear spins 1p: 25, 50, 75 and 100 respectively. The experimental values are same as those shown in figure 5.1. ........................... Calculations for n,- from the sequential feeding calculation Tcm = 4 MeV, fimc = 6 and (am)nc = 2.5 are shown as the solid line in the 93 94 98 figure. Experimental values for n,- are depicted by the large solid points.105 Coordinate system used to describe the a-decay of particle unstable excited states of 1CB. 0,; and (23d are the decay angles as defined in the 5.5 5.6 5.7 5.8 5.9 5.10 5.11 Relative energy spectra for the decay 1°B—> a+6Li at different values of the decay angle, 0.; . .......................... Relative energy spectra for the decay 1OB—+ a+6Li for different values of the decay angle dad. .......................... Relative energy spectra for the decay 6Li—* a+d for values of the decay angle, 9d ................................... Relative energy spectra for the decay 6Li——> a+d for values of the decay angle, aid. ................................. The 0d dependence of the decay angular distributions are shown for various excited states of 10B. The vertical scale is normalized to the average value of the distributions for each case. The dashed line shows the prediction from an isotropic decay. ................. The (pd dependence of the decay angular distributions are shown for various excited states of 10B. The vertical scale is normalized to the average value of the distributions for each case. The dashed line shows the prediction from an isotropic decay. ................. The 9d dependence of the decay angular distributions is shown for the first excited state of 10B. The vertical scale is normalized to the average value. The predictions from statistical calculations with Ip = 25, 50, 75,100 are shown by dotted, dashed, dot-dashed and solid lines respectively. ................................ xvii 109 112 113 115 116 118 6.1 6.2 6.3 6.4 6.5 6.6 Results of the least-squares analysis for four groups of fragments. The solid lines depict xi calculated for a combination of population prob- abilities and the ratios of population probabilities. The dashed lines show x3 when just the ratios of population probabilities are included. Results of least-squares analysis for a combination of all fragments. The dashed, dash—dotted and solid lines depict x3 calculated for the population probabilities, the ratios of populations probabilities, and the summation of the population probabilities and ratios of population probabilities respectively. ........................ Experimental values for 68,”, and Tapp are shown as the solid points for excited states of Li and Be isotopes. The histograms represent the results of sequential feeding calculation with an initial temperature Tem = 4 MeV ................................ Experimental values for flap}, and Tapp are shown as the solid points for the groups of excited states of B and C isotopes described in table 3.3. The histograms represent the results of sequential feeding calculation with an initial temperature Tem = 4 MeV. ............... Experimental values for [33”, and T8”, are shown as the solid points for the groups of excited states of 13N and 1“N described in table 3.4. The histograms represent the results of sequential feeding calculation with an initial temperature Tcm = 4 MeV. .................. Experimental values for flaw and Tam, are shown as the solid points for the groups of excited states of 16O and 180 described in table 3.5. The histograms represent the results of sequential feeding calculation with an initial temperature Tem = 4 MeV. .................. xviii 121 123 126 127 128 A.1 The electronics diagram for a pair of Light particle and Heavy fragment "7 telescopes .................................. 13 I xix Chapter 1 Introduction I Motivation The emission of intermediatemass-fragments [IMF ’s, 6 S A S 30] is an important decay mode of highly excited nuclear systems. This decay mode has been observed in proton-nucleus and nucleus-nucleus collisions for a broad range of incident energies [Gelb 87a, Lync 87]. Dynamical [Bert 88] and statistical [Gelb 87b] models suggest that a variety of mechanisms could be responsible for fragment production. For ex- ample, IMF emission has been related to the occurrence of adiabatic instabilities [Bert 83, Schl 87, Snep 88, Boal 89a] which may lead to the liquid-gas phase sepa- ration of highly excited nuclear matter [Lope 84b, Finn 82, Jaqa 83]. Other models which do not incorporate a phase transition have been equally successful at reproduc- ing many features of the fragment data. To distinguish between the many models of fragment emission for a given reaction, one must determine whether binary or multi- fragment breakup configurations are predominant, whether thermal approximations may be appropriate, and determine the density and excitation energy (or tempera- ture) at breakup. It may be necessary to invoke different models to describe fragment production for different reactions. For example, models which may be appropriate to describe fragment production at low incident energies may be inappropriate for the most violent nuclear collision and vice versa. For nuclear reactions at low incident energies (E /A S 10 MeV), the dominant reaction process occurs through the formation and decay of a fully equilibrated com- pound nucleus. Fragment emission from such compound nuclei has been observed [Soho 83, Soho 84] and can be described by the sequential decay mechanism that pro- ceeds through binary decay configuration [Frie 83, More 75]. The excitation energy or temperature of the compound nucleus required by these models can in principle be deduced from the slope of inclusive kinetic energy spectra of evaporated light parti- cles. As the incident energy increases to about E / A = 20 MeV, formation of a unique compound nucleus becomes unlikely; and more importantly, faster fragment produc- tion mechanisms become important. As an example, figure 1.1 shows the kinetic energy spectra of 10B nucleus emitted in a reaction of 1“N on Ag at E /A = 35 MeV. The data are presented for four angles as listed in the figure. The slopes of the exper- imental energy spectra become steeper with the increase of the emission angle. The feature of the data seems to be quite different from the solid curves which represent a parameterization that assumes 10B nuclei are emitted from an equilibrated single moving source formed by the fusion (mom,cc = can and T = TON) of 14N and Ag, and characterized by a Maxwell Boltzmann distribution with a temperature comparable to that expected for the compound nucleus. Indeed, the data can be better described by a superposition [F iel 89] of more than one moving sources [see chapter 3]. The energy integrated cross sections are strongly forward peaked even in the center-of- mass frame, further indicating that most of the fragments are emitted prior to the attainment of statistical equilibrium of the full compound nucleus. Statistical models for such non-equilibrium processes often assume the existence of a local thermal equilibrium in the vicinity of the fragment at the time of its emission [Fiel 84, Fai 82]. Information is needed to assess the validity of this approximation dza/deE (ub/sr MeV) H o. I—* T I I IIIIIII II IIIIII I IIIII1_I - natAg(14N, 10B)X '_ E/A=35MeV I I I lITI I I I TIill'I IJIIIIII I 0 Q ‘Q —~ 0 - \. \ .. \Q Q .. Q ‘9 TI _ ”Q \0 t)\ ‘ ‘ \ \ \ _, \ Q \Q 0\ \ \ Q \\ \ \ \ _E -\\ 0\ ¢ 3 o . \ \ _ T \ $\ .\\ d \ ‘ -'.- / -/ -Cl/ / / -. I 0 “III 100 200 Elab (MeV) Figure 1.1: Inclusive differential cross section for 10B at the laboratory angles shown in the figure. The solid curves are described in the text. The dashed curves correspond to fits with a “three moving source” parametrization which will be discussed in chapter 3. and to provide appropriate values for the excitation energy. Temperatures have been estimated from fits to inclusive spectra which assume thermal emission from a sub- system. Such fits usually provide higher source velocities and temperatures than would be consistent with the compound nucleus [F iel 84, F iel 86]. Temperatures extracted from such fits are likely to be misleading, however, because of their sen- sitivity to the Coulomb barrier fluctuations [Ban 85], sequential feeding from higher lying states, as well as strongly time dependent phenomena such as collective motion [Siem 79, Tsan 84, Tsan 86], and equilibration [Frie 83, Fiel 84]. Information about the intrinsic excitation of the fragmenting system at breakup may be obtained alternatively from the relative populations of ground and excited states of emitted intermediate mass fragments. Statistical models frequently assume that the intrinsic degrees of freedom are fully thermalized and the asymptotic excited states of these fragments are populated statistically with weights determined by the excitation energy or “temperature” of the emitting system [Gros 82, Gros 86, F rie 83, Rand 81, Fai 82]. If the internal excitation energy of the system is large at freeze- out, many of the fragments are emitted in excited states; if the internal excitation energy is small, few fragments are excited. The relative populations of states of a given fragment therefore provide a measure of the internal excitation energy of the fragmenting system at freezeout. The ratio 721/712 of the populations of two relatively narrow excited states of a fragment is given approximately by fl_(_2_{i1_) (-95) (1.1) n2 _ (2.12 + 1) ex Tem Here AE = E; - 5, J,- and E; are the spin and excitation energy, respectively, of the i-th state of the fragment, and Ten, is the “emission temperature” which characterizes the internal excitation energy of the system at freezeout. If the excited states are thermally populated and the feeding from sequential decay of heavier nuclei is not significant, one may in principle, determine Tem from the population of two states of a fragment via equation (1.1). The method of measuring emission temperatures from the relative populations of states have been applied to decays from particle stable excited states by 7-rays [Morr 84, Morr 85, Xu 86, Xu 89] and to decays from particle-unstable states [Poch 85a, Chit 86, Poch 87, Chen 87a, Chen 87b, Chen 87c, Fox 88, Deak 89]. Figure 1.2 sum- marizes results obtained from the measurement of relative populations of particle unstable states in Li and Be isotopes in three different reactions [Chen 88a]. Two striking features are immediately evident from the figure. First of all, the emission temperatures derived from the ratio of populations is about 4 — 5 MeV, which is significantly smaller than the temperatures (T w 12 —— 18 MeV) one extracts by fit- ting the kinetic energy spectra. This difference could be due to complications arising from collective motion [Tsan 84, Tsan 86] which influences the slopes of kinetic en- ergy spectra [Frie 89, Boal 89]. A more interesting aspect of the measurement is the fact that the emission temperatures obtained in the three reactions are very similar, even though the incident energies vary widely. If one takes the emission tempera- ture in figure 1.2 to be the temperature of the system at freezeout and allows for an adiabatic expansion dynamics, this implies that freezeout occurs at nearly constant temperature rather than constant density as assumed by certain models. One would like to know whether this is a general phenomenon which would find its manifestation in the relative populations of excited states of heavier IMF ’8. All previ- ous measurements of the emission temperature derived from the relative populations of states (with the exception of [Xu 89]) were based on comparison involving only few states of a single fragment. Such comparisons do not test in detail the internal consistency of the approach. More stringent tests of the freezeout assumption can be performed by comparing the measured population probabilities of a large number of states of a single isotope to statistical model predictions. This requires the study of o 1“o 94 MeV 197Au+ U‘°Ar, E/A= so MeV 0 1‘N L35 MeV } BB“ :0: ].u( was ] 201 LLLIILLIILIILIlLIlLflllL 2 4 6 8 1 0 1 2 APPARENT TEMPERATURE (MeV) -O- .0— 5 —a—. ‘9 lm was“) . .0. _g._ + .0— -CI- Figure 1.2: Apparent emission temperatures extracted from the relative populations of states of ‘He,5Li,5Li, and 8Be nuclei extracted for the three reactions indicated in the figure [Chen 88a]. IMF ’s which have many well resolved excited states. This dissertation research was undertaken to investigate the relative populations of the particle unstable excited states of intermediate mass fragments. For this pur- pose a position sensitive detection array was designed and the population of particle unstable excited states of intermediate mass fragments with 2 < Z < 9 were measured for the 14N + Ag reaction at E / A = 35 MeV. Detailed sequential feeding calculations were performed to assess the influence of sequential decay on the measured excited state yields. The comparison between calculated and measured excited state popula- tions revealed non-equilibrium effects inconsistent with the concept of local thermal equilibrium. II Organization This thesis is organized as follows. An overall description of the position sensitive high resolution hodoscope, the energy and position calibrations of its individual de- tector telescopes, particle identification, and other experimental details are given in chapter 2. In chapter 3, single particle inclusive spectra and two particle coincidence cross sections are presented. Methods used for extracting the relative populations of states of particle unstable nuclei are also described in this chapter. Sequential feeding from high lying states has a significant effect on the observed populations of excited states of fragments. Since it is not possible to accurately determine the amount of feeding experimentally, it has to be calculated. We have performed extensive calculations to determine the effect of feeding starting with a thermal distribution of primary fragments. The details of the statistical calculations which assess the influence of sequential feeding are presented in chapter 4. In chapter 5, tests of the freezeout assumptions using particle-unstable states of 10B nuclei are discussed in detail. The measured populations of these states differ significantly from those predicted by statistical models which include the sequential decay of heavier particle unstable nuclei. Here it is also discussed whether angular momentum effects due to rotation of the emitting system can account for the dis- crepancy between experimental data and model predictions. Experimental results for the decay angular distributions of the decays from 10B nucleus are presented in this chapter. These angular correlations suggest that rotational effects do not significantly influence the excited states populations. In chapter 6, apparent temperatures for 40 groups of particle unstable states of Li, Be, B, C, N, and O isotopes are extracted and compared to the predictions of statistical feeding calculations. Using a least squares analysis, global comparisons between experimental data and results from statistical calculations are obtained and presented. Finally, the thesis is summarized in chapter 7. Conclusions and suggestions are provided. The electronics set up and details of the efficiency calculation for the position sensitive high resolution hodoscope are given in the Appendices. Chapter 2 Experimental Setup A position sensitive high resolution hodoscope [Mura 89] was designed for measuring the populations of particle unstable states of intermediate mass fragments. Since the cross sections and the‘energies separating the excited states of these fragments are often small, the detection apparatus must have both a high efficiency and a high excitation energy resolution. Computer simulations revealed that the excitation energy resolution of the hodoscopes is limited primarily by the angular resolution of the detectors which detect the coincident daughter fragments from the particle decay of the excited nucleus. To achieve both high efficiency and resolution, we have constructed a position sensitive detection array which can be placed rather close to the target. An overall description of the various components of the detection array is given in the next section. Details of the construction and operation of the position sensitive gas detector elements are presented in the second section. The choice of filling gas and its importance for controlling aging eflects in the gas detectors are also discussed in this section. The position calibrations of the proportional counters is discussed in the third section, and the energy calibration is discussed in the fourth section. In the last section of this chapter techniques used for particle identification obtained are presented. 10 I Description of the Detection Array A schematic front view of the detection apparatus is shown in Figure 2.1. The ho- doscope consists of nine light particle telescopes (LP) and four heavy fragment tele- scopes (HF). One light particle telescope is situated at the center of the array. The four heavy fragment telescopes are situated above, below and to the left and right of the central light particle telescope. At the periphery of the array are situated eight additional light particle telescopes. The light particle and heavy fragment telescopes have solid angles of 4.5 msr and 5.7 msr, respectively. The angular separation between adjacent telescopes is 8". A cross sectional view of the array including the central light particle telescope is shown in Figure 2.2. The light particle and heavy fragment telescopes are located at different distances from the target. Expanded views of the two types of telescopes are shown in Figure 2.3. Both light particle and heavy fragment telescopes consist of two independent single wire proportional counters, providing position information along two orthogonal coordinates (here denoted by .1: and y), followed by triple element energy loss telescopes. The defining apertures for the telescopes were located between the x-y position sensitive proportional counters and the triple element telescopes and were situated at distances of 27.3 cm and 20.3 cm from the target for the light particle and heavy fragment telescopes, respectively. The staggering of the light particle and heavy fragment telescopes allowed a maximization of the detection efficiency because it minimized the dead area between telescopes normally occupied by the detector cases and mounts. For the detection of light particles, a non-planar 200 um silicon surface barrier detector of 450 mm2 surface area was used for the first element, a non-planar 5 mm thick Si(Li) of 500 mm2 surface area was used for the second element and a 10 cm ll Luiiuulmimimi 012345 Scale in cm Figure 2.1: Front view of the hodoscope showing all the nine light particle(LP) and four heavy fragment (HF) telescopes. The actual dimensions of heavy fragment de- tectors are displayed. Since they are closer to the target, however, they cover larger solid angles than suggested by this projection. 12 /:'// \ HF Figure 2.2: Schematic cross sectional view of the hodoscope. 13 X Counter Y Counter 200m Si Detector 5mm Si(Li) Detector No.1 Detector l '////////'" Q 5mm Si(Li) Detector 100nm Si Detector 75pm Si Detector Y Counter X Counter 0 1 2 3 4 5 Sceleincrn Figure 2.3: Expanded drawing showing the LP (top) and HF (bottom) telescopes. 14 thick NaI(Tl) scintillation detector was used for the third element. The 5 mm Si(Li) detectors were fabricated with a total dead layer less than 15 pm [Walt 78]. The silicon detectors for these telescopes were mounted on the front and the NaI(Tl) detectors on the rear of a mounting plate consisting of a spherical section subtending a half angle of 16°. For the detection of heavy fragments, planar 75 pm and 100 pm silicon surface barrier detectors of 300 mm2 surface area and 1.5 ‘70 thickness uniformity were used for the first and second elements; a 5 mm thick Si(Li) detector of 400 mm2 was used for the third element. The heavy fragment telescopes were positioned in front of the light particle telescopes by cylindrical rods which were bolted to the mounting plate. The experiment was performed at the National Superconducting Cyclotron Labo- ratory of Michigan State University using 14N beam at E/ A = 35 MeV from the K500 Cyclotron. The experiment was set up in the 60 inch diameter scattering chamber. A natural silver target of 0.5 mg/cm2 areal density was placed in the target ladder at the center of the chamber. The hodoscope was placed on the base table of the chamber with the center at an angle of 35° with respect to the direction of the beam and at an angle of 16° above the plane of the scattering chamber as shown in Figure 2.2. Consequently, the target was rotated by 35° to the beam axis and 16° in vertical direction so that target plane is parallel to the vertical plane of the hodoscope. A photograph of the assembled detection array is shown in Figure 2.4, and a photo- graph of the actual setup in the scattering chamber is shown in Figure 2.5. Cables connecting to the silicon and gas detectors and the urathane tubes supplying gas to the proportional counters can be seen in these figures. The preamplifiers for the gas detectors were placed in vacuum close to the detectors. The block diagram of the electronics is given in appendix A. 15 Figure 2.4: Photograph of the assembled hodoscope. The four HF telescopes are in the foreground. Eight out of nine LP telescopes can be seen at a larger distance from the target. 16 Figure 2.5: Photograph the full experimental set up in the scattering chamber 17 II Position Sensitive Gas Proportional Counters Position information for each individual telescope was obtained with two single wire gas proportional counters each providing one coordinate of a two-dimensional Carte- sian readout. A photograph of the individual components of these counters is shown in Figure 2.6. Each counter was cylindrical in shape with length of 1.2 cm and diam- eters of 3 cm and 2.3 cm for LP and HF detectors, respectively. The anode wire was situated in the middle of each counter along the circular diameter and insulated from the detector case by G-10 feedthroughs. The position along each wire was obtained by resistive charge division. This readout scheme appeared to be more linear and more space efficient than a comparable drift chamber configuration. The front and rear windows consisted of 6 pm Mylar ((C10H304)n) aluminized on the interior to provide a cathode surface. A 1.5 pm Mylar foil, aluminized on both sides, separated the a: and y position counters. The anode wire was made by 7.6 pm Nichrome wire having total resistances of approximately 600 and 400 Q for the light particle and heavy fragment telescopes, respectively. Choice of Filling Gas and the Aging Rates of Gas Counters The efficiency and long term stability of the gas counters were tested with a variety of gas mixtures and pressures. Isobutane ((CH3)2CHCH3) offered both high efficiency and high resolution, but the performance of the gas detectors with isobutane was degraded seriously after about 4 x 108 counts, with the gas gain decreasing by at least a factor of 2. Such deterioration in counter performance is caused by hydrocarbon polymerization on the electrodes [Saul 77, Vavr 86]. Detector lifetimes can be improved by adding non-polymerizing quenchers, such as isopropyl alcohol ((CH3)2CHOH) or methylal (CH2(OCH3)2), to the gas mixture. 18 Figure 2.6: Photograph showing individual components of the HF position sensitive gas detector. From left to right one sees the entrance window, front cylinder, middle foil, back cylinder, and exit window which also serves as a mount for AE and E silicon surface barrier detectors. A ruler provides the scale in inches. 19 We tested our detectors with different mixtures of isobutane and methylal. For prac- tically all mixing ratios, the counters had good efficiency - even for pure methylal. Better resistance to aging effects appeared at higher methylal concentrations. Since high methylal concentrations may adversely affect counter and gas handling system components [Vavr 86], a mixture of 20% methylal, 80% isobutane was used in the actual experiments. During experiments, the heavy fragment telescope was operated at a pressure of 40 torr and a voltage of 900 V, while the light particle telescope was operated at 100 torr and 1250 V. At these pressures, the detection efficiencies for light particles and heavy fragments were 100% over the energy range of interest (5 MeV S E/ A _<_ 40 MeV). These high operating voltages correspond to the upper portions of the proportional regime approaching the domain of limited proportionality. A constant gas flow rate was maintained for all telescopes such that 20 % of the counter gas was replaced every minute. III Position Calibrations of the Gas Counters The position spectra of the gas counters were calibrated with the 5.805 and 5.763 MeV 0: particles from a 1 mm diameter 244Cm source which was placed at the tar- get location. A calibration mask with holes of 1 mm diameter, separated by 1.5 mm, was placed in front of the hodoscope at a distance of 16.5 cm from the target center. Because of the higher energy loss of low energy a—particles, the operating voltage of the light particle telescopes was lowered to 1150 Volts during the calibra- tion. Non—linearities of the 23-3; position spectrum were corrected by the empirical transformations; X = a0 + ale + agYm + ang, + a4Xm Ym + a5Y,,21 20 + anS, + 07X?" Ym + angYflf + 091/3, (2.1) Y 2 b0 + I’I‘Xm + b2Ym 'I' bBszn + b4XmYm 'I' bSYr: + bGXf; + b-,X,3,Ym + bsmef, + b91153. (2.2) Here, Xm and Ym denote positions directly obtained by the charge division method, and X and Y represent the actual positions. Coefficients a,- and b,- were determined via fitting the position spectrum measured with the mask. Figure 2.7 shows the two dimensional calibration spectrum for a heavy fragment telescope after correction for non-linearities. This spectrum and the spectra of other heavy fragment telescopes are consistent with a position resolution of 0.33 i 0.02 mm F WHM. The spectra for light particle telescopes are consistent with a slightly worse resolution of 0.50 :l: 0.01 mm FWHM. The position resolution was limited primarily by the preamplifier noise; it scales inversely with the signal height and therefore inversely with the energy loss in the detector gas. An image of the full calibration mask is shown in figure 2.8. The missing points seen in the spectra correspond to holes that were blocked in the mask in order to identify and establish the orientation of different telescopes. The proportional counters proved to be rather sensitive to electrons and soft pho- tons produced by the beam in the target. The corresponding background could be reduced to a tolerable level by installing 5 mg/cm2 Au foils in front of the gas detec- tors and adding a magnetic electron suppression system midway between the target and the detector array. IV Energy Calibrations Computer simulations described in the Appendix B, indicate that accurate energy calibrations of the detector telescopes are much more critical than good energy reso- lutions to the achievement of optimal excitation energy resolution. Before and after 21 15:— Emé — - a as; ,x assesses; ; - isséeéeeé - t eaéiéeéss 1 - ééfifiiii T’ ii as . - ass - Y (mm) - o L l l l i L L L l l l I l l I 1 LL . 0 5 10 15 X (mm) Figure 2.7: Two dimensional position spectrum of the calibration mask for one of the heavy fragment detectors. The missing points were used to identify and establish the orientation of the different detectors. 22 Figure 2.8: Image of the full calibration mask for all the telescopes. The missing points seen in the spectra correspond to holes that were blocked in the mask in order to identify and establish the orientation of difl'erent telescopes. 23 the experiment the detectors were calibrated at low energies with 2’“Am and 21”Po (1 sources. These calibrations were extrapolated to energies of several hundred MeV by injecting a signal from a precision BNC pulser into the input stage of the preampli- fiers. In this fashion, relative calibrations of all the silicon detectors were established to an estimated accuracy of about 0.5%. Dead layers of the silicon detectors were measured with an 241Am source by rotating the detectors with respect to the direction of the incident 0 particles. Most light particles originating from the decay of nuclei with 10 S Z S 16 were stopped in the second (5mm) element of the light particle telescope. Light particles from the decay of lighter nuclei (A S 9) frequently penetrated the 5 mm detector and stopped in the NaI(Tl) detector. The NaI(Tl) detectors have energy resolutions of about 1-2%, adequate to resolve the states of the lighter nuclei. Calibrations for the NaI(Tl) detectors were obtained by converting the AE information from the 5 mm Si(Li) detectors to corresponding energies. To assess the accuracy of this conversion, the thicknesses of the 5mm Si(Li) detectors were measured by the method of X—ray attenuation. Calibrations were cross checked by the measurement of energies of recoil protons backscattered from a polypropylene target by a 490 MeV l“N beam. The energy calibrations of NaI(Tl) detectors are estimated to be accurate to within 5%. Gain shifts of the NaI(Tl) detectors were stabilized by using the AE information produced by Si(Li) detector [Poch 87]. V Particle Identification For ions with E/ A Z 5 MeV, the heavy fragment energy loss telescopes, constructed with planar silicon surface barrier detectors, provided accurate charge and mass iden- 24 tification via the empirical relationship [Goul 75], R 0‘ Eb/[qufl], (23) where E, R, [If and qeg denote the energy, range, mass and effective charge state of the fragment, respectively, and b is an adjustable constant with a typical value of about 1.7. For a planar AE detector of thickness T and a stopping E detector, one obtains from Eq. (2.3) Mr]:fir o< ( (E + AE)b — Eb ) /T. (2.4) Following Shimoda et. al. [Shim 79] we have adopted a number of empirical im- provements. Particle identification (PID) is obtained, instead, using = ln(b AE) + (b — 1) ln(E + cAE) — b 1n(300), (2.5) AE[MeV] b = 1.825—0.18 , Tlflml c = 0.5. Figure 2.9 shows the particle identification achieved for particles which stop in the second (100 ,um) element of the telescope. Isotopic resolution is achieved for all elements displayed. Similarly, good resolution is obtained for heavy fragments which stop in the third element (5 mm Si(Li)) of the telescope. The PID resolutions were also adequate to separate Helium isotopes (not shown). Experimental data for the decay of 5Li, 6Li, 7Li, 8Be were also obtained by analyzing helium ions stopped in the heavy ion telescopes. Because of cost-efficiency reasons, non planar fully depleted detectors of 200 mm thickness were used as first elements of the light particle telescopes. The detec- tors were fabricated by a technique producing convex shaped Si wafers with non- uniformities of up to 25%. If the variation of detector thickness is a function only of 25 4 10 I l r T I y I r g I ' f I I 1 : L1 Be B c N 3 O . 103 m l .1 E l a 2 l « d d 6 i ‘ \ 1o2 1 In 1 a -l g : O 'i o 'l 101 100 , 1 L l l ,, 7 P .1 o 200 400 600 PID Figure 2.9: Sample particle identification spectrum for a heavy fragment telescope. 26 the radial distance from the center of detector p (mm), we can replace T in Eq. (2.4) by T(p) = Tof(p), (f(p) S 1) (2-6) where To is the thickness of the detector at the center. Good particle identification can still be achieved by correcting for the thickness variation using the position in- formation provided by the gas detectors. Taking into account the dependence of the PID on thickness, one obtains PID = ln(bAE)+(b—1)1n(E+cAE)—bln(300)—ln(f(p)), (2.7) AE [MeV] b = 1825—018 , Talflml c = 0.65. Figure 2.10 shows the particle identification in the central region of a typical detector as a function of p2, obtained with f(p) = 1. The PID values decrease linearly with ,02 by 28% from the center to the periphery of the active area. All the first elements of the nine light particle telescopes displayed similar thickness variations ranging from 25% to 30%. We adopted the functional form f(p) = exp(-Ap2) (2-8) with A = 5.3 x 10'3 mm”. This improved the particle identification in Figure 2.11. Moderately clean isotope resolution was obtained for helium isotopes. 27 6O IIIfierFIlIIrTTITo-IIIT-r AAAAAAAA d o ooooooooo coo-W A A AAA“A AAA AAA A 0...... o no... one... 00...... AA .1 .0...- “coo-cocooooooooomu ‘; “““““““““ ‘ .0-...sauna-ouoooooooooeooooo“ ...... : I...- O..- C. A AA AAAAAAAAAAAAAAAAAAAAAAAAAA O C... C... .- C.“-. Q . . .' ' I.’ ' I." "2'"??? .............................. H OOOOOOOOOOO C C C O O O O 0.. C. 0.... 0...... O. I 0" trim”“'““':.:.::.:.::::.: ..... '.: ........ :°. ° W...“ l 30 5mm ......... P .. P. OOOOOOOOOOOOOOOOOO .0 r ". ' I '. '2 2': IfZIZ'I'IZZ'IZIIZIIIZIZI?I: P . ' ° . . " I ' ' 3' 33:33:22: 2':- 20 — ' ° ° — I- 4 l L l L l l 1 1L 1 1 l l L l l I 1 L l l l J‘ Figure 2.10: Particle identification in the central region of a light particle telescope as a function of p’, where p is the radial distance from the center of the detector. 28 105 fiIIMTYMTTIIITIIVIrT‘IrTI—r Without correction p at 1.04 d t. 103 "LJ .21.) With correction Counts / Channel '5 N 104 103 102 ‘ .HlnllrllLLLAL A] o 450 100 150 200 250 PID Figure 2.11: Upper part: particle identification spectrum without correction.- Lower part: particle identification spectrum for a light particle telescope after correcting for non-uniformity of AE detector. Chapter 3 Data Analysis and Reduction In this chapter, the data for single particle inclusive spectra and two particle coinci- dence cross section will be presented. In the first section we discuss the data for single particle kinetic energy spectra, and fits to the data using a “moving source” parame- terization. The second section of this chapter deals with the two particle coincidence cross section. The detection and resolution of the hodoscope will be described as well as the details of fitting the resonance curves using compound nucleus R-matrix theory. We next present the data for the particle decay of excited lithium, beryllium, boron, carbon, nitrogen and oxygen isotopes. There we describe details relevant to the extraction of the relative populations of particle unstable states in these nuclei. 29 30 I Single Particle Inclusive Cross Sections Single particle inclusive energy spectra for hydrogen and helium isotopes are shown in figure 3.1. Kinetic energy spectra for selected isotopes of lithium, beryllium, boron, carbon, nitrogen, and oxygen are given in figure 3.2. All the spectra in figures 3.1 and 3.2 are Maxwellian in shape, display maximum at energies close to the exit channel Coulomb barrier and then decrease exponentially at higher energies. Single particle inclusive energy spectra have been measured for 1“N induced reactions on Ag over a wide angular range and for a variety of incident energies [Fiel 89]. These measurements demonstrate that the emission from the equilibrated compound nucleus makes only a small contribution to the energy spectra measured at forward angles. For the present data set, this can be illustrated‘by fitting the energy spectra using a “moving source” parameterization given by (1:122:13 = ; Nq/E - Uc exp{—[E — Uc + E, — 2(/E,-(E — Uc) cosO]/T,-} (3,1) where, N,- is a normalization constant, U6 is the kinetic energy gained by the Coulomb repulsion from the residue assumed for simplicity to be stationary in the laboratory system, and T,- is the kinetic temperature parameter of the ith source. E,- = §mv§, where m is the mass of the emitted particle and v, is the velocity of the ith source in the laboratory system. Fits to the data are shown by the solid lines in figures 3.1 and 3.2, obtained with the use of three “moving sources”, and the parameter values for the fits are listed in Table 3.1. In calculations of the efficiency for detecting decay of the unstable fragments, the angular distributions of the excited fragments are assumed to be the same as that for the corresponding stable nucleus. Therefore, accurate fits to the single particle kinetic energy spectra are required for the extraction of the relative populations of the excited states of IMF ’3. These fits have also been used in the simulations of the backgrounds 31 “atAgr(”N, X), E/A= 35 MeV H O H '11 dza/dOdE (,ub/sr MeV) 3... 5;. ....., H... ....., 1..., . Tr‘ o 100 200 o 100 200 Elab (MeV) Figure 3.1: Inclusive differential cross sections for H and He isotopes as shown for laboratory angles listed in the figure. The solid lines represent “moving source fits”. 32 natis.g(1‘*N,X), E/A = 35 MeV ° 30.6° ° 358° ‘3 41.7° 45.8° 10-1 10“2 102 101 10° 10‘1 10"2 dzo/deE (pb/sr MeV) 0 100 200 O 100 200 i 0 100 200 0 100 200300 Elab (MeV) Figure 3.2: Inclusive differential cross section for selected isotopes of Lithium, Beryl- lium, Boron, Carbon, Nitrogen and Oxygen are shown for laboratory angles listed in the figure. The solid lines represent “moving source fits”. 33 Table 3.1: Source parameters of three moving-source fits. The Coulomb repulsion energies UC and the temperature parameters T,- are given in units of MeV, and the normalization constants N, are given in units of pb/(sr MeV3/2). Uc T1 Ul/C N1 T2 UQ/C N2 T3 v3/c N3 meat: 6.23 8.56 8.33 13.08 12.82 15.54 16.00 15.81 20.66 20.24 30.41 29.86 40.97 40.67 40.15 46.15 45.87 50.74 50.25 3.46 4.04 5.49 5.35 5.38 6.14 9.17 19.57 8.97 10.24 9.09 9.09 7.72 7.38 9.53 10.43 10.43 12.22 12.22 0.036 0.03 0.035 0.04 0.045 0.043 0.064 0.023 0.06 0.055 0.053 0.053 0.054 0.053 0.051 0.061 0.061 0.057 0.057 33490 4372 1421 530.5 11060 96.32 74.54 79.10 14.98 32.98 0.63 24.62 7.12 33.17 7.31 0.57 6.21 4.25 1.43 9.27 12.07 12.24 12.80 12.91 14.49 16.73 4.64 18.77 17.97 18.90 18.90 16.82 16.28 14.97 16.69 16.69 3.42 3.42 0.168 0.12 0.14 0.158 0.138 0.116 0.114 0.089 0.107 0.114 0.113 0.113 0.105 0.091 0.101 0.118 0.118 0.091 0.091 618 164 892.7 411.1 2101 28.82 50.96 121.2 14.87 12.53 0.40 9.33 2.46 7.15 1.25 0.13 1.41 31.08 19.36 3.98 7.30 6.11 4.96 6.43 9.56 11.22 12.17 11.08 10.89 11.33 11.33 12.02 13.75 16.11 9.88 9.88 12.34 12.34 0.27 0.223 0.242 0.26 0.232 0.193 0.207 0.139 0.198 0.200 0.207 0.207 0.193 0.176 0.155 0.194 0.194 0.114 0.114 4159 1862 804.5 1601 4289 32.16 106.2 83.81 57.83 41.32 2.54 55.98 13.62 12.57 0.84 2.06 11.93 1.24 0.06 34 to the coincidence yields from particles which are emitted independently and are not the decay products of a heavier particle unstable IMF. Details of the calculation of the efficiency function and the coincidence background are given in the next section. Although the fragment kinetic energy spectra are rather well described by the superposition of the contributions from three sources, the range of angles covered in this experiment was not sufficient to unambiguously establish the parameters of these sources. Indeed, the representation of these spectra by the superposition of an equilibrium plus two non equilibrium sources is an approximation which we justify mainly by the accuracy of our fits. As an illustration of the decomposition into equilibrium and non equilibrium sources imposed by our fits, we show the measured energy spectrum for 10B fragments as the solid points in figure 3.3 along with the full three moving source fit (solid line) and the best fit assumptions for the equilibrium fit (dashed line). Consistent with [Fiel 89], these fits suggest that equilibrium emission plays only a minor role in the emission of the more energetic fragments. The precise magnitude of the equilibrium contribution, however, can not be established without additional measurements at backward angles. 35 ““Ag(“N,1°B)x, E/A=35 MeV ....,....r..--,. 102 5- —] A I I > . . Q) 2 101 ,- _: I" ' : Q E : .0 ' o ' 3} O 0 \ ‘ m 10 =- o 41.7° \‘ce‘ I. e -: g : I 45.s° \\\\\ ,. 3 I'd : \\\\\\ + ., \ " \\\\\ + I - Nb —1 ‘\\\\ “d 10 :— \\\\\ T _2‘ I ‘\\\ 2'. : \\\\ I 2 . \\\\ . \\\\ . \\\\\ .. 10—2 1 I l 1 l 1 l 1\\-\L\l 1 l L ll 0 H O O N O O (D O C Figure 3.3: Inclusive differential cross section for 10B fragments. The solid curves describe the full “three moving source” fits and the dashed curves show the emission from a slow moving “target like” source. 36 II Two Particle Coincidence Cross Sections A Detection Efficiency and Resolution Products from the decay of particle unstable nuclei are detected as coincident par- ticles. The energies of the coincident particles are combined to obtain the relative energy and, by accumulating all the measured events, the relative energy spectrum 14°41'25”) is obtained, E“ me, being the measured excitation energy. This total excita- tion energy spectrum has contributions from the following two parts : Kermit...) = chEr'La) + YbaCk(E:nea) (32) where Yc is the yield from the decay of the particle unstable nucleus, and Yback is the background yield due to coincidences which do not proceed through the decay of the particle unstable nucleus being investigated. The coincidence yield, YC can be related to the normalized excitation energy spec- trum |dn( E ‘) / dE" la in the rest frame of the unstable fragment for decay into channel 0 by the equation, dn(E‘) dE‘ (3.3) mea mea YC(E‘ )=/dE“e(E‘,E‘ ) C where 6(E‘,E;m) is called the efficiency function, E‘ being the actual excitation energy. The decay yield |dn(E‘)/dE“|c is normalized so that f°° dE"|dn(E")/dE"'|C is the total yield into channel c divided by the total yield of the corresponding particle- stable nucleus. A detailed description of the decay yield will be given in the next subsection. The efficiency function is calculated for the complete detector geometry of the hodoscope by taking into account the position and energy resolutions of the tele- scopes. It also includes the target beam spot size, multiple scattering and energy loss in the target and the gas detector windows. This calculation assumes that the 37 particle unstable nucleus decays isotropically in its rest frame, and the energy and angular distributions of the excited nucleus are identical to those measured for the corresponding particle—stable nucleus. Details of the efficiency calculation is given in Appendix B. As an example, let us consider calculations for the decay 14N —»13C+p for 14N induced reactions on ““Ag at E/ A = 35 MeV. In these calculations, the energy spectra and angular distributions for particle unstable l4N nuclei are assumed to be the same as those measured for stable l4N nuclei, shown in figure 3.2. The geometry and resolution of the hodoscope elements, and target and detector foil thicknesses were taken from conditions encountered during the experiment. Results of calculations for the total efficiency em) = / dE.:... 453213;...) (3.4) and the root mean square resolution tot e u a s- : 2 1/2 UE‘ = ( demea 6(El ’Emea) (Emea — E ) ) (3'5) are shown in Figure 3.4. The total efficiency (shown in upper part of the figure) is normalized to 1 at the relative kinetic energy of 0.42 MeV, which corresponds to the 2'(E" = 7.97 MeV) excited state in 1“N. The resolution shown in Figure 3.4 is mainly limited by the position resolution of the individual telescopes. The position resolutions of the gas counters for the LP and HF telescopes were adjusted for getting optimum fits to the coincidence yields. We have used position information from the gas detectors in the expression for PID (see equation 2.8) in order to achieve good particle identification for H and He isotopes using the LP telescopes. The regions close to the periphery of the silicon detectors, where only poor isotopic resolution could be attained, were avoided by utilizing software gates on position information. The efficiency function, turned out to be somewhat sensitive 38 2.0* vvvlvrnln..,...rT....l-fi. natAg(14N’p 13C)X E/A= 35 MeV 6“.=38.4° q 1 1 1 5 - O 4 1 1.0 ' 0.5 ' A Efficiency (arb. units) ALL.lALLLIALLLIALIAILLLLILLLI . V'r'l'VVVlt'V'IfT'T'rr" 71" 100 OLannanALLlLLnLILAALIAALLLLLAL 0 1 2 3 4 5 6 Era (MeV) Figure 3.4: Calculated total efficiency (upper part) and rms resolution (lower part) for the detection of p-“C pairs resulting from the decay of particle unstable 1“N. The efficiency has been normalized to 1 at E“; = 0.42 MeV (E‘ = 7.97 MeV). 39 to the position resolutions of the gas counters. The uncertainties in the efficiency calculations due to the uncertainties in the position resolution of the gas counters were therefore, estimated and included in establishing the uncertainties in the excited state yields. The background yield, mek(E' ) which appears in equation (3.2) can be written men in an approximate form as Yback : C'12 0102“ + Rback(Erel)la (3'6) where C12 is a normalization constant, 01 and 02 are the single particle inclusive cross sections for particles 1,2 interpolated by moving source fits as discussed in the last section, End is the relative energy of the two particles, and [1 + Rbuk(Ere1)] is the background correlation function. The background correlation function is assumed to vanish for Erel —> 0 and to go to unity at large E"; where final state interactions can be neglected. To get an approximate description of the background, we have parameterized the background correlation function as 1+ Rback(Erel) = 1— exp{—(E' — Eb)/Ab} . (3.7) where Eb is the threshold energy for an excited nucleus to decay by a given decay channel and the fit parameter Ab governs the width of the minimum at Eb. The accuracy of the above approximation can be easily assessed by constructing the total correlation function, [1 + Rtot(E,.el)], defined by Kot(Erel) = C'12 0102 [1 'i' Rtot(Erel)] (38) and investigating the correlation function at relative energies for which no particle unstable states exist, and consequently at those energies R,0,(E,e1) = Rback(E,¢1). The experimental correlation function [1 + R,0,(E,._.1)] is obtained by summing both sides of the above equation over all values of energies of the two particles corresponding to 40 a fixed relative energy Erel and choosing Cu such that the total correlation function is unity at large relative energies. As an example, Figure 3.5 shows the experimental total p-13C correlation func- tion. Between 7.55 S E" S 10.27 MeV, 16 states decay only by proton emission. The distinct structures observed at E“ = 7.97, 8.49, "~90, z9.4 and $810.1 MeV corre- spond to groups of excited states with J 2 2; additional states in this region with J = 0 are not strongly populated. Consistent with Equation (3.8) the correlation function is very close to unity between the peaks and at large relative energies where the background correlation is dominant. It also decreases to zero for small Era. The shape of the background correlation function resulting from the above parameteri- zation (equation 3.7) is shown by the dashed lines in the figure. From this shape, Rback(Erel) may be determined and the background yield can be subtracted from the total yield. The sensitivity of the excited state yield to uncertainties in the back- ground subtraction may be explored by making different choices for the background. One such choice is depicted by the dotted lines in the figure. Details of the calcula- tions for the correlation functions and the backgrounds are provided in the appendix B. 41 Figure 3.5: p-“C correlation function. The excitation energy in the 1‘N nucleus is indicated on the top. The dashed curve indicates an estimated background and solid curve is a fit described in the text. The dotted curve shows an alternate description of the background. 42 B Fits to the Resonances : R-matrix theory To describe the experimental yield for particle unstable nuclei resulting from two par- ticle coincidence cross sections, one needs the excitation energy spectrum |dn(E')/dE‘ lc. For this purpose, one needs to be able to describe the population of an excited state which can have a total width that is comparable to the temperature of the ensemble of such fragments. To find this expression we must consider the modifications of the phase space density of the decay products due to their mutual interactions. To illustrate these modifications, we consider the interactions of two spinless non-identical particle. The density of two particle states containing one of each of the decay products can be written as was) = mu?) - M) (3.9) -o where pT(P) is the density of states associated with the motion of the center-of-mass of these particles, and p,((j’) is the density of states for the relative motion of the two particles. Here 131, [9'2 are the momenta of the two detected particles, cf is the relative momentum and P is the total momentum. The density of states for center of mass motion pT(13) is not affected by the mutual interaction of the two decay products. We need consider only modifications of the density of states for the relative motion p,.(cj'). If one considers the number of states in a box of volume V about the center of mass of the two particles and requires the relative wave function to vanish at the boundaries of the box, one can obtain ME) = pom + AM) (3.10) where pom = .5— (3.11) 43 is the density of states for non-interacting spinless particles, and Apm = $2321 + 1);!!- (3.12) describes the modification of the phase density due to the interactions between the two particles [Huan 63]. In this expression, 6; is the scattering phase shift for the partial wave with orbital angular momentum 1. Additional quantum numbers are in general associated with the phase shifts. Each of these phase shifts can contribute to Ap. If one assigns an index i to each phase shift, one can generalize eq. (3.12) for particles with non-zero spins : 06,- 2(2J;+1)5;. (3.13) >IIH AIM) = If the two particles are in contact with a thermal reservoir with a tempera- ture T, the phase space will be populated in accordance with the Boltzmann fac- tor exp(—E" / T). For the phase space of relative motion, one expects a probability distribution which has the form [Land 80] : .. q2 _, q? .. qz pr(<1)exp( mil-Ii.- ) = po(q)exp( v2? ) +Ap(q)exp( -2#—T ), (3-14) where p is the reduced mass of the two body decay channel. The latter term in eq. (3.14) arises from the interactions between the two fragments. If one isolates the portion 6,”, of the total phase shift 6,- which corresponds to the modifications of the two particle phase space due to long lived resonant interactions between the two fragments, one obtains an expression for the population of resonant excited states. For a system with a single open channel, the expression for the decay spectrum of the excited nucleus becomes a6i,res n- : 1 dnfE l = stable Zexp< —-1; ) ;r-(2J, -+- 1) 8E" , (3.15) dE“ where Cstable is a constant fixed by the requirement that f°° dE"|dn(E"‘)/dE"‘|c is C the total decay yield into channel c divided by the total yield for the corresponding 44 particle-stable nucleus. Practical details of the evaluation of Gable are given in the discussion of the relative populations later in this section. To proceed further, we need an expression for 86,-,"8/613“. We must also consider the possibility that more than one decay channel may be open for the excited states we encounter. Most of the phase shifts for the formation of particle unstable light nuclei are already experimentally known. Many are parameterized using the R-matrix theory of nuclear reactions [Lane 58]. We now recapitulate the essential elements of this theory. Central to this theory is the R-matrix, RW: which is the multichannel analog to the logarithmic derivative of the radial wave function 112”. One can relate the external solutions of the Schrodinger equation to the internal solutions using the R-matrix via the equation. (Myau)'1/2 z/2V(a,,) = Z(M,,:a,,:)‘1/2 Ru”! ay’%¢V’(r‘/’) J (3.16) where My is the reduced mass of the decay channel, a, is the matching radius (channel radius) which is usually channel dependent, and 212,, is the radial wave function for that part of the total wave function which is in channel 11. The symbol V is a shorthand which denotes the many quantum numbers (e.g., c, l, m, channel spins etc.) required to completely specify the decay channel. The index c designates two specific daughter isotopes produced by the decay of the particle unstable nucleus. Due to the existence of particle unbound states in the fragment, RW: is often expressed as a sum of poles : 7M7»! W: —— 3.17 . 2w < > corresponding to resonances at E“ z E,\. The terms 7)”, are the reduced widths which contain information about coupling of the resonance A to the decay channel V. In principle, the locations of the poles EA correspond to the energy eigenvalues of eigenstates z/i,\,,(r) which satisfy Schrodinger equation at r < a” in addition to a 45 boundary condition = BV¢»A,,(a,,), (3.18) r=ay d .3; [ W) 1 at the channel radius a,,, with the boundary value 3,. Within the R-matrix theory, the scattering matrix S,,,,; is given by a matrix ex- pression s = (ka)1/20-1[1 — R(L — B)]‘1[1 — R(L" — 13)]1 (ka)‘1/2 (3.19) (kafi = (mats... 0‘1 = 036..., L = LAM, B = BV6VU'3 and I = I,,6,,,,: (3.20) are matrix representations for channel dependent quantities. Here, 16,, is the channel wave number, 0,, and 1,, are the outgoing and incoming solutions of the radial equation for channel V, and _ fl _ UL)" L,, — a,, 0,, — a,,(1u)* (3.21) is the corresponding logarithmic derivative. Values for a,, and 8,, are not apriory specified by the R-matrix theory. In practice, for charged particle decay channels, a,, is often chosen sufficiently large that the outgoing and incoming radial wave functions 0,, and 1,, can be accurately approximated by 1,, = (Gu—iFu)exp(iw,,) (3.22) ~ 0,, = (Gu+iF,,)exp(—iw,,) (3.23) 46 where F and G are the regular and irregular Coulomb wave functions and 02,, is the reduced Coulomb phase shift. The choices for B,, and a,, are not by themselves important, but they do define a convention which must be constantly followed because 8,, and a,, are coupled to the values of E A and 7;, obtained from fitting the equation (3.19) to low energy scattering data. As a consequence, the parameters of a resonance are not completely specified by EA and 7” alone, and one must consistently follow the conventions for B,, and a,, when fitting R—matrix expression to the experimental data. Little can be gained by further discussing the R-matrix theory in its full generality. One must now choose limiting cases which are relevant to this dissertation. One-level approximation When E" is near an isolated resonance at energy E A, the R-matrix is often approxi— mated by RVU’ : RSV“ ”I'M; ° 7111' = __ 3.24 E; - E“ ( ) where the pole reflects the influence of the resonance at E" = EA. Substituting equation (3.24) into (3.19), the one-level formula for the S-matrix becomes i(I‘AuI‘/\u’)1/2 . 3.25 EA+A,\—E'—%I‘,\ ( ) 8111/ = eXP[i(0~'u + wu’ _ (by _ 4512’“ 6qu ‘i' where (b is the hard sphere phase shift. Here 6W: is the Kronecker delta function. The width FM, and the energy shift A)“, can be expressed in terms of the reduced width 2 ’7» as follows 1“,, = 2P,~,§,,, r, = 21“,, (3.26) A, = = —— 2(5), — B,)7§,,. (3.27) 47 Here I‘), is the total width of the resonance. P,, is called the penetration factor which is related to the probability that the particles in the exit channel escape from the interaction region. Mathematically P,, and 5,, can be expressed in terms of F and G, the regular and irregular solutions of the radial wave equation in the external region and their derivatives, all evaluated at channel radius a,,. One obtains Pu = PAL-zirmzu and u = PA;1(8AV/apllr=au (3°28) where A,, = F3 + 0,2,, and p = kr. The inclusion of the factor A, in equation (3.25) has the consequence that the level energy E A is different from the resonance energy Em of the level /\ and is given by [Bark 72] : E) = Ere, + A). (3.29) From equation (3.25) it is clear that the S-matrix has off-diagonal terms which mix channels V and V’. To obtain the modifications of the phase space density due to unbound resonances, the S-matrix must be diagonalized. In the diagonal representa- tion, the S-matrix in the resonant channel becomes EA+AA—E'+%FA = . . 3.30 EA+Ai—E’—%I‘A ( ) S = exp(2i6,\,res) Using eq. (3.30) in eq. 3.15, one can obtain a thermal expression for the excitation energy distribution of this isolated level : dnA,tot(E.) _ E. (QJA +1) (IE. — Cstable eXP( - T ) 71' PA/2 [ CIA; EA-I-AA—E” (IPA _ 3.31 X(EA+Ai—E')2+%I‘i dE‘+ I‘A dE‘ ( ) This state will decay to all available channels V. The branching ratio which governs the decay to the original channel V is equal to the absolute value of the coefficient which 48 describes the contribution to the resonant channel from the Vth original channel. Using the S matrix of the eq. (3.30) we obtain the branching ratio BR,, for the Vth channel F 1. BR, = A . (3.32) FA For the excited states considered in this dissertation, a given pair of final decay products, 0, are emitted with a unique partial wave 1,, and channel spin zc. Thus the index V becomes redundant and the decay spectrum for the channel c becomes dn,(E-) = N. exp( _E" ) (2.11 +1) dE“ c T 7" FAc/2 [ LIA) EA+A,\—E' (IF), X (EA + AA — E’)“ + iri 1— (113‘ + PA dE“ ”3.33) where the constant Cmble has been replaced by another constant N A which depends specifically on the level A. In the absence of sequential feeding from heavier particle unstable nuclei, the value of N A should be equal Cmble for all states. Values for N A for individual excited states can be assessed from fits to the experimental data, and compared to the prediction of statistical model calculations. Further details of these fitting procedures are given below in the discussion of the experimental extraction of the relative populations of excited states. In many cases the resonance parameters I", and A1 depend only weakly on the energy, then a Breit-Wigner description of the S-matrix is frequently used. In this case, F), and AA are constants, and |dn/dE"'|c becomes dn)((E") dE* (3.34) =N ex (_E") (2J,+1) r,/2 I}; A p T 7r (Em-E‘)2+§P§ r,‘ C where Eres is the resonance energy for the level A. 49 The Two-level approximation The analysis of overlapping levels with the same spins and parities is more compli- cated. For the purpose of this dissertation, however, it is only necessary to obtain the appropriate expressions for the case of two overlapping levels and two open decay channels. The R-matrix for this case is given by ([Lane 58] page 329) : _ 71V71V’ 72V72V' RW, _ _El _ E + —E2 _ E' (3.35) The relationship between R-matrix and S-matrix given in equation (3.19) can be written in the form S = 9W0 (3.36) where n = 11/1’0-1/2 (3.37) and the components of the matrix W in the case of two levels with two open channels are W11 = 1 + 2iP,[R11— L3(R11R22 — Rf2)]d‘l, (3.38) W22 2 1 + 2iP2[R22 — L?(R11R22 — Rf2)]d’1, (3.39) W12 = W2, = 23P,‘/°R,,P,‘/°d-1, (3.40) where d = (1‘ RuLl’Xl — 322143) ‘ LiRing (3-41) with L3 = L,, — 8,, = 5,, + iP,, — B,,. (3.42) To find stationary wave solutions in both the channels, we need to solve the eigenvalue equation (a = we (3.43) 50 for the eigenvectors 5' corresponding to the eigenvalues C. This yields the two possible eigenvalues : 1 c1 = 5{ W.. + W22 — \/(W.. + W22)? — 4(W11W22 — W12W21) } (3.44) 1 C2 = -2—{ W11+ W22 + ,/(W.. + W22)? — 4(W11W22 — W12W21) } (3.45) for the two levels considered. By substituting these eigenvalues in equation (3.43), the eigenvectors ii are obtained. The branching ratios for the decay from one of the levels A by a channel V(=l,2) are then (BR), _ = _ W?“ (3.46) ““1 IQA — I’Vul2 + III/'12]2 ICA — I’Vul2 ( ),\,u_2 I“ _ Wlliz + |W12|2 ( ) The two-level decay spectrum for the decay into channel V is given by dn(E"‘) _ 13* (2J+1) 1dr, iii } dE' — Nexp( — T ) _‘27ri { C1 dE‘ C2 dE*(BR)2” . (3.48) (BR)1V + V where J is the spin of the levels considered. We use a single normalization constant N for this case because the experimental data do not allow a separate determination of the emission temperature T and two normalizations. Evaluation of the Population Probability In general, the decay spectrum consists of a sum of contributions from the various levels : dn(E") dnA(E"') dE" dE’ By summing the decays from one of the levels in eq. (3.49) over the open decay = Z (3.49) C A C channels, one obtains the excitation energy distribution of the level considered : dTl,\(EI) dnA.tot(E*) _ Z: dE“ . (3.50) dE‘ C C 51 If the branching ratios to the various channels are known, a measurement of a single dnA,tot(E‘) dE' Following [Naya 89], one can define a “population probability”, n )1, for this level. decay channel is sufficient to evaluate By integrating over excitation energy 1 (In; tot(E*) = —— dE“ —’————. . "A (2.1, + 1) / dE‘ (3 51) The spin degeneracy factor (2J1 + 1) in the denominator of the eq. (3.51) reflects an unfortunate choice of notation adopted in [N aya 89] which must be kept in mind during subsequent discussion of the measured and calculated population probabilities. For the majority of the excitation energy spectra considered in this dissertation, the excited states are relatively narrow and the Boltzmann factor exp(—E" / T) varies little over the resonance line shape. Then the Boltzmann factor can be approximated by exp(-E,es / T), and taken out of the integral. The population probability becomes n), = N, exp(—E‘/T), (3.52) . . . dn(E‘) . . . and m the limit that E can be approx1mated by a set of Brett Wigner resonances, one obtains, dn(E") (2J1 +1) FA/Z PAC __ = —, 3.53 dB“ . 2,: * r (E... - E')2 + m r. ( ) and n; can be evaluated directly. Regardless of the form of the fitting expression, YC(E;;M) is obtained by folding drift) against the efficiency function 6(E‘, Egg) according to eq. (3.3). Because of the manner in which parameterization of the single particle inclusive spectra are used to evaluate the efficiency function, the population probability n1 is equal to the yield for the state A divided by the total yield of the particle stable nuclei for the isotope being considered. 52 III Excitation Energy Spectra for Particle Un- stable Nuclei Particle Unstable States of 5Li Figure 3.6 shows the correlation function for the decay 5Li —+ a+p. The lower scale in the figure gives the relative energy of the proton and a particle, and the excitation energy of 5Li is given in the upper scale. At low relative energies, there is a narrow peak [Poch 85b] at E,e1=0.19 MeV due to the two stage decay of 9B, where 9B“, -¥ p+8Be8,,. —>p+(a+a). To estimate the contamination due to the 9B3, decays, a Breit Wigner resonance of width I‘ = 0.055 MeV was included in the fit. The broad peak at 1 MeV S Ere; S 3 MeV is due to the decay of particle unbound ground state of 5Li (J1r = §-,I‘ = 1.5 MeV, I‘p/I‘ = 1.0) [Ajze 88]. Because the state is rather wide, we explicitly included the Boltzmann factor in fitting this peak. A value of T = 3 MeV was assumed in the fit. The population probability was extracted according to equation 3.51 and by using the Breit-Wigner formalism given by equation 3.34. The value of n A extracted for this state are not very sensitive to the value of T used in this fit. Because 5Li has no particle stable states, the efficiency was calculated using the energy spectrum for particle stable 6Li. As a consequence the population probabilities given in table 3.2 are defined relative to the particle stable yield of 6Li. The solid line in the figure shows fits to the data assuming the background depicted by the dashed line. The uncertainties in this yield were assessed by varying the background and also by varying the position resolution assumed in the calculation of efficiency. One such alternate background is shown by the dotted line in figure 3.6. 53 E*(5L1) (MeV) 0 2 4 e 2.0 ...-.....,.... I 3/2- natAg(14N,p(X)X ‘ ”$0 1.5 — — Lt ' .. £131 . m . + ‘. 0 .00 ‘ H 1 O ...::..__.._._.._._.._..;.._: _ ; _ i I _.._L _ r°- it 1- / J / . I Figure 3.6: Correlation function as a function of relative energy for a-p. The solid curve is the fit to the data assuming the background designated by the dashed line. The dotted line shows an alternate background. 54 Table 3.2: Spectroscopic information for Lithium and Beryllium isotopes which was used to extract excited state populations. Branching ratios Fc/I‘ are given in per- centage. Except for 5Li, relative populations n ,\ are defined relative to the particle stable yields for the same nucleus. The group structure is explained in the text. ] TGroup ] E‘(MeV) ] J1r ] I‘cm(MeV) I Pairs ] I‘c/I‘ I Relative population, m] 5Li 1 g.s. %— 1.5 a-p 100 a) 0.347 :l:0.03 2 16.66“ =3“ 0.20 3Hed 86") a>5.3><10-34 1.4><10--3 6Li 1 2.186 3+ 0.024 a-d 100 0.154001 2 4.31 2+ 1.7 a-d 97 0.0594002 5.65 1+ 1.5 a-d 74 7Li 1 4.63") g‘ 0.093” a-t 1005) 0.04742.5x10-3 2 6.68”) 3‘ 0.875”) a-t 1006) 0.034 7x10-3 7.466) 3' 0.089”) wt 18”) 3 11.24 3‘ 0.272 6He-p 59 4.8x10-341x10-3 7Be 1 457") g‘ 0.175“ a-3He 100 0.05245x10-3 2 6.73”) 3‘ 1.25) a—3He 1005) 0.0314001 7.21") 3‘ 0.5” a-3He 3b) 3 7.21 3’ 0.5 6Li-p 97 0.021435x10-3 “l Values of n A for 5Li are defined relative to the the particle stable yields of 6Li. I” Analysis performed using R-matrix parameters given in the text. 55 The d-3He correlation function is shown in figure 3.7. The relative energy of d-3He is shown in the lower scale and the upper scale gives the excitation energy of 5Li. A pronounced peak corresponding to the 16.66 MeV state in 5Li (J1r = %+,I‘ = 0.20 MeV) [Ajze 88] can be seen at Ere, 20.4 MeV. We used the R-matrix expression for decay from a single level (equation 3.33) to describe this state which has two decay channels : 5Li—>d+3He and 5Li—>p+a. The resonance parameters for this state are E,\ = 129 keV, 72(d) = 780 keV, 1,, = 0, ad = 7 fm, 72(p) = 12 keV, [p = 2, ap = 7 fm [Ajze 79], with boundary conditions 8,, = Bp = 0. The resonance parameters however, gave a peak in the excitation energy spectrum which occured at about 280 keV lower in relative energy than the peak observed experimentally. Because the d and 3He have different charge to mass ratios, distortions of the excitation energy spectra can result from Coulomb final state interactions with the residual nucleus [Poch 86b], but such effects have not been explored qualitatively for the d-3He system. Also included in the fit are contributions from the wide state at E" = 20 MeV. The solid curve in the figure shows fits to the data by assuming the background given by the dashed line. One alternate background is shown by the dotted line. As in the case of the 5Li ground state, the population probability n), of ”this state listed in table 3.2, is defined with respect to the yield of stable 6Li nuclei. The uncertainties in the population probability reflect uncertainties due to background subtraction, and uncertainties in the efficiency due to uncertainties in the position resolutions of the gas counters. Particle Unstable states of 6Li The correlation function for the decay 6Li—>d+or is shown in figure 3.8. An iso- lated peak corresponding to the 2.186 MeV state of 6Li (J1r = 3+,I‘ = 24 keV, I‘a/I‘ = 1.0) [Ajze 88] is observed at Ere, z 0.71 MeV. Two overlapping peaks at 4.31 56 E*(5L1) (MeV) 18 20 22 24 3 "'I"r'l""1v*"1 . (3/2”) natAg(14N,d3He)X ' ¢,.. -+ . 4 . ., ¢¢ + ..................... +4... Figure 3.7: Correlation function as a function of relative energy for 3He-d. The solid curve is the fit to the data assuming the background designated by the dashed line. The dotted line shows an alternate background. 57 nan"1.g(1‘*N,cxd)X 25 ' . . <— 2.186MeV . + . 2.186MeV (3) j . (3+) - 20f l 4.31MeV f (2*) 5.65MeV _~ | ((+) . ’7'; 15V 3... p m _ ................. v _ e: - 1] H 10.— 4.31MeV 5.65MeV 5 T (2*) (1*) Erel(MeV) Figure 3.8: The d-a correlation as a function of the relative energy. The fits to the resonances is shown by solid lines assuming the background shown by dashed line. The dotted curve shows as an alternative form of the background. 58 MeV (J1r = 2+,F =1.7 MeV, I‘a/F = 0.97) and 5.65 MeV (J1r =1+,I‘ 21.5 MeV, Fa/I‘ = 0.74) [Ajze 88] were associated with the second maximum in the 6Li spectra. Both the states at 4.31 MeV and 5.65 MeV are sufficiently wide, and are affected by line shape distortions coming from the Boltzmann factor. We therefore fitted the full spectrum by using a fixed value for T = 4 MeV using one normalization value for the state at 2.186 MeV and a second one for the states at 4.31 MeV and 5.65 MeV. The population probabilities were extracted according to equation 3.51 using the Breit-Wigner formalism given by equation 3.34. The fit shown by solid curves in the figure was obtained by assuming the background shown by the dashed line. One alternate background, used for assessing the systematic error, is depicted by the dotted line. Values for the extracted population probabilities ni, with respect to the ground state yield of 6Li, are listed in table 3.2 for the two groups of states. The un- certainties associated with these probabilities include uncertainties in the background estimation, and also the uncertainties arising from the efficiency calculation because of the uncertainties in the position resolutions in the gas detectors. Particle Unstable states of 7Li The correlation function for the decay TLi—r a+t is shown in figure 3.9. The peak marked by 7/2‘ and located at Ere) = 2.1622 MeV, corresponds to the 4.630 MeV excited state of 7'Li (J1r = §_,I‘ = 93 keV, I‘a/I‘ = 1.0) [Ajze 88]. This peak is fitted by using the R-matrix theory for decay from a single level (equation 3.33). The relevant parameters for the resonance are EA=2.80 MeV, 72 = 1.3 MeV, I = 3, a = 4 fm, B = —3 [Spig 67]. A broad structure can be seen in the a+t spectra of figure 3.9 corresponding to two overlapping states at E‘ = 6.68 MeV (J1r = g: F = 875 keV, I‘a/I‘ = 1.0) and E“ = 7.46 MeV (J’r = %-,I‘ = 89 keV, I‘a/I‘ = 0.18) [Ajze 88]. The state at 6.68 MeV has only the a+t channel open. The threshold for neutron decay 59 ““Ag(“N,at)x. E/A=35MeV I '7 . 1|: 4 8 ;_ L1 -> t + or ; , 4.133ro 2.0 :- , (7/3') C l 1.5 Figure 3.9: t-a correlation function as a function of relative energy. Location and spins of particle-unstable states in 7Li are indicated. The insert gives an expanded view showing the second maximum. The solid curves are the fits to the data assuming the background designated by the dashed line. The dotted line shows an alternate background. 60 is at E' = 7.25 MeV, and the state at 7.46 MeV decays by both a+t and 6Li+n channels. Because these states are overlapping and have the same spins and parities, the phase shift for these states were analysed by using the R-matrix formalism for two overlapping levels, equation 3.48. For simplicity, we designate the levels at 6.68 MeV and 7.46 MeV as levels 1 and 2 respectively in equation 3.48, a and neutron channels as channels 1 and 2, respectively. The R—matrix parameters are (E; = 5.730 MeV, 72(0) = 0.98 MeV, 10, = 3,00, = 4.4 fm) [Ivan 68] for the level at 6.68 MeV, and (EA 2 5.188 MeV, 72(0) = 0.024 MeV, IO, = 3,aa = 4 fm, 72(n) = 1.2 MeV, 1,, = 1,a,, :2 4 fm) [Spig 67] for the level at 7.46 MeV. These resonance parameters were obtained with the boundary conditions 8,, = —3 and Bn = —1. The excitation energy spectrum was fitted with two normalization parameters, one for the state at 4.63 MeV and another for the doublet at 6.68 and 7.46 MeV. The solid curve in the figure shows the fit to the data assuming the background designated by the dashed line. The resonance at E“ = 9.67 MeV was included in the fit to better describe the data. The dotted line shows an alternate choice for the background which was used for the estimation of systematic errors. The population probabilities n,\ are listed in table 3.2. The uncertainties in the population probability reflects uncertainties in the background as well as uncertainties in the efficiency due to uncertainties in the position resolution of the gas detectors. Figure 3.10 gives the correlation function for 7Li—-+6He+p. The peak seen at En.) = 8.77 MeV correspond to the proton decay of a state at E“ = 11.24 MeV (J1r = g: F = 0.272 MeV, I‘p/I‘ = 0.59) [Pres 69, Ajze 88]. This peak is fitted using the Breit-Wigner formalism (equation 3.53). The solid curve shows the fit to the data assuming the background depicted by the dashed line. The dotted line shows an alternate choice of the background used for the estimation of systematic errors. The population probability n1 is listed in table 3.2. The uncertainty in m reflects both 61 E*("Li) (MeV) 10 11 12 1 2,5_....,....l.,,,?,,,,1f1,, 3/2- natAg(l4N,p6He)X 2.0 - I i—a U‘I l l 1 I l l 1 l l l 1 l I 1+ROI-73ml) S; l 1 1.011; _____ f++ + : ................. ,m. . .. +. . . . H 0.5 1'1 1 I I 1 1 1 1 I l 1 1 1 I 1 1 I 1 I Figure 3.10: p-“He correlation function as a function of relative energy. The excitation energy in 7Li is indicated on the top. Location and spin of a particle-unstable state in 7Li is shown. 62 the uncertainty in the background estimation and the uncertainty in the detection efficiency as discussed earlier. Particle Unstable states of 7Be The correlation function resulting from the coincidence of 3He+a is. given in figure 3.11. Because of isospin symmetry, the analysis of the states in 7Be is similar to the analysis of the 7Li states. The peak at E" = 4.57 MeV (J’r = §-,I‘ = 175 keV, Fa/l" = 1.0) [Ajze 88] was analysed by using the R-matrix formalism for decay from a single level (equation 3.33). The corresponding R-matrix parameters are (E; = 3.885 MeV, 72(0) = 1.595 MeV, 10:3, ac, = 4 fm, Bo, = —3) [Spig 67]. The states at E“ = 6.73 MeV (J’r = g-J‘ = 1.2 MeV, Pa/I‘ = 1.0) and 7.21 MeV (J’r = g-, F = 0.5 MeV, Fa/l‘ = 0.03) [Ajze 88] were analysed by using the R-matrix formalism for decay from two nearby levels (equation 3.48). The relevant parameters for the level at 6.73 MeV are (EA=9.007 MeV, 72(a)=3.1 MeV, 10, = 3,00, = 4 fm), and for the level at 7.21 MeV are (E; = 5.993 MeV, 72(0) = 0.023 MeV, Io, = 3,010, = 4 fm, 72(p) = 1.2 MeV, [p = 1,ap = 4 fm) [Spig 67, Bark 72]. The corresponding boundary conditions are of Ba, = —3 and BD = —1. Solid curve in the figure shows the fit to the data assuming the background designated by the dashed line. The dotted line shows an alternate choice of the background used to estimate the systematic errors. The excitation energy spectrum was fit assuming one free parameter for the normalization of the state at 4.57 MeV and another for the doublet at 6.73 and 7.21 MeV. The relative populations In for the first state and the second group of states are listed in table 3.2. The uncertainties in n ) reflect the uncertainties due to the background estimation and uncertainties in the efficiency calculations due to the uncertainties in the position resolutions of the gas detectors. The correlation function for 6Li+p is given in figure 3.12. A clear peak can be seen 63 ““Ag(“N,a3He)x, E/A=35MeV : 6.73uev 6 _ 4.57““ 7.21MeV - (”2'") 1.5 Q4. .4 o _ g * i 0:: i 05 i”. : ' o 4' 6 s 2 -— 6.73uev 7.2111ev _ (5/2‘) “I f— (W?) i ,...-:.:'..'.'..-:.T.:..-..:7..-..7..=...-...-.:.:7..-. ................. b 1 O l l l l i l l l l i I l l l l l 1 l l l l l O 2 4 6 8 E:rel (MeV) Figure 3.11: Correlation function as a function of relative energy for 3He-a. The solid curves give a fit to the data with the background shown by the dashed lines. The dotted line shows an alternate background. 64 E*("Be) (MeV) 6 8 1o 12 Figure 3.12: p-“Li correlation function as a function of relative energy. Location and spins of a particle-unstable state in 7Be is indicated. The solid curve shows a fit to the data with the background designated by the dashed line. The dotted line shows an alternate background. 65 corresponding to the state at E' = 7.21 MeV (.]’r = g-J‘ = 0.5 MeV,I‘p/I‘ = .97) [Ajze 88] of 7Be. This peak was fitted with the Breit-Wigner formalism (equation 3.53). An additional state corresponding to E‘=9.27 MeV was included in the fit, but the population probabilities was not extracted from this. The solid curve depicts the fits to the data assuming the background given by the dashed line. The dotted line shows an alternate background used for estimating the systematic error. The population probabilities m are listed in table 3.2. The uncertainties in the population probabilities include the uncertainties due to the background estimation and the uncertainties in the efficiency calculation. Particle Unstable states of 8B The correlation function for 8B -+7Be+p is shown in Figure 3.13. The relative energy of 7Be+p and the excitation energy in the 8B nucleus are indicated in the lower and upper scales respectively. Two pronounced maxima corresponding to the excited states of 8B at E" = 0.774 MeV (J = 1+,I‘ = 37 keV, I‘p/I‘ = 1.0) [Ajze 88] and E” = 2.32 MeV (J’r = 3+,I‘ = 350 keV, I‘p/I‘ = 1.0) [Ajze 88] are clearly seen. The spin of the 0.774 MeV state is taken to be same as the corresponding state in the mirror nucleus 8Li. For the 2.32 MeV state, I‘ = 310 keV was used instead of 350 keV in the fit which gave a better description of the data. These two peaks were analysed by using Breit-Wigner formalism (equation 3.53). The solid curves show fits to the data corresponding to the background depicted by the dashed line. An alternate description of the background is shown by dotted lines. The population probabilities n A are given in Table 3.3. The associated uncertainties reflect the uncertaintias in the background estimation and also the uncertainties in the efficiency calculation as was discussed earlier. 66 E*(BB) (MeV) 1 2 3 4 5 6 F I r I i r I I I l mi I rl r I r rl fit I I I I I I I l. : nat 1 7 3 4— 1“ Ag( 4N,p Be)X _ 0 "' .. h —— 3+ __ LTJ 3 ' l ' v r- .1 m t _i + __ H 2 ' .. 3 ’ +1 + 3 1 :— ....... T 0:: ...................... 1 ....¢... o.: J O ' 1 1 1 i 1 1 4 1 l 1 L 1 1 14 1 LL] L L 1 1 i 1 1 1 LT O 1 2 8 4 5 Erel (MGV) 03 Figure 3.13: I’B —»7Be+p correlation function. The excitation energy in 8B is indi- cated on the top. The solid curve shows a fit to the data assuming the background depicted by the dashed line. The dotted line shows an alternate background. 67 Table 3.3: Spectroscopic information for 8B, 10B, and 11C isotopes which was used to extract excited state populations. The branching ratios are given in percent, and n A are defined relative to the particle stable yields for the same nucleus. The group structure is explained in the text. I IGroup IE‘(MeV) I J’r I I‘m(keV) I Pairs I Fc/I‘ I Relative population, n,\ I 8B 1 0.774 1 37 7Be—p 100 0.152i0.016 2 2.32 3+ 310 7Be—p 100 0.212i0.085 108 1 4.774 3+ 8.4x 10-3 6Li-a 100 001340.001 2 5.1103 2- 0.98 Gm: 100 9.6x10‘3i1.5x10‘3 5.1639 2+ 1.76x10‘3 6Li-az 13 5.180 1+ 110 6Li-a 100 3 5.9195 2+ 6 6Li-o: 100 0.0142h0.002 6.0250 4+ 0.05 6Li-a 100 6.1272 3’ 2.36 6Li-a 97 4 6.56 4’ 25.1 6Li-a 100 1.0X10'2zlz2JX10'3 5 7.430 2" 100 9Be-p 70 4.2x 1Wi8x 10'4 7.467 1* 65 9Be—p 100 7.478 2+ 74 gBe-p 65 7.5599 0+ 2.65 gBe-p 100 ~ 6 7.67 1+ 250 gBe-p 30 6.1 x10'3:1: 2.1><10--3 7.819 1- 260 QBe-p 90 8.07 2"” 800 9Be-p 10 7 8.889 3- 84 gBe-p 95 3.2x 10'321: 4.6x10-4 8.895 2+ 40 9Be-p 19 w I 11C 1 8.1045 0.011 7Be-a 92 5.80x10'321: 4.3x10“ 5 8.420 3‘ 0.015 786.6 80 5.67x10'321: 4.3x10“ 3 8.655 g“ 5 736-0 94 5.93x10'3zi: 3.4x10-4 8.701 3* 15 786-0 100 68 Particle Unstable states of 10B Relative populations 12; of particle-unstable states in 10B nuclei were measured by de- tecting the coincident decay products for the channels 10B—76Li+a and 10B—>9Be+p. The measured coincidence yields, Y(E‘) are shown Figure 3.14 as a function of the excitation energy of 10B. A number of distinct peaks are identified. In spite of the good excitation-energy resolution of the hodoscope some states couldnot be resolved and were analysed as a group. Within a given group of unresolved states, the popula- tion probability n1 is assumed to be same for all states. The upper part of the figure shows the 6Li+a coincidence spectrum. The first peak corresponds to an excited state at 4.774 MeV with (J1r = 3+,I‘ = 8.4 eV, I‘a/I‘ = 1.0) [Ajze 79, Ajze 88, Albu 66]. The second group consists of three states at 5.1103 MeV (J’r = 2',F = 0.98 keV, Fa/I‘ = 1.0) [Ajze 79, Ajze 88, Fors 66, Meye 58], 5.1639 MeV (J1r = 2+,I‘ = 1.76 eV, Fa/I‘ = 0.13) [Ajze 79, Ajze 88, Fors 66, Meye 58, Albu 66, Spea 79] and 5.18 MeV (J1r = 1+,1" = 110 keV, I‘a/F = 1.0) [Ajze 79, Ajze 88, Dear 62]. The small shoulder after this group could be explained by the decay of 8.889 MeV and 8.895 MeV states of 10B to the 3.563 MeV excited state of 6Li" and 0:. These two states were included in the fits, but were not analyzed further. The third group is made of states at 5.9195 MeV (J1r = 2+,l‘ = 6 keV, Fa/I‘ = 1.0) [Ajze 79, Ajze 88, Dear 62, Fors 66, Youn 69], 6.0250 MeV (J’r = 4+,F = 0.05 keV, I‘a/F = 1.0) [Ajze 79, Ajze 88, Fors 66, Youn 69], and 6.1272 MeV (J1r = 3‘,I‘ = 2.36 keV, I‘a/l‘ = 0.97) [Ajze 79, Ajze 88, Fors 66, Youn 69, Meye 67, Blan 80]. The fourth peak in this spectrum is an isolated state at 6.56 MeV (J'1r = 4‘,I‘ = 25.1 keV, Fa/I‘ = 1.0) [Ajze 79, Ajze 88, Fors 66, Youn 69, Meye 67, Blan 80, Bala 71]. All the groups of states were analysed by using the Breit-Wigner formalism (equation 3.53). The solid lines in figure 3.14 depicts a fit using the background shown by the dashed lines. An alternate background used for the estimation of systematic errors is shown 69 14N+.1.g, E/A=35MeV, 60:38° IIIIIIIIIIIIIIIIIII'IITIIIIII _ - 3+ : 6000 T 1 2- 2+ — ~ I Ir" 3 4000 :— . —‘ 1 : 2000 E- ‘ { 600 400 Y(E*) (counts/20 keV) 200 W I I I I I I I I I I I I I I 0 .........11111 4 5 6 7 8 9 10 E*(1°B) (MeV) Figure 3.14: 6Li-l-a (upper part) and °Be+p (lower part) excitation energy spectra. Location and spins of particle-unstable states in 1“B are indicated. The solid curves show the fits to the data assuming the background depicted by the dashed line. The dotted lines indicate an alternate choice for the background. 70 by the dotted line. The population probabilities are given in table 3.3. The uncer- tainties in the population probabilities n1 reflect the uncertainties in the background subtraction, and uncertainties in the efficiency for detecting products of the particle unstable 10B nucleus due to uncertainties in the resolution of the position sensitive detectors. The lower part of Figure 3.14 gives the coincidence spectra of 9Be+p. The first group indicated in the figure is a combination of four states at 7.43 MeV (J1r = 2", F = 100 keV, I‘p/F = 0.70) [Ajze 79, Ajze 88, Moze 56, Sier 73, Auwa 75, Mo 69], 7.467 MeV (J1r = 1+,F = 65 keV, I‘p/I‘ = 1.0) [Ajze 79, Ajze 88, Sier 73, Auwa 75, Hara 80, Bala 71], 7.478 MeV (J1r = 2+,I‘ = 74 keV, I‘p/I‘ = 0.65) [Ajze 79, Ajze 88, Auwa 75, Mo 69, Hara 80,.Horn 64, Elli 62, Rohr 73], and 7.5599 MeV (J " = 0*, F = 2.65 keV, I‘p/I‘ = 1.0) [Ajze 79, Ajze 88, Moze 56, Auwa 75, Mo 69, Rohr 73, Elli 62, Horn 64, Hara 80]. The second group is made of three states at 7.67 MeV (J1r = 1+,I‘ = 250 keV, I‘p/I‘ = 0.30) [Ajze 79, Ajze 88, Mo 69], 7.819 MeV (J1r = 1",F = 260 keV, I‘p/I‘ = 0.90) [Ajze 79, Ajze 88, Mo 69, Rohr 73], and 8.07 MeV (.]’r = 2+,I‘ = 800 keV, I‘p/l" = 0.10) [Ajze 79, Ajze 88, Mo 69]. The last group in this spectrum consists of two peaks at 8.889 MeV (J1r = 3",F = 84 keV, I‘p/F :2 0.95) [Ajze 79, Ajze 88, Oele 79] and 8.895 MeV (J7r = 2+,I‘ = 40 keV, I‘p/I‘ = 0.19) [Ajze 79, Ajze 88, Kiss 77]. In addition, there are two neighboring peaks near the threshold at 6.873 MeV (J7r = 1‘, = 120 keV,) and 7.002 MeV (J1r = 2+,I‘ = 100 keV,). These states were not analyzed because the branching ratios are not well known. All groups of states in this spectrum were analysed by using Breit- Wigner formalism (equation 3.53). The solid line in figure 3.14 depict the fit to the spectrum assuming the background indicated by the dotted line. An alternate choice for the background is shown by the dotted line which was used to estimate the systematic errors due to background subtraction. The population probabilities 71 n,\ for the different groups of states are given in table 3.3. The uncertainties in the population probabilities reflect the uncertainty in the background and from the uncertainties associated with the efficiency calculation. Particle Unstable states of 11C The excitation energy spectra of 11C obtained from the coincidence cross section of 7Be +a is given in figure 3.15. The relative energy of 7Be and a, and the excitation energy of 11C are indicated in the lower and upper parts of the figure respectively. The positions of the first three groups of excited states in 11C and their spins and parities are indicated in the figure. The first peak is at 8.1045 MeV and corresponds to (.I"r = $3-, I‘ = 11 eV, I‘O/I‘ = 0.92) [Ajze 85, Hard 84]. The second peak shown in the figure is at 8.420 MeV and corresponds to (J" = %-,I‘ = 15.2 eV, Fa/I‘ = 0.80) [Ajze 85, Hard 84]. The third group consists of two peaks at 8.655 MeV (J1r = $21“ = 5 keV, I‘a/I‘ = 0.94) [Ajze 85, Wies 83] and 8.701 MeV (J’ = gfir = 15 keV, Fa/F = 1.0) [Ajze 85, Wies 83]. Although the state at 8.701 MeV is slightly proton unbound, it decays predominantly by a-particle emission [Wies 83]. Excited states of 11C at E‘ = 9.20, 9.65, 9.78, 9.97, 10.083, 10.069, 11.03, 11.44 and 12.65 MeV were also included in fitting the experimental yield. But we did not extract population probabilities from these because the spectroscopic information for some of these states are uncertain. The fits to the data were obtained by using Breit Wigner formalism (equation 3.53). The solid curve in the figure shows the fit corresponding to the dashed background. An alternate background is shown by the dotted line. The population probabilities m for the first three groups are listed in table 3.3. The systematic uncertainties in these quantities reflect the uncertainties in the efficiency calculation due to uncertainties associated with the position resolutions of the gas detectors and by the uncertainties associated with the background determination. 72 E*(“C) (MeV) 8 9 10 11 12 F j r I I I l I T T I T I r I I I I l I I I FIG % h '7/2+ 5/2+ 1 " fil— nat 1 7 ' x 600 L Ag( 4N,a Be)X — o : ' cu _ i 6 - 3 4.) -— _ a 400_ .. S _ .. O - q 0 _ ‘ 200 - "" A 1- .1 III 12:: ' ‘ v " J >4 0 1— ...lao.—.—4-.v-.-R.nofin. r1?" 2 3 4 5 Erel (MeV) Figure 3.15: Excitation energy spectrum of 11C obtained from the coincidence} cross section of 7Be+a. The excitation energy in 11C is indicated on the top. The solid line is a fit to the data assuming the background depicted by the the dashed line. The dotted line shows an alternate background. 73 Particle Unstable states of 13N The excitation energy spectrum of 13N obtained from the coincidence cross section of 12C and proton is given in Figure 3.16. The lower scale in the figure gives the relative energy of 12C and proton and the upper scale gives the excitation energy of 13N assuming the 12C is emitted in its ground state. Two groups of states were analysed for extracting relative populations. One group consists of two overlapping states at 3.511 MeV (J" = 3‘, r = 62 keV, I‘po/I‘ = 1.0), and 3.547 MeV(J1r = gfir = 47 keV, PpO/F = 1.0) [Ajze 85] states of 13N. This group is indicated by the pair of spins %‘ and %+. Here the subscript p0 refers to the decays to the ground state of 12C and a proton, and p1 refers to the decays to the first excited state of 12C (E‘ = 4.44 MeV) and a proton. A second peak indicated by 7/2+ in the figure corresponds to the decay of the 7.155 MeV(J1r = gfir = 9 keV, 1‘pl /r = 1.0) [Ajze 85, Bark 63] state of 13N which decays to an excited ”C‘ in the 4.44 MeV excited state and a proton. Additional excited states of 13N at E‘ = 2.3649, 6.364, 6.886, 7.376, 9.00, and 9.476 MeV were included in the fit to the experimental data, but population probabilities are not provided for these states either because they lack statistics or because we lack the necessary spectroscopic information. The analysis was performed by using Breit-Wigner formalism (equation 3.53). The solid lines in the figure shows a fit obtained by assuming the background indicated by dashed line. An alternate background shown by the dotted line was used to estimate the systematic error in the background subtraction. Table 3.4 gives the population probabilities, ml. The uncertainties in 77). include uncertainties in the background subtraction, and also the uncertainties associated with the efliciency calculations caused by the uncertainties in the position resolutions of the gas counters. 74 E*(13N) (MeV) 200 —- nat4“48(1‘11‘1.1.I>120)X ~— . I F2221 3 100- I — Y(E*) (counts/20 keV) Erel (MeV) Figure 3.16: Excitation energy spectrum of 13N obtained from the coincidence cross section of 12C-p. The solid line is a fit to the data assuming the background depicted by the dashed line. The dotted curve shows an alternate background. 75 Table 3.4: Spectroscopic information for 13N and 1“N isotopes which was used to extract excited state populations. Branching ratios Fc/F are given in percent, and n; are defined relative to the particle stable yields for the same nucleus. The group structure is explained in the text. I IGroup I E“(MeV) I NI I‘cm(keV) I Pairs I I‘c/I‘ I Relative population, n; I 13N 1 3.511 3' 62 ”Op 100 0.1101002 3.547 3* 47 12Op 100 2 7.155 4+ 9 120p 0) 100 0.071002 MN 1 7.9669 2- 2.5x10-3 130p 99 7.3x10'3i1.4x10‘3 2 8.062 1- 30 ”Op 100 5.3x10-3 i 1.7x10-3 3 8.4899 4- 3.46x10‘5 13C-p 79 9.8x10'3zt1.1x10'3 8.6197 0+ 3.8 13C-p 100 8.776 0- 410 130p 100 4 8.9118 3- 16 130p 100 6.32x10-3 1: 7.9x10-4 8.9638 5+ 6.25x10-6 13Op 80 8.9804 2+ 8 130p 100 5 9.1289 3+ 18.9x10-6 130p 81 I5.7x10’3:l:1.2x10'3 9.1723 2+ 0.135 l3C-p 95 6 9.3893 2- 13 130;) 100 3.8x10-3 :1: 7.1x10-4 9.509 2- 41 13C-p 100 7 10.079 3+ 10 ”Op 100 5.4x10’3 1: 1.0x10-3 10.101 2+ 12 130;) 100 8 10.812 5+ 0.39x10-3 E0;) 96 6.2x10‘3:1:1.5x10‘3 9 11.05 3+ 1.2 13C-p 100 4.2x10-3 2: 1.3x10-3 “l Branching ratio for decay to an excited ”C“ nucleus (E‘ = 4.44 MeV) and proton. Particle Unstable states of 1“N The correlation function resulting from the decay l4N —+13C+p was presented in figure 3.5. Here we show the the experimental yields for 1“N ~113C+p in Figure 3.17. The relative energy of 13C and proton and the excitation energy Corresponding to the excited states of 1“N are indicated in the lower and upper part of the figures, respectively. We have analysed nine groups of states which are identified in the figures. The first group corresponds to an isolated state at 7.9669 MeV (J1r = 2‘,I‘ = 2.5 eV, I‘p/ 1‘ = 0.99) [Ajze 86a]. The second state is at an excitation energy of 8.062 MeV (J1r = 1‘,l‘ = 30 keV, I‘p/I‘ = 1.0) [Ajze 86a]. The third group is formed by overlapping states at 8.4899 MeV (.1" = 4‘,I‘ = 3.46 x 10'5 keV, Fp/F = 0.79) and 8.6197 MeV (J"' = 0"“, 1‘ =‘3.8 keV, I‘p/I‘ = 1.0). The fourth group is made of three overlapping states at 8.9118 MeV (J1r = 3',I‘ = 16 keV, I‘p/I‘ = 1.0) [Ajze 86a], 8.9638 MeV (.11r = 5+,I‘ = 6.25 x 10-6 keV, rp/r = 0.80) [Ajze 86a], and 8.9804 MeV (J’r = 2*, I‘ = 8 keV, I‘p/I‘ = 1.0) [Ajze 86a]. The fifth group consists of two states at 9.1289 MeV (J1r = 3", = 18.9 x 10‘6 keV, I‘p/I‘ = 0.81) [Ajze 86a] and 9.1723 MeV (J7r = 2", F = 0.135 keV, I‘p/I‘ = 0.95) [Ajze 86a]. The sixth group is a combination of two overlapping states at 9.3893 MeV (J1r = 2‘, F = 13 keV, I‘p/I‘ = 1.0) [Ajze 86a], and 9.509 MeV (J1r = 2‘,I‘ = 41 keV, Fp/I‘ = 1.0) [Ajze 86a]. The seventh group is made of two peaks at 10.079 MeV (J1r = 3*, F = 10 keV, Fp/I‘ = 1.0) [Ajze 86a], and 10.101 MeV (J1r = 2+,F = 12 keV, Fp/F = 1.0) [Ajze 86a]. The eighth group is an isolated state at 10.812 MeV (J1r = 5"”,1‘ = 0.39 x 10'3 keV, I‘p/F = 0.96) [Ajze 86a]. The ninth. and last group we have taken into consideration is an isolated peak at 11.05 MeV (J1r = 3+,F = 1.2 keV, Fp/F = 1.0) [Ajze 86a]. The fits to the experimental data were performed by using Breit-Wigner formalism (equation 3.53). Excited states of 1“N at E" = 9.703, 10.226, 10.432, 10.534, 11.761, 12.2, 12.408 were included in fitting the spectra, but population probabilities from 77 E*(”N) (MeV) 8 9 10 11 12 300 Hump I T I I I I I I I I I T‘ I I I I N I Y(E*) (counts/ 20 keV) Figure 3.17: Energy spectrum resulting from the decay of particle unstable l4N. Solid curve is a fit described in the text assuming the background shown by dashed curve. The dotted curve shows an alternate description of the background. 78 these were not extracted. Fits assuming the background depicted by the dashed line in figure 3.17 are shown as solid curves in figures 3.5 and 3.17. The dotted curve shows an alternate background used to estimate the uncertainty in background subtraction. The extracted relative populations m and the associated uncertainties are listed in table 3.4. The uncertainty in n; reflects both the uncertainty in the background subtraction and the uncertainty due to the efficiency function. Particle Unstable states of 160 We next consider the excitation energy spectrum for 160 resulting from the coinci- dence spectrum of l2C+oz, shown in figure 3.18. The scale on the bottom gives the relative energy of 12C and a, and the top scale gives the excitation energy for 160. Four groups of states are identified in the figure. The first peak labelled by 2' in the figure corresponds to the 12.53 MeV state (J1r = 2‘, F = 0.097 keV, Fol/F = 0.74) [Leav 83] of 16O which decays to a 12C" nucleus in its 4.44 MeV excited state plus an a particle. The subscripts 00 and al refer to the decays to the ground state and the 4.44 MeV state of 12C, respectively. The second group of peaks at about 9.9 MeV of excitation energy, is a combination of four states. One of these corresponds to the 9.845 MeV (J1r = 2+,I‘ = 0.625 keV, Foo/F = 1.0) state of 160. In addition, there are three states at 14.1 MeV (J’r = 3‘,F = 750 keV, Fol/F = 0.8), 14.399 MeV (J1r = 5+,I‘ = 27 keV), and 14.302 MeV (J1r = 4‘,F = 34 keV) which decay to an excited 12'C‘ nucleus (E' = 4.44 MeV) and an a particle. Since the branching ratios for these latter two decays are not known, the sensitivity of our analysis to these states is explored by varying the branching ratios for these states between 0% and 100%. These variation in the branching ratios causes variations in the population probabilities for the states, and we use the range of such variations as an estimation of the systematic uncertainties associated with the unknown branching ratios. The 79 8 9 10 11 12 13. > EnatAg(14N alzC)X + 1 q) .- 9 4 _I .3: 100 r 5+ ...... 2 5‘ (\2 L. 2 '3 ES .2 \ 80 ’ 3'“: I 333 i 3 h . .. . : :1 60 :- 2- 1 1 :3 ’ - 0* 1 O I . 1 3 40f 1 {K : . : m 20 :- . (I ‘ ...’.... .. 0.00. .o.o.o....o...... 0.00. .;‘ v - «i 1 —- >3 * .Qt.1.1..1.1....1....11... 1 2 3 4 5 6 Erel (MGV) 0 Figure 3.18: Excitation energy spectrum of 1°C obtained from the coincidence cross section of 12C-+—a. The solid curve describes a fit obtained by assuming the dashed line as one possible background. The dotted curve shows an alternate background. 80 third group of states which is seen around 10.4 MeV is a combination of five peaks, the 10.356 MeV state (J1r 2' 4+, 1‘ = 26 keV, Foo/1‘ = 1.0) of 16O, and the 14.620 MeV state (J1r = 4*, F = 490 keV, Pal/1‘ = 0.2), the 14.660 MeV state (J’r = 5",F = 670 keV, Fol/1‘ '2 0.06) the 14.815 MeV state (J1r = 6+,I‘ = 70 keV, Fai/F = 0.65) and the 14.926 MeV state (J" = 2+,l‘ = 54 keV, I‘m/l" = 0.58) [Ajze 86b] of 16O. The last four states in the third group decay to an excited 12C" (E‘ = 4.44 MeV) and an a particle. The fourth peak in 16O excitation energy spectrum is seen around 11 MeV in the figure and has contributions from four states of 16O at 10.957 MeV (J" = 0‘,l" = 8.21 x 10'5 keV, Foo/1‘ = 1.0), 11.080 MeV (J1r = 3+,I‘ = 12 keV, Poo/F = 1.0), and 11.097 MeV (J1r = 4+,I‘ = 0.28 keV, Poo/F = 1.0) [Ajze 86b]. The state at 15.408 MeV (J’r = 3‘,F = 132 keV) which could contribute to this group has a very small (R1170) al branch [Ajze 86b]. 160 states corresponding to E“ = 8.8719,9.585,11.52,11.6,12.049,12.440 MeV which decay to the ground state of 12C and an a particle, and E“ = 12.796, 12.97, 13.02, 13.09, 13.129, 13.259, 13.664, 13.869, 13.98, 14.032, 15.196, 15.26, 15.785 and 15.828 MeV which decay to an ex- cited 11’0“ (E‘ =4.44 MeV) and an a particle were also included in fitting the spectra, but unanalyzable either because they lack statistics or because we lack the necessary spectroscopic information. All the states were analysed by using the Breit-Wigner formalism (equation 3.53). Fits assuming the background depicted by the dashed line in figure 3.18 are shown by the solid curve. The dotted line shows an alternate choice of background used to estimate the uncertainty in the background subtraction. The relative populations n A and the associated uncertainties are listed in table 3.5. The uncertainty in n A reflects the uncertainty in the background subtraction, the uncertainty due to the efficiency function, and the uncertainties due to the unknown branching ratios to the first excited state of 12C. 81 Table 3.5: Spectroscopic information for 16O and 180 isotopes which was used to extract excited state populations. Branching ratios I‘c/I‘ are given in percent, and n ,\ are defined relative to the particle stable yields for the same nucleus. The group structure is explained in the text. I I Group I E‘(MeV) I J1r Tch(keV) IPairs I Fc/F I Relative population, n; I 160 1 12.530 2- 0.097 12Ca 0) 74 1.89><10‘3:i:7.2x10‘4 2 9.845 2+ 0.625 ”C-a 100 7.4x10'3i5.2x10’3 14.1 3- 750 120-0 a) 80 14.302 4- 32 120-5 14.399 5+ 27 12Ga 3 10.356 4+ 25 12Ga 100 4.5><10-3:1:1.0x10-3 2 7.864 5- 8 “Ga 100 6.1x10‘3i1.5x10"3 3 8.039 1- 2.5 14C-6 100 2.93x10‘3zl:6.7><10‘4 8.125 5- 1 ”Ga 100 8.213 2+ 1.6 “C-a 99 8.282 3- 8 ”C-a 89 “) Branching ratio for decay to excited l2C" nucleus (E‘ = 4.44 MeV) and an a particle. 82 Particle Unstable states of 18O The excitation energy spectrum for 180 obtained from the coincidence cross section of 1“C+a is shown in figure 3.19. The lower scale in the figure indicates the relative energy of 14C and a, and the upper scale shows the excitation energy of 180. Three groups of states are identified. The first peak is at 7.1169 MeV (J1r = 4+,F = 2.6 x 10'5 keV, I‘a/I‘ = 0.53) state of 18O [Ajze 87, Gai 87]. The second peak is identified as the 7.864 MeV (J1r = 5",1‘ = 8 keV, Fa/I‘ = 1.0) [Ajze 87, Gai 87, Beck 73] state of 18O. The third group consists of four states at 8.039 MeV (J1r = 1', I‘ = 2.5 keV, I‘a/I‘ = 1.0), 8.125 MeV (J’r = 5',I‘ = 1 keV, I‘a/I‘ = 1.0), 8.213 MeV (J1r = 2+,I‘ =1.6 keV, I‘a/I‘ = 0.99) and 8.282 MeV (J1r = 3',F = 8 keV, I‘a/I‘ = 0.89) [Ajze 87, Gai 87, Beck 73]. These peaks are fitted by using the Breit-Wigner formalism (equation 3.53). Fits assuming the background depicted by the dashed line in figure 3.19 are shown by the solid curve. A zero background assumption is used as an alternate choice to estimate the uncertainty in the background subtraction. The relative populations n ,\ and the associated uncertainties are listed in table 3.5. The uncertainty in m reflects the uncertainty in the background subtraction and the uncertainty due to the efficiency function. 83 8*(180) (MeV) 7 8 9 10 11 I I I I I I I U I I T I I I I T l r I I l I l I 00 O l l l 1 l l l I l 1 l L l H O Y(E*) (counts/20 keV) 10 O ""l""l""1 5 >0“ "i .2“ Q 3 3 ix: 11+: O C Figure 3.19: Excitation energy spectrum of u'0 obtained from the 1“C-01 coincidence cross section. The solid curve describes a fit obtained by assuming the dashed curve as one possible background. Chapter 4 Sequential Feeding from Higher-lying States Measurements of the relative populations of excited states of emitted fragments pro- vide a measure of the intrinsic excitation energy of the emitting system at freezeout. If the excitation energy is thermally distributed, then the population probabilities n for excited states within a fragment would follow a Boltzmann distribution. However, the observed populations of excited states are influenced by the sequential decay of heav- ier particle unstable nuclei, [Poch 85a, Xu 86, Sobo 86, Hahn 87, F iel 87, Come 88, Xu 89, Deak 89] and the populations and decays of many of these unbound states are not known experimentally. Since one does not usually know the feeding corrections experimentally, they must be calculated. We have performed calculations to deter- mine this effect of feeding on measured values of population probabilities n. In the calculations, the states of primary fragments are assumed to be thermally populated characterized by a temperature, T m [Xu 86, Hahn 87, F iel 87]. The primary elemen- tal distributions were adjusted to ensure consistency between the calculated final and experimental distributions. In this chapter, we describe the essence of the sequential feeding calculation. In the first section we discuss how various fragments and their excited states are 84 85 included in the calculation. A method for choosing unknown spectroscopic factors of low lying states is also discussed. In the second section we give the expressions for primary populations of states. In the third section, details of decay calculations will be discussed. We describe the results of calculations in the fourth section. I Levels and Level Densities To determine the feeding corrections to the measured relative probability, we per- formed sequential decay calculations for an ensemble of nuclei with 35 Z 313. A lookup table containing excitation energies, spectroscopic factors and different decay channels with corresponding branching ratios for approximately 2600 known levels for isotopes within this charge range [Ajze 84, Ajze 85, Ajze 86a, Ajze 86b, Ajze 87, Ajze 88] was constructed. Since the spins, isospins and parities of many low-lying particle bound and un- bound levels of nuclei with Z311 are known, the information for these lighter nuclei was used in the sequential decay calculations. For known levels with incomplete spec- troscopic information, values for the spin, isospin, and parity were chosen randomly according to primary distributions obtained from the non—interacting shell model. The shell model program ‘OXBASH’ [Brow 88] was used to calculate the number of states at a given spin, parity and isospin for energies up to 2hw. Single particle energies, obtained from the Nilson diagram [Tabl 67] were combined to obtain the final energies for a particular particle-hole configuration. The energy of the lowest level with appropriate spin, parity and isospin was taken as the energy of the ground state. The distributions were then smoothed out to obtain the level density distribu- tion as a function of excitation energy above this ground state. For a level of a given excitation energy but unknown spin, parities or isospin, we randomly selected the 86 unknown values of spin, parity or isospin according to the level density distributions. The calculations were repeated with diflerent initial values for the unknown spec- troscopic information until the sensitivities of the calculations to these spectroscopic uncertainties were assessed. The low-lying discrete levels of heavier nuclei with Z212 are not as well known as those of lighter nuclei. To calculate the decay of these heavier nuclei for low excitation energies, E" S 60(A,, Zg), we used a continuum approximation to the discrete level density [Chen 88], modifying the empirical interpolation formula of ref. [Gilb 65b] to include a spin dependence: . _ _ 1 ._ (2J.+1)exp1—(J.-+ 92/2031 p4 / . 3:3 . m _ ‘ a 4 Q) ’ . Q 20" - '5 * < > ’ . Q) I4 4 - J 10' .. :« 15. . ‘ n-.1.... - FFL O l l...l-nn.l .L Ex (MeV) Figure 4.1: The level density of 20Ne as a function of excitation energy [Chen 88a]. The histogram gives the number of known levels whereas the solid curve shows results of level density predicted by eq (4.3). 89 where P0(A.-, Zg) denotes the population per spin degree of freedom of the ground state of a fragment and Tem is the emission temperature which characterizes the thermal population of states of a given isotope. (This temperature is associated with the intrinsic excitation of the fragmenting system at breakup and is, in general, different from the “kinetic” temperature which may be extracted from the kinetic energy spectra of the emitted fragments.) The initial populations of states of a given fragment were assumed to be thermal up to excitation energy of E‘ a = A- constant. cuto This cutoff was introduced to explore the sensitivity of the calculations to highly excited and short-lived nuclei, some of which may be too short lived to survive the evolution from breakup to freezeout. Calculations were performed for cutoff values of Egmofi/A = 3 and 5 MeV corresponding to mean lifetimes of the continuum states of 230 fm/c and 125 fm/c, respectively [Stok 77]. The calculations were qualitatively similar for the two cutoff energies. All the results presented here were done with Egutofl/A = 5 MeV. For simplicity, we parameterized the initial relative populations, P0(A,-, Z.) by P0(A, Z) oc exp(—fV(;/Tem + Q/Tem), (4.9) where V0 is the Coulomb barrier for emission from a parent nucleus of mass and atomic numbers AP and Zp and Q is the ground state Q-value vc = 2,.(2, — z.)e2/{ro[A3’3 + (A. — [tr/31} (4.10) and Q = [B(Ap — 141,219 — Z.) + 8.] — B(A,, 2,). (4.11) We used a radius parameter of ro=1.2 fm, Ap=122, Zp=54. The binding ener- gies, B(A, Z), of heavy nuclei were calculated from the Weizsacker mass formula [Marm 69]. Z2 _ 03M — 2.2)2 A1/3 ——A——, (4.12) B(A, Z) = COA — CIA?” — 02 90 with 00:14.1 MeV, C1=13.0 MeV, C2=0.595 MeV, and 03:19.0 MeV. For the emit- ted light fragments we used the measured binding energies, Bg, of the respective ground states [Waps 85]. At each temperature Tm, the parameter, f in Eq. (4.9) was adjusted to provide optimal agreement between the calculated final fragment distributions (obtained after the decay of particle unstable states) and the measured fragment distributions. This constraint reduced the possibility of inaccuracies in the predicted primary elemental distributions at high temperatures [Hahn 87, Fiel 87]. The values of f obtained for different Ten, are discussed in the last section of this chapter. III Details of the Decay Calculations The branching ratio for a state to decay by different channels has to be known for decay calculations. If known, tabulated branching ratios were used to describe the decay of particle unstable states. If unknown, the branching ratios were calculated from the Hauser-Feshbach formula, with additional constraints on isospins and pari- ties. The branching ratio for a channel c in the original Hauser-Feshbach formula is [Hans 52], I} 0.; F = 2' G- (4.13) where Z=IS+jI l=|J+Z| Ge: 2 Z T,(E). (4.14) Z=|S-jl lzlJ-Zl Here, J and j are the spins of the parent and daughter nuclei, Z is the channel spin, 5 and I are the intrinsic spin and orbital angular momentum of the emitted particle, and T1(E) is the transmission coefficient for the lth partial wave. By incorporating the parity and isospin conservations, we can write Ge as 91 CC = < T1,DTI.FT(3)1,DT(3)I.F|T1,PT(3)I,P >2 Z=|S+j| l=|J+Z| X Z Z {l1 + WPWDWFf-llll/Z} Tz(El- (4-15) Z:|S-j| l=|J—Z| The factor, [1 + 1rp1rD7rp(—l)’]/2 enforces parity conservation and depends on the parities 7r = i1 of the emitted fragment and the parent and daughter nuclei. The Clebsch-Gordon coefficient involving T1,p,T['D, and TLF, the isospins of the parent nucleus, daughter nucleus, and emitted particle, likewise allows one to take isospin conservation into account. For decays from states when the kinetic energy of the emitted particle is less than 20 MeV and I S 20, the transmission coefficients were interpolated from a set of calculated optical model transmission coefficients. For decays from continuum states when the kinetic energy of the emitted particle exceeds 20 MeV, the transmission coefficients were approximated by the sharp cutoff approximation; T1(E) = 1, for ($10 = 0, otherwise, (4.16) with 10 = (27r/hl7‘0l/‘l.U3 + (Ap - Aill/3l\/2#(E - VC), (4-17) where p is the reduced mass, and h is Plank’s constant. The calculation was restricted for the decays via n, 2n, p, 2p, d, t, 3He, 0: channels. The decays through 7 rays were taken into account directly to calculate the final particle stable yields. 92 IV Results The calculation was performed for Tam: 2, 3, 4, 5, 6, and 8 MeV. The measured fragment elemental and isotopic distributions and calculated final elemental distribu- tions for Egumfl / A = 5 MeV are compared for different values of Tea, in figures 4.2, 4.3 and 4.4. The solid points correspond to the fragment yields at 0 = 38° summed over all measured energies. The dashed lines in Figure 4.2 show the calculated isotopic distributions of primary fragments assumed for each temperature. The fitted param- eters, f, are indicated in the figure. The solid lines show the calculated final isotopic distributions obtained after the statistical decay of particle unbound fragments. The parameter, f, was adjusted at each temperature so that the calculated final isotopic distribution closely follows the trend of the measured isotopic distribution. Since these parameters, f, have been adjusted to reproduce the isotopic yields measured in this experiment, one must be very cautious about applying the results of these calculations to other reactions. The solid histograms in figure 4.3 and 4.4 represent final isotopic distributions obtained for each temperature. In general, the trends of the isotopic distributions are reproduced. Calculated values for the relative population probabilities for excited states of fragments were determined at each temperature Tcm from the calculation with full feeding taken into account. These values are compared in chapter 5 to the experi- mental data for a variety of emission temperatures and to determine whether these calculations can explain the observed relative populations. 93 14N+Ag, E/A=85MeV, 60=BB° I I I I I I I 2 MeV '1‘em = 5 MeV ass r==2 —-q —1 _ _ H O .fi *— _ _ —_ _ _ _ F __l l — .1— _ _ _— _ a H “’1 N Y'Ifi U'Vfi 'I'fi 'U'fi 1" H °u .p ~1— Yield (arbitrary units) K __ (D <: a a 11 II CD I: (D < ..J 10—2 10‘3 10—4 1 1 1 1 1 1 1 1 1 1 1 1 1E "I‘fi .1111 'V'fi U'U'fi II I I l l Figure 4.2: Element yields at 0 = 38° summed over measured energies. The dashed and solid histograms show the primary and final yields of particle stable fragments produced by the feeding calculations. Results for Tm=2, 3, 4, 5, 6, and 8 MeV with the corresponding parameters f are given in the figure. 94 1 4N+Ag, E/A=35MeV, 90=sa° r .l . I I I I I I I I I I I 561.1 UL: 738 IOBeBB 118 138 12c 14C MN ION 160 1803 100 f- 7Li “Li ”Be 1188103 123 “C 13C 13N 15N 150 170 .1 10-1 T,m 2 MeV 1 = 0.85 5 Yield (arbitrary units) 10‘4 Isotopes Figure 4.3: Comparisons of measured and calculated isotopic yields at 0 = 38°. The solid histograms show final fragment distributions for feeding calculations at Tem=2, 3, and 4 MeV. 95 1 4N+Ag, E/A=35MeV, ao=as° I l I I I °Li 6U 788 IOBeBBIIIBIIGBIIZCIHCIIGNIIGNIIOOIIBO: 7“ 9L1 988 “Be 103 128 11C 13C 13N 15N 150 170 ‘ rm = 5 MeV r = 1.67 Yield (arbitrary units) Isotopes Figure 4.4: Comparisons of measured and calculated isotopic yields at 0 = 38°. The solid histograms show final fragment distributions for feeding calculations at Tm=5, 6, and 8 MeV. Chapter 5 Nonstatistical Excited-State Populations Most models for fragmentation and emission of particle unstable complex nuclei in in- termediate energy nuclear reactions use statistical concepts to explain the experimen- tal observables such as the fragment mass distributions or the populations of ground and excited states of the fragments. One stringent test of these statistical models can be performed by measuring the population probabilities of a large number of states in a single fragment and comparing those to the predictions of statistical calculations. In this chapter, we present a series of comparisons involving the particle-unstable states of 6Li and 10B nuclei. Additional results for other nuclei will be discussed in chapter 6. We will compare yields of excited states of 10B to statistical calculations in the first section of this chapter. In the second section, we consider effects of rotation of the emitting system on the calculations of the population for high spin states. To obtain an independent measure of rotational effects, we have investigated the spin alignment of the emitted fragments by studying the angular distributions for the de- cays of particle unstable states in 6Li and 10B nuclei. This will be given in the third section. A short summary of the chapter will be given in the last section. 96 97 I Non-statistical Populations of States in 10B The data for particle unstable states of 10B nuclei were obtained from the coincident measurements of 6Li+a and 9Be+p as discussed in chapter 3. The excitation energy spectra of 10B which was given in Figure 3.14 showed the data from our measurement and the fits to the coincidence yields. In total, seven groups of states are considered, and within a given group of unresolved states, the population probability n is assumed to be the same for all states. Figure 5.1 shows the final relative populations for the different groups of states as functions of excitation energy. The solid points indicate the relative populations which are normalized so that £42.11. + l)n;c = 1, if the summation is restricted to the particle stable states of 1"B. The error bars reflect uncertainties of the background subtraction which were estimated by making different assumptions about the background coincidence yield and also the uncertainties in the efficiency which arises from the uncertainties in the position resolution of the gas proportional counters. If the intrinsic degrees of freedom of the system are thermalized at low density, the initial populations of the excited states of intermediate-mass fragments should be proportional to the Boltzmann factor exp(—E"/Tem), where Tea, is the temperature of the system at freezeout. The measured relative populations deviate significantly from this monotonic behavior. Indeed, the group of states at 6.0 MeV even exhibits a population inversion with respect to the lower-lying states at 5.2 and 4.8 MeV. The observed populations for 10B excited states can be compared with the results obtained from the sequential feeding calculations to determine whether it can account for the discrepancy. Calculations discussed in chapter 4 were performed that included the continuum states of fragments with Z S 13 for excitation energies up to E'=5A MeV, where A is the mass of the fragment. The results of these calculations are 98 “N+Ag. E/A=35MeV, 00=3s° 11111 ....... ZMeV 0.020 . ——4MeV« ---sMev 0.005 . l +§\\ 0.002 111111111111111f'411].11L111 4 5 6 7 8 9 E‘(‘°B) (MeV) Figure 5.1: Relative populations, ng, of different groups of particle-unstable states in 10B are plotted as a function of excitation energy. The vertical scale is normalized so that 2,,(2J1. + 1)n)¢ = 1, where the summation is restricted to the particle-stable states of 10B. 99 shown as the shaded bands in figure 5.1 for an initial temperature of 4 MeV. These bands depict the range of values for n,- obtained for different assumptions for the spins and parities of states with incomplete nuclear structure information. Clearly, these calculations do not reproduce the non-monotonic dependence of n.- upon excitation energy and the uncertainty due to unknown spectroscopic information is much less than the observed enhancement of the experimental populations at E‘=6 MeV. Thus the measured populations of particle-unbound states of 10B are inconsistent with thermal fragment distributions at the instant the fragment separates from the rest of the system. In the next two sections we consider if rotational effects on high spin states of emitted fragment can account for such deviations. II Angular Momentum Effects on Populations of States Angular momentum effects due to the rotation of the emitting system can cause the populations of high spin states of emitted fragments to be selectively enhanced. Such effects are not only relevant for compound nuclear emission; they can also influence observables for multifragment breakup processes as well [Snep 88]. We have explored this effect in the context of a compound nucleus model, and compared the prediction to our measured values. A Rotational Effects: Statistical Theory of Compound Nu- cleus Decay Let us first discuss these issues within the contest of the statistical theory of compound nucleus decay. In the statistical theory of the compound nucleus, the yield Y.- of an 100 excited state of an emitted fragment can be written as [Lu 72]: 00 110'” Z+Ii Y.- = 0.: z z: / 4E1. 41223.10) IKE; + 0.... - E: - Ea). (5.1) 1:0 Z=|Ip-I| ID=IZ-I.| Here Co is a factor independent of the spin and excitation energy of the excited state of the emitted fragment, I is the orbital angular momentum, Z is the ichannel spin, I1, and E; are the spin and excitation energy of the parent nucleus, ID and E5 are the spin and excitation energy of the daughter nucleus, 1, and E: are the spin and excitation energy of the emitted fragment and Q8... is the ground state Q-value for the decay. p(Eb, I D) is the level density of the daughter nucleus and T; is the transmission coefficient for the emitted fragment. For the purposes of these illustrative calculations, the level density of the daughter nucleus can be written in an exponential form [Eric 60] which is approximately valid for the range of temperatures considered here : E‘ Er0 NEBJD) = C (210 +1) eXI)( D - -—t‘) (5-2) TD TD where C is a constant and TD is the temperature of the daughter nucleus : _ /8Eb TD_ A0 (5.3) where we have taken the level density parameter a = 8 (MeV)“, and we approximate E23 by EI) = E; + Qgs _ E: — Vcoul(Ru)s (5.4) where ,u is the reduced mass of the fragment plus daughter nucleus system, and Vcoul(Ru) is the Coulomb potential when the fragment and daughter nucleus are sep- arated by a distance R“. Em, is the rotational energy associated with the daughter nucleus and is approximated by, 2(ID + %)2 Erot = (hC) QIDC2 (5.5) 101 where ID is the moment of inertia of the daughter nucleus. For simplicity, we assume a rigid body moment of inertia ID = 2/5 mDRf) where my and RD = 1.2/1}),3 (fm) are the mass and radius of the daughter nucleus. We also assumed a sharp cut-off transmission coefficient T; given by It I: It a In . h2(l + %)2 T1(Ep + Q54. ‘E.’ ‘E0) = 9(Ep + Q34. ‘5.- —ED - Vcoul(Ru) — W). (5-5) u The measured quantity in our experiment is the population probability n,- defined by : Y,- 2114-1, (5.7) 11,: which has to be calculated. Combining the information given above and integrating over energy, an explicit expression for n.- in the limit of full spin coupling is : e‘E'/TD oo Ip+I 2+1, (1+ '1')2h2 (10+ 'f'%)2h2 120 Z Z (2ID+1)CXP[_{—2#2Ri + 210 }/TDl = oz: lip-Illa: 12- II "i: (5.8) where Bo is a constant which, like Co is independent of spin and excitation energy of the fragment. Values for n,- were calculated for the excited states of 10B assuming Ip = 25, 50, 75 and 100 and assuming a mass Ap = 118, charge Zp = 50, and excitation energy E; = 200 MeV for the parent nucleus. The overall normalization constant N (1,9) for the calculated values of n,- was determined at each value of I, by minimizing the function xi Np..." x3= Z ("”‘P‘; "““V (5.9) N pomt c xp { where Npoim is the total number of data points, new, and n“)..- are the experimental and calculated values of the population probabilities, respectively, and Gem,- is the ex- perimental uncertainty. The results are shown in figure 5.2 along with the experimen- tal values of 71,-. Values of x3 = 2.4,1.8,1.5,1.7 were obtained for 1,, = 25,50, 75,100 102 l‘N+.4g, E/A=35MeV, 00=ss° IIITIIIIIIIITIIITTIITIIIIIT 0020 ~ '1 0.010 - — s" - 1 0.005 ~ — 0.002 ‘1‘111111111111111l11L1111 4 5 6 '7 8 9 E‘(‘°B) (MeV) Figure 5.2: Calculations for n,- in the limit of full spin coupling are shown as dotted, dashed, dot-dashed and solid lines for parent nuclear spins 1,: 25, 50, 75 and 100 respectively. The experimental values are same as those shown in figure 5.1. 103 respectively. Thus the agreement with experimental data is improved slightly for larger Ip. We see that larger values of the parent nucleus 1,, lead to larger enhance- ments in the populations of high spin states of the emitted 10B nuclei. 1, = 75 and 100 show enhanced populations of high spin states at E: z 6.0, 6.6, and 8.9 MeV, but the effects are nevertheless small compared to the experimental variations in Tlg. Larger rotational effects are predicted for larger values of Ip, but values of I, greater than 1,, = 88 are inconsistent with the conditions of stability for a metastable equili- brated compound nucleus calculated with the liquid drop model. These calculations also suggest that it is not possible by rotational effects to enhance the populations of the group of states at E: = 6.0 MeV without likewise enhancing the high spin state at E: = 6.56 MeV or the high spin doublet at E: z 8.9 MeV. Therefore we conclude that while rotational effects may play some role in the description of heavy fragment production, inclusion of these effects appears insufficient to describe the population probabilities experimentally observed. B Rotational Effects : Sequential Feeding Calculations In the last chapter, we have described a calculation to assess the effect of feeding on primary populations of states. The primary population for a fragment of mass A, charge Z, spin J, and excitation energy E“ was taken to be (equation 4.8) P oc P0(A, Z) (2J + 1) exp(—E'/Tcm), where Tam is the initial temperature, and the factor (2J + 1) signifies that the m- substates of spin J are equally populated. To explore rotational effects we performed calculations with enhanced populations of selected m substates. In these calculations, we approximated the primary population by J (m—fiz)2 PocPo(A,Z) ( Z exp{————- 2 20m 1) exp<—E‘/T.m). (5.10) m=-J 104 where m are the m-substates (—J S m S +J) of a given J. Here, fit and am describe the centroid and width of the distribution, respectively. The centroid and width of the m-substate distribution was chosen to be proportional to the rigid body moment of inertia. For simplicity we express fn and 0m in terms of the corresponding values for 12C fragments, i.e., (5.11) _ I m = mrzc Inc I Inc and am = (0m)izc (5.12) where I is the moment of inertia for the specific fragment being investigated, and 772(12C) and am(12C) are the centroid and width parameters for 12C fragment. Rotational effects were explored for a variety of values for fimc and (0m)izc and by using equation (5.10) to provide the primary distribution and following the sequential decay process as outlined in chapter 4. As for the calculation outlined in chapter 4, P0(A, Z) was adjusted so that the calculated and measured particle stable yields were in agreement. Calculated values for the final population probabilities n, are obtained from the complete feeding calculations which use these primary distributions. The calculated values for n.- nearest to the experimental data were obtained for 172120 = 6 and (0m)nc = 2.5. These calculations were presented by the solid line in figure 5.3. The populations of high spin states are enhanced by this calculation, but the enhancement for the high spin triplet of states (J = 2,3,4) at E: = 6 MeV can not be reproduced without simultaneously overpredicting the population of the high spin state (J = 4) at E; = 6.56 MeV and the spin doublet (J = 3,2) at E: = 9.0 MeV. In this respect, the results of these calculations are qualitatively similar to those obtained for compound nucleus expression and presented in the last subsection. 105 1"‘N+Ag. E/A=35MeV, so=3s° TITIIITfrlIIUIIrITIIIFFTIT] 0.020 - . 0.010 — + _ 0.005 - / . 0.002 11111 4 5 6 7 8 9 E*(1°B) (MeV) Figure 5.3: Calculations for n,- from the sequential feeding calculation Tm = 4 MeV, 771.120 2 6 and (Um)1zc = 2.5 are shown as the solid line in the figure. Experimental values for n,- are depicted by the large solid points. 106 III Decay Angular Distributions Most fragmentation models assume isotropic spin distributions for the outgoing frag- ments. When rotation becomes significant, enhanced populations of angular mo- mentum substates parallel to the axis of rotation can be expected. This issue has been explored via the measurements of the angular distribution and circular po- larization of coincident y-rays which accompany the emission of non-equilibrium intermediate-mass fragments [Tsan 88]. These experiments have shown that target- like residues which accompany the emission of intermediate-mass fragments are both strongly aligned and highly polarized with their spins parallel to the reaction normal, 7‘2 1. || (fibem x fins). Spin alignments of the non-equilibrium mass fragments cannot be precluded. Such spin alignments can be explored by the measurement of their decay angular distributions. Previous measurements have shown that the decay of 6Li-+ a+d in the reaction of 40Ar+197Au at E /A = 60 MeV is isotropic, consistent with avanishing spin alignment of the excited 6Li [Poch 87]. In this dissertation, the spin distributions of the heavier fragments have also been explored, and decay angular distributions for particle unstable states of 6Li and 10B are presented. A Experimental Angular Correlations for 6Li and 10B De- cays Figure 5.4 defines the angle convention used for investigating the angular correlations for the decay 1°B—> a+6Li. The reaction normal f1 J_ is a unit vector which is perpen- dicular to the reaction plane defined by the beam axis and the momentum of the 10B fragment. A polar angle 0d is defined to be the angle between the reaction normal fl; and the direction of the velocity v"; of the outgoing a-particle in the center of mass 107 'ler Figure 5.4: Coordinate system used to describe the a-decay of particle unstable excited states of 1"B. 0.; and «A; are the decay angles as defined in the text. 108 frame of the decaying particle unstable nucleus. Mathematically one has _1 17a ' (Ebeam X 6cm.) [170] lifbeam X 17c.m.| 0d = cos —1‘f‘(‘7beamxp) 14116..me1’ (5.13) =3 COS where q" and I3 = 151 + p} are the relative and total momentum of the a particle and 6Li, designated as particles 1 and 2 in this case; and fibem is the beam velocity. An azimuthal angle qbd is defined as the angle between the projection of the vector 6°, on the reaction plane and the direction of the total momentum of the 10B nucleus. Mathematically, one has -o 1 Cf") lq‘llPlsinfld 05d = cos’ if sin 0d # 0 (5.14) Thus, 05d and 9d distributions correspond to correlations in the reaction plane and correlations as a function of the angle with respect to the reaction normal, respectively. We have analysed both 0.5 and 03,1 angular distributions for particle decays from the excited states of 6Li and 10B. For 1OB—+6Li+a, relative energy spectra are obtained for specific gates on 0.1. Figure 5.5 shows the relative energy spectra for 10B —» 0+6Li for 0.; values ranging from 0° — 180° in steps of 20°. The solid circular points with error bars show the data points. The relative energy spectrum for 0d = 80° - 100° was fitted with the Breit-Wigner resonance parameters as described in chapter 3. The solid curve shown in the panel on figure 5.5 for this angular range shows a best fit to the data assuming the dotted curve for the background. Using the fitted parameters such as the relative population 12,, C12 and Ag, obtained from the angular range 0.3 = 80°—100°, calculated energy spectra were obtained for the other values of 9.1 using the appropriate efficiency function calculated for these angles. The solid and dotted curvas in the other panels show these calculations for the relative energy spectra and backgrounds, respectively. Y(E') (counts/20 keV) 109 Theta Angular Distribution for 1oB"—>01+“Li 100‘ ’ r f I 0 80 1’ 94:13-20" 300}- 0d=21_4oo 400 _:_ od=41—60° .. z .45 200 L— : 20 1 ‘ i . 0 f I ' ’5“ ‘ L : k I .. 94=61~80° : DESI—100° : 0.=101—120° : ‘°° l : 1 E 3 I 200 _ l 200 :‘l .1 1 l E E ‘ 1 200 ) L 100 _- . 100 F4 -E o n ”LU-4. . _---- . k _ ..... 1 E j 9 =121-140" 300; = _ o : 250 d ; 94 141 160 150 L 94=161-180° 150 . 100 50 0 Figure 5.5: Relative energy spectra for the decay 1°B—4 a+°Li at different values of the decay angle, 04 . 110 The data are reasonably well reproduced for all cases as shown in the figure, indicating that the same values of the relative populations can account for the relative energy spectra at all values of 0d. This suggests that anisotropies in the decay angular distribution for 10B are small. Figure 5.6 shows the relative energy spectra of l0B-—)‘5Li-+-cr for different gates in the angle m. The solid and dotted curves in the figure show the relative energy spectra obtained by using the appropriate efficiency function assuming isotropic decay and the parameter 72;, 012, Ag, obtained in the fit shown in fig 5.5 at the polar angles 0d = 80° — 100°. The data are reasonably well reproduced for all values of 45d, suggesting again that the anisotropies in the decay angular distributions are small. The decay angular distribution for 6Li—> a+d was also investigated. Figures 5.7 and 5.8 show the relative energy spectra of a and deuteron obtained for different gates on 0d and 03d by using parameters 71,-, C12 and Ag, obtained by fitting the relative energy spectrum for gate 0.1 = 80° —100° shown in the center panel of figure 5.7. These fitted parameters have been used for other ranges of angles and provided the solid and dashed curves for the relative energy and background spectra respectively. It can be seen from this comparison that the anisotropies in the angular distribution for the decay 6Li —+ a+d are also small. Similar comparison have been performed for other nucleus and no significant anisotropies with the decay angular distributions were observed. B Comparison with Statistical Calculations More detailed and precise measurements of the decay angular distributions for 10B were determined by fitting each of the relative energy spectra in figures 5.5 and 5.6 to obtain 11,- as a function of 9d and (25d. Uncertainties in the extracted population probabilities n,- were estimated using different assumptions for the background and Y(E') (counts/20 keV) 111 Phi Angular Distribution for 10B"-v01+5L_i 300- + < . 1' L ' zoo =0-20° . _ _ : ‘ _ _ . : M 300 — “’21 40° 300 _- WP“ 60 ‘. 200 I . l , 150 * 200:— 200} 9 -: 100 0 t i 100 L— o 100 .— " so i ' d 0 * r 4 i < : ’ t 300 El : 200:- l . 100 :— i ‘ O : f _ i 3'. -a. 250 (11,,=121-140o 250E- ¢¢=141-160° m; f 4.4—101 180° 200 200 C- l i 150 150:- 100} 100 i moi—lL : 50 1— 50 : .kj’u- Ada—‘__ db... ....... . . 1°. 1.; A L . E £7103) (MeV) Figure 5.6: Relative energy spectra for the decay 1°B—» a+°Li for different values of the decay angle (#4. Y(E') (counts/20 keV) 112 Theta Angular Distribution for 6I.i"->oz+d aooL = - ° = - ° : I 0,, O 20 500 94 21 40 2000} Od=41—60° I 400 i E 200 1— I 1600 :- l I l 300 : I C 1000}- 100 — zoo I E 100 5°° :' o I D: l I A A h- . 4‘ -- 3000 I" .f O 1) . ; 0561—80 1500 , 0381-100“ 1.‘ 95101—120" . ‘ q l m 1— | I ; 1000 1o .. I '50 a» I“? I‘ 600 I no i .. o A A . A A A A A A.._ _. A A A A A A -._-. _ 0‘ ‘——.. .. _.. 1250: 500 I . = ._ 0 = — o E 9.1 121 140 m 94 141 160 1.. od=1s1-100° 1000 ,- l i 750 ,- °°° ‘ 100 500 E- 1' s so 1 250 1:- 10' 0 . m- h-_~_. _ '“ tut-.4..-4_- M... ~-- .- z a 4 5 s a z 3 4 5 AAA E:(6L1) (MLV) . Figure 5.7: Relative energy spectra for the decay 6Li——» a+d for values of the decay angle, 04. 113 Phi Angular Distribution for 6I_.i"->01+d C 2500 1500f ¢d=O—20° : ( 2000 L a 10007- 1500 I 1000 600:- ? 1 °°° 0 L .8: 0g ~~~ 1 Ar. “-- o P Q 000? ¢a=61—50° 000 .3 000.- G : g m, 0 I v , 200 A 0 E1 0 >4 2000 ¢4=121-140° 1500 ¢d=21_4o° 1360 ‘ ¢d=41-60° 1000 I 760 ¢d=81-‘100° 4:101_120° .l ¢d=141-160° 1260 1000 ¢,=1s1-1so° 2 3 4 5 0 3 3 4 5 0 E*(°Li) (MeV) Figure 5.8: Relative energy spectra for the decay 3L3... a+d for values of the decay angle, 03.1. 114 by considering possible uncertainties in the efficiency function. Angular correlations are extracted for the first three groups of a-unstable states of 10B corresponding to excitation energies of 4.66 — 4.92 MeV, 5.0 — 5.26 MeV, 5.66 - 6.36 MeV respectively. In figures 5.9 and 5.10, we present the decay angular correlation as a function of 0d and 45.; respectively. These angular correlations were normalized to average values of unity. The excitation energy ranges and the spins of the states which contribute to a group are indicated in the figures. The anisotropies in these angular correlations are small for the first state and negligible for the second and third groups of states. The decay angular distributions were calculated using the statistical theory of the compound nuclear decay as discussed in section (IIA). In general, anisotropic decay angular correlations require non-uniform m-substate populations of the fragment ex- cited states. To explore this issue, m-substate populations were calculated. Within the statistical theory of compound nuclear decay, the population for each m-substate is given by n,(m,~) = Z e'Ei/TD < I,m,-IDmD|Zu >2< lmzZullpIp >2 l,m(.Z,u.ID,mD (1+1)°h2 + (ID +1)“2 2pr‘ 21' The various quantities are as defined before. Next we calculate the decay probability X910 + 1) eXPl-f }/TD] (5.15) from the fragment nucleus 10B by two coincident particles 01 and 6Li. This is given by Eli : Z < [1m112m2IZfl/f >2< lfmfol/fllgmg >2 d“ I;,m,.Zf.l/f.m1 Tzii z 114,...(12112—1—Tg 1P.(m.->1 (5.16) z 4727 where the subscript 1 refers to the values for 10B fragment, and the subscripts l and 2 refer to the relevant values for the emitted particles 0 and 6Li respectively. I f and 115 10B""—>01+6Li L41 . (a : 3“) E°=4.66-4.92MeV : 1.2 ‘ + "a .9 -__§_l ______ --—+ 0.6b....ln...l....l j'VIUYIU ’3 E (b : 2*,2‘,1+) E°=5.0-5.2snev ES; 1J3? on I c: 1.0 :- - - - - - J- - \\. Z A - 'U 0137 Q . v ' A a 0.6:414111...l..1.l..: g (c : 2*,4+,3‘) E°=5.66-6.36MeV : 1.2 _- 1 1.0:— - - - -+—-+—+ --§ 1 T 1’3 0.8 :- .1 0.6:..L11....l....lr.: O 50 100 150 6c1 (deg) Figure 5.9: The 0.1 dependence of the decay angular distributions are shown for various excited states of 1"B. The vertical scale is normalized to the average value of the distributions for each case. The dashed line shows the prediction from an isotropic decay. 116 °B*->a+°L1 1.4_ g (a: 3+) E'=4. ss- 4.92Mev: 1.2 _- ++ _: 1.0 3- ‘ —4— +- 4 ----- +—+ _F 0.6 E-f q 0.65 1 . L l 1 1 . . I 1 . . . I . 1 ‘8— E (b = 2+.2'.1+) E‘=5.0—5.2snev : V 1.2 _- _‘ g . : 1:. L _ _ _ ._ .9- \ 10* f * +3-4- f'TU : v . 5:: 0.6: . . . 1 A . . . 1 . . . . 1 . . g (c : 2*,4*,3") E°=5.ss-s.ssuev : 1.2 _- T E 9 3 1.o;+-+- -+- wr- (a 0.8 I— -1 06 D . . 1 A l 1 1 1 . l . . . 1 I . 0 50 100 150 ¢d (deg) Figure 5.10: The 4).; dependence of the decay angular distributions are shown for various excited states of 10B. The vertical scale is normalized to the average value of the distributions for each case. The dashed line shows the prediction from an isotropic decay. 117 m I are the orbital angular momentum and corresponding m-substate value for the decay of 10B by a and 6Li; Z I and u; are the channel spins and the corresponding m-substate values for this decay. P,(m,~) is calculated according to equation (5.15). The transmission coefficients T" are obtained from the optical potential calculations as discussed in chapter 4. The decay angular distributions for loB—r a+°Li using the parent spins Ip=25, 50, 75, 100 were calculated by assuming all the m-substates of the fragment spin to be populated according to equation 5.15. The results for 0d-angular correlations for the first group of states are shown in figure 5.11. The small anisotropy seen for the first group is consistent with the prediction for 1,, to be between 50 and 75. The data from other group of states are consistent with the prediction of isotropic population of m-substates. The constraint 1,, < 75 suggests that the rotational enhancement of n.- should be small, and cannot account for the large deviations of experimental relative populations from statistical calculations in which rotational effects have been neglected. IV Discussion We find that the populations of particle-unbound states of 10B cannot be reconciled with the thermal excited state populations. The measurement of decay angular distri— butions reveal the anisotropies in the angular correlations to be small, and rotational effects cannot be accounted for the magnitude of the observed discrepancy. Since the mass of the 10B is relatively close to that of projectile, simpler non-statistical production mechanisms cannot be excluded with certainty. 118 10B : 4.774 MeV : Full spin Coupling 1.4 (ave) 931 d0 ‘/ d_P do 06 1 1 1 1 l 1 1 1 1 l 1 1 1 1 l 1 1 Figure 5.11: The 0.; dependence of the decay angular distributions is shown for the first excited state of 10B. The vertical scale is normalized to the average value. The predictions from statistical calculations with I, = 25, 50,75, 100 are shown by dotted, dashed, dot-dashed and solid lines respectively. Chapter 6 Emission Temperatures In chapter 3, we have presented experimental data for the population probabilities of particle unstable states of intermediate mass fragments. In chapter 4, we have described sequential feeding calculations to determine theoretical estimates of the population probabilities of states starting with a thermal distribution of primary fragments at an initial temperature, Tam. Information about the emission tempera- ture can be obtained by direct comparison of the measured and calculated population probabilities. Because the particle stable states of the fragments are strongly pop- ulated by sequential feeding, the sensitivity of these comparison to sequential decay correction may be somewhat reduced by comparing ratios of population probabili- ties of states within the same fragment. In this chapter, we present the results of such comparisons between the experimental and theoretical population probabilities calculated for a range of emission temperatures. Experimental population probabilities for 40 groups of particle unstable states of intermediate mass fragments are presented in Tables 3.2, 3.3, 3.4 and 3.5 of chapter 3. Before comparing these results individually to feeding calculations, it is instructive to make overall comparisons between the measured and calculated population prob- abilities and ratios of population probabilities. To provide a global test for statistical 119 120 calculations, we have performed a least-squares analysis by computing 2 _ if: (31¢pr — ycaln') . (61) 1:1 012 for each initial temperature in the calculation. Here gem.- and yeah,- are the experi- mental and calculated values of the populations or ratios of populations and u is the number of data points. In the case of the ratios of population probabilities, these data points are not completely independent. This form of comparison was chosen in order to provide a measure of the agreement between measured and calculated quantities. Restricting the summation to only the mathematically independent quantities, such as the population probabilities would have made the x3, function unduly sensitive to the feeding correction to the population of particle stable states. The uncertainty 0,- in equation 6.1 is given by a? = ‘73pr + 0'3,” where (rem,- is the experimental uncertainty, and acm- reflects the range of calculated values obtained for different assumptions for the spins, isospins, and parities of low-lying states where these infor- mations are incomplete. The range of calculated values was determined by repeating the calculation with different spectroscopic assumptions until the sensitivity of the calculation to those uncertainties could be assessed. Values of xi according to equation (6.1), were computed for combinations of population probabilities and the ratios of population probabilities. The results are presented for four groups : Z = 3,4; Z = 5,6; Z = 7; and Z = 8, according to the fragment charge. Figure 6.1 shows values for X: as functions of temperature (Tem = 2 — 8 MeV) of the primary distribution in the feeding calculation. The solid lines depict values for X3 where both the independent population probabilities and all the ratios of populations have been included, and the dash-dotted lines show X: where just the ratios of population probabilities are included. Results for lithium and beryllium isotopes are shown in the upper left hand window of the figure. The x3 functions for these isotopes display a minimum at about T m z 3 MeV for only the 121 1"'N+.4g, E/A=35MeV, so=ss° 2 g—s OI IIVVY‘IIIUUIUUWT'IU" 11"! r Pf. I I r! I I I I I I V TI—r m_ 4; 03 0°C N 4:. C) CD Figure 6.1: Results of the least-squares analysis for four groups of fragments. The solid lines depict X5 calculated for a combination of population probabilities and the ratios of population probabilities. The dashed lines show X3. when just the ratios of population probabilities are included. 122 ratios of population probabilities and Tem z 4 MeV for all the quantities combined. Similar calculations for Boron and Carbon isotopes are shown in the upper right hand window. Minimum value of xi occur in the neighborhood of Tenn 2: 3 MeV for both cases. In the lower left hand window, the results for nitrogen isotopes are presented. For this case, a minimum in the neighbourhood of Ten. = 3 MeV is obtained for xi when both the population probabilities and the ratios of population probabilities are included. This minimum shifts to Tem z 4 MeV when xi is restricted tojust the ratios of population probabilities. In the lower right hand window, the results for oxygen isotopes are given. Here very few groups of states are detected, and the location of the minimum in the xi functions are not well determined. Comparisons of the temperature dependence of xi for different elements do not reveal any unambiguous trends. The values of Ten, that correspond to the minimum value of xi do not appear to be strongly dependent on the charge of the fragment. To get an improved measure for Tcm, we have combined the results for all fragments. Figure 6.2 shows the corresponding values of xi. The solid curve in the figure depicts the values of xi where both the independent population probabilities and the ratios of population probabilities have been included. In addition, the dashed line in the figure indicates the values for xi where the sum in equation 6.1 runs over only the independent population probabilities, and the dash-dotted line shows the correspond- ing values where the sum includes all the ratios of population probabilities which may be constructed. Minimum value of xi in these comparisons occur for emission tem- peratures of Tem z 3 — 4 MeV. Also shown as the dotted line in the figure is the xi value for the single comparison involving the 5Li ground state and 16.66 MeV excited state. Calculations indicate that the relative populations of 5Li excited states are rather insensitive to the sequential feeding from heavier particle unstable nuclei [Chen 88]. For the 5Li states, the minimum value of xi occurs at Ten, = 4 MeV, 123 14N+Ag, E/A=35MeV, so=ss° 20 ' [I ' ' I ' . I I . 1 e, _ \ .' _ ‘ . . 1 .. 1- ‘ ' 1 5 1 15 — 1 5 - .. I : \ . . q ( _ - l 4 1 01>? - 1 - 10 *- — Figure 6.2: Results of least-squares analysis for a combination of all fragments. The dashed, dash-dotted and solid lines depict xi calculated for the population (proba- bilities, the ratios of populations probabilities, and the summation of the population probabilities and ratios of population probabilities respectively. 124 consistent with the emission temperature extracted from the 1“N +197Au system at E /A = 35 MeV (see figure 1.2). Thus the emission temperatures of Tem = 3 - 4 MeV obtained from heavier particle unstable nuclei are in average slightly lower than those extracted from 5Li. Even for Tem z 3 — 4 MeV, the values of xi shown in figures 6.1 and 6.2 are rather large indicating significant discrepancies between measurement and calculation. This issue was investigated in greater detail for states of 10B in chapter 5. To explore this issue for other nuclei, we now present detailed comparisons between the experimental and calculated population probabilities and the ratios of population probabilities at Tem = 4 MeV. For these comparisons, we adopt the conventions in figure 1.2, in which an apparent temperature Twp is defined by the relation 11" = exp1—1E:— £714.41. (62) "j where fiapp = 1/Tapp. If j denotes the ground state, from equation (6.2) and the definition of population probability, we obtain (213.5. + 1),“ = €Xp(—E: ((39131))9 (6’3) where 1105- is the spin of the ground state. Equations 6.2 and 6.3 define Tam, in terms of ratios on measured or calculated values of n,- and n,-. Sometimes the values for Tapp provided by equations 6.2 and 6.3 are negative or infinite. To avoid this singular behavior of the apparent temperature, we will extract and assign an uncertainty to 3.,” rather than Tapp. In figures 6.3, 6.4, 6.5, and 6.6, we present values for flaw (on the lower axis) and Tam, = ”6,,pp (on the upper axis) for population probabilities and ratios of population probabilities for isotopes with Z = 3,4; Z = 5,6; Z = 7; and Z = 8, respectively. The solid points represent the values for flaw obtained for experimental population probabilities or the ratios of population probabilities. The histograms 125 represent corresponding values for (3991) obtained from sequential feeding calculations starting with an initial temperature Tem = 4 MeV. The uncertainties in the calculation are designated by the spread of the histogram which is shaded in the figure. In this figure, only those cases are plotted for which both the calculated or experimental uncertainties are smaller than the dynamic range of the figure. Values for (32:99 and Tapp obtained from the population probabilities of two groups of states in 5Li and 6Li, and three groups of states in 7Li and 7Be are shown in figure 6.3. (The relevant populations were given in table 3.2 of chapter 3). The experimental data in figure 6.3 for the population of states in 5Li, °Li, and 7Li are comparable to the results shown in figure 1.2 obtained in previous measurements at similar energies (see figure 1.2 and [Poch 87, Chen 88]). The effect of sequential feeding is minimal on the widely separated ground state and 16.66 MeV state of 5Li [AB = 16.66 MeV]. From these two states, an apparent temperature of 4.0 :I: 0.26 MeV can be obtained from the ratio of population probabilities. This value for Tapp is identical to the value of Tcm = 4 MeV which was used to compute the corresponding quantities in the sequential feeding calculation. In general, the calculated apparent temperatures are similar to the measured ones for most other transitions. Notable exceptions are the ratios involving the 6.64+7.47 and 11.24 MeV excited states of 7Li. The values for Ba”, and Tapp obtained from measured and calculated population probabilities for two groups of states in 8B, seven groups of states in 10B and three groups of states in 11C are presented in figure 6.5. For convenience of presentation, the ratios are labelled in the figure in terms of the groups of states discussed in chapter 3. Compared to the first group of excited states of 8B, the measured populations for the second group of excited states of 8B were larger than one would expect from the sequential feeding calculation. Much larger discrepancies are observed for the excited states of 10B. Measured ratios involving the second group of excited states at aPP 0° 4 3 2 1.33 1 I I I I I I ‘_°'_ 7Be7_21/7Be‘.57 ___—”"— 7Be13.73+'7.21/ 7394.57 7387.21/7Bemb14 .. 7Besxrswzr/WBB31.111141 + 7Be4.57/7393111111. —°_' 71411.24/7Lis.s4+7.47 1 + 714111.24/71-44133 ___".J— 7L16.64+7.47/7Li4.83 "' 71111.24/ 7Liltabla 7ms.s4+7.47/ 714131.111. . ’ , 7L1ass/7101:1111». + shamans/“mama ‘H"°— 6144.31+5.ss/6Limb1e + 61412.11311/ ol‘istable shiatsu/sun... .l...l...l.11.lrr..ll O 0.25 0.5 0.75 1 19.... 0491“) Figure 6.3: Experimental values for 6.”, and T.” are shown as the solid points for excited states of Li and Be isotopes. The histograms represent the results of sequential feeding calculation with an initial temperature Tem = 4 MeV. 127 Ta,pp (MeV) -43 00 55 £3 1 (LEV? ()15 I I I l grp_3/stable" grp_2/stable I nc grp_1 / stable _] srp_7/srp_6" srp_7/srp_5 ' srp_7/srp_4 ' grp_6/grp_4 I grp_5/grp_4 l srp_7/srp-3 1 srp_6/srp_3 1 srp_5/srp_3 | srp_4/srp_3 . srp_7/srp_2 I srp_6/srp_2 srp_5/srp_2 : r _4 r _2 s p /s p |1H53 I 1 1 1 1 I 1 I I 1 1 srp_3/srp_2 srp-7/srp_1 srp_6/srp_l sr9_5/srp_1 srp_4/srp_l srp_3/srp_1 srp_2/srp_1 grp_7 / stable grp_6 / stable 0 grp_5 / stable grp_4/ stable 0 grp_3 / stable 0 grp_2 / stable . grp- 1 / stable _1 + srp_2/srp-1 '0' , grp_2/stable:| 8B E 5 5L— grp_ 1 / stable 1 l l l i l l l l L L 1 l 1 L k l 1 l l l I l 1 l —0.5 0 0.5 1 1.5 2 11.... (MeV“) I l Figure 6.4: Experimental values for (6.1,, and T.” are shown as the solid points for the groups of excited states of B and C isotopes described in table 3.3. The histograms represent the results of sequential feeding calculation with an initial temperature Tm = 4 MeV. -23 9° 55 £3 1 (LET? (315 I if I I grp_9/grp_7 ”I srp-8/srp-7 grp_9/srp_6 grp_B/srp_6 srp_9/srp_5 srp-8/grp_5 grp_7/srp-5 grp_9/srp_4 grp_B/srp_4 srp_7/srp—4 grp_9/srp_3 srp-8/srp—3 srp_7/srp-3 grp_G/srp_3 srp_9/srp_2 grp_B/srp_2 srp-7/srp_2 srp-6/grp_2 grp_S/srp-2 grp_4/srp_2 srp_9/srp_1 grp_B/srp_1 srp-7/srp_l grp_B/srp-1 grp_S/srp_1 grp_4/srp_1 grp_9 / stable grp_B / stable grp_? / stable grp_6 / stable grp_5 / stable grp_4 / stable grp_3 / stable grp_2 / stable grp_ 1 fistableJ grp-2 grp_ 1 grp_Z/stable] 13N rp_ 1 / stable A 1 l J J l l l L 1 L 1 l 1 L l 1 fi 1 1 L 1 l l J —O.5 O 0.5 1 1.5 2 5.... (MeV“) Figure 6.5: Experimental values for H.” and T.” are shown as the solid points for the groups of excited states of 13N and 1‘N described in table 3.4. The histograms represent the results of sequential feeding calculation with an initial temperature Te... = 4 MeV. “N -0J5 129 0:5 0133 I grp_B/srp_1 grp_Z/srp_1 grp_3/stable grp_Z/stable grp_l/stable srp-4/srp_3 grp_4/grp_2 grp_B/srp_1 grp_Z/grp_1 grp_4/stab1e grp_3/stable grp_Z/stable grp_l/stable l l l I l 160 |_________________l|__.____l 23 Figure 6.6: Experimental values for ,3.” and T..pp are shown as the solid points for the groups of excited states of 16O and 180 described in table 3.5. The histograms represent the results of sequential feeding calculation with an initial temperature Tem = 4 MeV. 130 E " z 5.1 MeV and the third group of excited states at E“ as 6 MeV are very strongly in disagreement with the calculations. In fact the ratios between the third and second groups give large negative apparent temperatures in contrast to the predictions from the calculation. The groups of states for 11C on the other hand, are well described by the calculations. Because of the large uncertainties in the calculation for the ratios between excited states of 11C, however, these ratios are not plotted. In figure 6.5, we present Em, and Tapp for two groups of states of 13N and nine groups of states of 1“N. The structure and the population probabilities of these groups were listed in table 3.4. The experimental populations for the first group of 13N are in agreement with the calculations. The deviation for the second group is large. The population probabilities which are defined with respect to particle stable yield in case of 1“N deviate significantly from the calculation for Tem = 4 MeV. For these cases, the calculation predicts somewhat more feeding to the particle stable states than observed. Slightly better agreement for the population probabilities are obtained for Tem = 3 MeV. For the ratios of population probabilities, however, the agreement is actually better for Tem = 4 MeV, and with the exception of the ratio of group 6 to group 3, the overall agreement is rather good. In figure 6.6, we have plotted 3,”, and Tam, for the populations of four groups of states in 160 and three groups of states in 180. The structure of the groups and the corresponding population probabilities are given in table 3.5. The overall agreement between the data and calculation in case of 180 states is somewhat better than that of the states of 16O. The second and third groups in 160 are combinations of states that are far apart in excitation energy. The median energies obtained for the groups are rather close to each other which make the discrepancy large. To summarize, it is observed that about half of all the experimental population probabilities and and one-third of the ratios of population probabilities showed signif- ._ *3 131 icant deviations from the predictions of statistical calculations. The largest discrep- ancies in the ratios of excited state population probabilities are observed for 10B, and for the population probabilities, the largest discrepancies were observed l“N. Whether these discrepancies would be less in experiments with heavier or lighter beam where simple fragment production modes are suppressed is an open question which should be addressed by future investigations. Chapter 7 Summary and Conclusion In this dissertation, we have presented a detailed study of the relative populations of particle unstable states of intermediate mass fragments for the reaction 1"N on m“Ag at E /A = 35 MeV. In many thermal models, the populations of excited states at freezeout are expected to follow a Boltzmann distribution weighted by the emission temperature of the system, Tm. Tests of this freezeout assumption were made by comparing relative populations of a large number of particle unstable states to the predictions of statistical calculations. Experimentally, the populations of particle unstable states were obtained by mea- suring the decay products in coincidence using a new high resolution position sensitive hodoscope. Numerical techniques were developed to model the detection efficiency of the hodoscope. Experimental population probabilities were extracted by fitting the spectra for the true coincidence yield to an appropriate R-matrix or Breit-Wigner formalism. Even with good energy resolution of the hodoscope (50 keV for 4.774 MeV state of 10B), it was not possible to isolate each of the excited states in the experiment, and some neighboring states were grouped together statistically. In this fashion, relative populations with respect to the corresponding particle stable yields were obtained for 40 groups of states in Li, Be, B, C, N and O isotopes. 132 133 Extensive calculations were performed to predict the relative populations of these states while taking into account the sequential feeding effect from heavier particle unstable nuclei. In these calculations, discrete and continuum states of nuclei with Z < 13 were thermally populated and allowed to decay sequentially. Unknown spins, parities and isospins of lower lying discrete states were assigned according to pri- mary distributions obtained from the non-interacting shell model. Calculations were repeated with varying assumptions until their sensitivity to unknown spectroscopic information could be assessed. Comparisons were made between the measured relative populations of particle unstable states and the corresponding calculated values for different initial temper- atures, T em. To allow a global comparison between the experimental data and the calculated results, xi functions were computed for the population probabilities and for the ratios of population probabilities between states of the same fragment. By examining the temperature dependence of these x2 functions, the best agreement between calculated and measured quantities occured at Ten, = 3 — 4 MeV. Even for emission temperature Te", 2 3—4 MeV, the magnitude of the x2 functions were rather large suggesting that many of the excited states of intermediate mass fragments may not be thermally populated. The relative populations of the excited states were studied in detail for excited 10B nuclei. Large discrepancies between the calculated and measured population probabilities were observed. Calculations were performed to see if rotational effects when imbedded in a statistical description can account for this deviation. These calculations indicate that rotational effects can make the population probabilities deviate significantly from calculations which neglect rotational effects. The trends induced by rotation however, still differed from the the trends observed experimentally. Rotational effects were further explored by measurements of the decay angular distributions of 10B fragments. The anisotropies 134 of these decay angular distributions were observed to be small. When this information was used to construct the calculated values of the population probabilities, it could be considered that rotational enhancements of high spin states are also likely to be small. All this evidence indicates that rotational effects are not likely to be the sole explanation for the discrepancies between the measured population probabilities and statistical calculations which neglect rotational effects. Assuming an emission temperature Ten, 2 4 MeV, comparisons were made be- tween the measured and calculated population probabilities and the ratios of popu- lation probabilities for states of other fragments. For roughly half of the population probabilities and one third of the ratios of population probabilities, the disagreement between calculated and measured quantities were substantial. This observation is not presently understood. Several explanations can be offered. 1) Since the masses of the fragments considered in our analysis are close to the mass of the projectile, simple non-thermal production. mechanisms cannot be excluded. These mechanisms may not thermally populate the fragment excited states. This possibility should be explored via additional measurements with heavier or lighter projectile nuclei. 2) There is a possibility that the spins or branching ratios of some of the states ana- lyzed in this dissertation may be incorrectly assigned in the literature. The extracted populations are sensitive to this spectroscopic information. Incorrect spectroscopic information will result in incorrect extraction of the corresponding population prob- abilities. For the states of 10B where large discrepancies were observed, however, the relevant spectroscopic information appears well established and the discrepancies ap- pear to be real. 3) Some of the measured peaks could contain background peaks from three body decays or from the decays to daughter fragments in particle stable excited states. 135 Additional measurements with improved excitation energy resolutions would help to clarify this issue. 4) Some heavier particle unstable nuclei could decay to nuclei we observed with branching ratios which differ significantly from those predicted by the Hauser-Feshbach model of statistical decay. This could lead to an enhancement or a depletion of the populations of selected excited states. 5) It is conceivable that the excited states of the fragments could be thermally popu- lated at a high density where the energies of the levels differ significantly from their asymptotic values. If the evolution of the system to zero density is adiabatic, the level population could be preserved while the ordering of the levels could be changed leading to the appearance of non-thermal populations of the isolated fragments. It is not presently clear how to best address questions 4 and 5, and therefore the question remains open. The best overall agreement between the measured and calculated population prob- abilities occurred for emission temperatures of about Tem % 3 — 4 MeV. The emission temperature extracted for 5Li fragments is slightly higher (Tem a: 4 MeV), and is consistent with the systematic incident energy dependence of emission temperatures extracted from 5Li fragments previously reported. Appendix A Electronics The block diagram of the electronics set up for a pair of Light particle and Heavy fragment telescopes is shown in figure A.1. The analog signals from the :1: - 3; position sensitive detectors, silicon detectors and NaI detectors were preamplified, shaped and amplified, and then were sent to the peak sensing ADC’s. Logic signals were extracted from fast signals derived from the second element of the Si telescopes. These logic signals were split into a two-way splitter. One signal from the splitter was sent to a discriminator with high threshold which provided the energy threshold for different particles. The other signal was sent to a constant fraction discriminator with low threshold, which was used to obtain the timing information. The output signals of both discriminators were sent to a coincidence unit and the output from this unit was fanned out to generate telescope logic signals. One of the signals was sent to a downscale unit to get particle inclusive data. The second signal was used to generate input signals for TDC stops and bit registers. The third output from the fanout was sent to a 32 channel majority logic unit which provided a coincidence output for coincidence between any two pair of light particle or heavy fragment telescopes. The fourth signal from the fanout was sent to a logic OR unit whose output was sent to a coincidence module, which generated an output for coincidence between light particle and heavy fragment telescopes. In this way the trigger levels could be adjusted for 136 137 .—x-y— L111 '; m AIII LIGHTPARTICUETELESCOPE * .L i _ * ISI 1 9 - X-Yu-I HEAVY FRAGMENT TELESCOPE Figure A.1: The electronics diagram for a pair of Light particle and Heavy fragment telescopes. 138 individual signals if the rates of different types of signals are different. In practice, the rates turn out to be not so different in our experiment and a majority logic unit was used for all triggers. The trigger for the experiment consisted of downscale telescope events, coincidence between light particle detectors, coincidence between heavy fragment detectors, and coincidence between light particle and heavy fragment detectors. A dead time circuit (not shown in the figure) was used to inhibit the CAMAC data acquisitions system while the computer was busy. Appendix B Details of the Efficiency Calculations The efficiency functions €(E‘, E51”) are usually obtained by performing Monte Carlo simulations for the emissioh and decay of the respective particle unstable nuclei. For most detection geometries, such simulations are very time consuming because of low detection efficiencies. We have avoided the inherent inefficiencies of such Monte Carlo simulations by calculating the efficiency function through direct integration of the two particle co- incidence cross section over the detector geometry. For simplicity we assume the decay to be isotropic in the rest frame of the particle unstable nucleus and the lab- oratory production cross section of the particle unstable nucleus to be independent of the excitation energy E‘. The laboratory two particle coincidence cross section can then be given in terms of the center of mass excitation energy spectrum and the “common” laboratory production cross section for the particle unstable nucleus da(Etot1 Qtot)/dEtotthot by d0'(EH, 0“, EL, 9L) = 6(Erela ”cm, Etot, ntot) d0'(Etoty fltot) LM (B 1) dEHdnl-l, EL, dQL) C(EH, {2“, EL, 0L) dEtotdntot 411' (IE: 1 . where 6(Eml, flan, Em, 0.01/MEWS)“, ELQL) is the Jacobian for the transformation from the center of mass coordinates Em, 9cm and the laboratory coordinates EM, 0.0. 139 140 of the parent particle unstable nucleus to the laboratory coordinates for the detected heavy (H) and light (L) decay products. This Jacobian is given by a(Erela 0cm» Etots ntot) = prl-I(EL ‘I' mL)(El-I + mH) 3(EH, 9H. EL. QL) PimptodEfim + WILKES" + ms) x [1-{{(Pf.+13'L'I3'H)(EH+mH) 0221+ 151-50113. + m.) 1’ } x { (EL +13“ +mL+mH)2(EL+mH) >< (EH +mH)(Erel+mL+mH)2 }-1 — { (paw-5141112.. +7744)? + (p11 +141. 713101121. +m1.)2 } x {(EL+EH+mL+mI-I)2 X (EL + mL)(EH + mg) }-l J. (3.2) Here mL and mu are the masses and EL and EH are the kinetic energies of the light particle and the heavy fragment, respectively. To convert the two particle cross section into the measured two particle distribution function dN/dEHmdQHdeLmdflLm at the measured laboratory angles, Slum and fle, and the deduced laboratory energies, EH,“ and ELm, which include a correction for energy loss in target and detector foil, one must consider the distribution of interaction points, h(fi8t), in the target beam spot which causes the actual emission angles (IL and {In to differ from the values, QLC and Que, deduced by assuming the reaction to occur at the center of the target. In addition, one must account for the difference between the corrected energies, EL and E3, of the particles after the entrance foil of the detector telescopes (calculated by assuming the reaction to occur at the center of the target) and the original energies, EL and EH, inside the target. These differences are represented by the distributions, AL(EL, EL, tm) and AME“, Emily), of energies losses (including energy loss straggling) in the target and entrance foil of the telescope, where ttg, 141 is the position inside the of the target. Likewise, the angles, (IL and (In, of the particles at the entrance foil of the detector telescopes differ from the angles inside the target according to the distributions, 6L(QLC,S~IL,t,g,) and 6H(QHC,OH),ttg,, for multiple scattering in the target. Finally, one must consider the detector angular resolution functions, Again, 0“,“, E3) and AL(f2L, QLm, EL) (which include the effects of multiple scattering in the entrance foil of the telescope and the energy dependence of the position resolution), the detector energy resolution functions, RH(EH,EH,,,) and RH(EH,EH,,,), and the detector efficiencies DL(EL,I~IL) and DH(E'H,fZH) which account for the loss of efficiency in the telescope due to multiple scattering in the telescope stack. In terms of these quantities, one obtains dN ._ dEHmdQHdeLmdQLm - QN,,, / da,.dt.,.dEHdELdoHCdnLCdEHdELdQHdQL >< Damn. QL)DL(EL, fiL)RH(EHa EHm) X RL(EL, ELmlAHmH. 011m. EnlALmL, 9L1... EL) x AME“, EH,t,8,)AL(EL, EL,1.,,)5H(0H.,0H,1.,.) X5L(QLC. (IL. ttgt)j(QH1 an)j(QL. 91.12) d0'(EH, 9H7 EL, QL) dEHdQHddELdQL ’ (B3) (this) where Q is the number of beam particles which traverse the target during the experi- ment, th, is the number of target nuclei per unit area and J (flu, Duo) and J (91,, QLC) are the J acobians of the transformation from the spatial coordinate system centered at point of interaction in the target and the coordinate system whose origin is at the center of the target. To obtain the yield Y,,(E;m) experimentally, one bins the data with respect to the measured energies and angles, calculates the mean excitation energy E3,“ correspond- ing the energies and angles of these bins and stores the data in the correct element of the array YC(E" mea ) corresponding to calculated value of Er‘nea. We designate this 142 operation as dN E“ — dE de md mdfl m m . B.4 Y( "165)-; bin.- H L Q” H dQL dEHmdQHdeLmdQLm ( ) (E;=E mea) Finally, to simplify the calculation further, we approximate Eq. (3.3) by dn( E;+A/2 E. = a: It I- Y.( m...) 21:4 dEE; ”L 51.4] -412 dE (E E...) EA “(Bil é(E; Es...) (3.5) j dE. E.=E. I Here E3?“ — E; = A, and A is chosen sufficiently small that this approximation is accurate. Then, the averaged efficiency E becomes 1 as; 3;“) = QN,,, 2L dELdeHmdflHmdQLm—A- (E:=Emea) ' E;+A/2 x / dE" s;—A/2 x { /d2f}gtdEHdELdflHchLchHELdQHdQL XDH(EH1(2H)DL(EL1S~2L)RH(EH1 E11m) X RL(ELa ELM/111mm 911m, EH)AL(QL1 QLma EL) >< A11(511. EHchclALIEL, ELattgt)6H(QHcafiH1ttgt) ><611(fcha (2L1 ttgt)j(nf‘la QHC)'j(QL, QLC) X h(7.‘. )a(Erela 9cm, Etot, 9,0,) d0(Etota ntot) _1_ wt 8(EH, QH, EL, QL) (“51“”det 47r . (B.6) This expression is relatively straightforward to evaluate. For the efficiency calcu- lations given here, the integrations over E', 77131. E3, EL, QHC, QLC, EH, EL, OH, {IL are performed by a Monte Carlo sampling algorithm. The cross section, da(E,o,,9,0,)/dE,o,dQ,o,, used in Eq. 17, was determined by fitting the inclusive 143 data for particle stable nuclei of the same mass. The detector resolution and depen- dence of this resolution on the particle energy and mass were determined from the experimental and calibration runs. Since no time is spent calculating the trajectories of particles which pass between detectors, direct integration proves to be considerably more efficient than Monte Carlo event simulation for calculating the efficiency. Direct comparisons between the two techniques have been made using calculations for an 18 element hodoscope used in measurements of 160 induced reactions on 197Au at E/ A = 94 MeV. The Monte Carlo event simulation was performed with the simulation program of Ref. 2. To better than 1% accuracy the present efficiency calculation agrees with calculations using the event simulation program. For this case, however, direct integration is about a factor of 20 faster than Monte Carlo event simulation. For determination of the background yield, mek( 13;“), it is necessary to perform an identical event binning for the product, our“, of single cross sections (see Eq. 13) as was performed in Eq. 17 for the coincidence yield. For position sensitive detectors, it is considerably easier to fit the singles cross sections 0L and on with a moving source parameterization and integrate the parameterized cross sections than to perform a mixed single-particle event analysis. Since the excitation energy E" is rather trivially related to the relative momentum Ap, it is equivalent and actually easier to define a correlation function [1 + Rbuk(E‘)] = [1 + Rback(Ap)], which satisfies the equation Ybuk(§H,5Ll = CUL(5L)0H(5H)(1+ Rback(E'))- (B-7) Summing both sides of equation for a fixed excitation energy E11,“ provides 144 Yback(E;e.) = Cl1+Rb8Ck(thea)l X Z / ‘ dEHdeLmdQHmdQLm (E :15 i me.) 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