J? 1 ..-‘ '.n<“, n... q,‘ ”,2...” I" .nl' 3 1293 00885 This is to certify that the dissertation entitled Multi-fragment disintegrations in intermediate energy nucleus-nucleus collisions presented by Yeongduk Kim has been accepted towards fulfillment of the requirements for Ph.D;. degree in Physics & Astronomy C. K. Gelbke W Major professor Date Oct. 29, 1991 MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 MULTI—FRAGMENT DISINTEGRATIONS IN INTERMEDIATE ENERGY NUCLEUS-NUCLEUS COLLISION S By Yeongduk Kim A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1991 592’635‘7/ AB STRACT MULTI-FRAGMENT DISINTEGRATIONS IN INTERMEDIATE ENERGY NUCLEUS-NUCLEUS COLLISION S By Yeongduk Kim The possibility for the highly excited nuclear matter to break into multi- fragment final states has been predicted by various theoretical calculations. Experimental information about low-density nuclear equation of state may be obtainable from detailed studies of multi-fragment emission processes. A new low-threshold 41: charged particle detector array, the MSU Miniball, has been constructed for the study of multi-fragmentation. Intermediate mass fragment(IMF) emission has been studied using beams of 14N at E / A=50MeV and 36Ar atE / A=20,35MeV on 238U and 197Au targets. MostIMFs are emitted in central collisions characterized by large charged-particle multiplicities. Energy spectra and angular distributions indicate the non-equilibrium nature of IMF emission. Time-scales for multi-fragment emission, key in distinguishing the various reaction mechanisms, have been studied using two-fragment correlation functions. The experimental two-fragment correlation functions were found to depend mainly on the reduced relative velocity, vr = Vrel/ JZ 1+Z2 , of the ed fragment pairs. It is therefore possible to sum over different pair combinations with little loss in resolution. From the comparisons of the inclusive experimental correlation functions with the classical expression of Koonin-Pratt formula, mean -M $.— . it: emission times of the order of 100-200 fm/c have been extracted with about 50% uncertainty. Three-body Coulomb trajectory calculations were also performed to estimate the uncertainty from the interaction with the heavy residue. A more detailed study indicates that fragment emission in central collisions begins at the very early stages of the reaction and continues throughout the later equilibrated stages. ACKNOWLEDGEMENT I would like to express my heartful thanks to my advisor, Professor Claus- Konrad Gelbke, for his invaluable supervision over my research and support. Without his continuous encouragement, intuitive suggestions, and enthusiasm for best experiments, it would not have been possible for me to finish this thesis. I have learned how to do physics experiments right here NSCL from him. His advices at each time step of my research period have been always appropriate and critical. Professor William Lynch deserves a great deal of recognition for his valuable advice during many experiments I worked on and helpful criticism on my research. Dr. Romualdo de Souza contributed a lot on this thesis through the construction of Miniball. I especially appreciate very much for his letters of recommendation. It was very pleasant to work with Dr. Nelson Carlin and I appreciate very much his help and suggestions on the Miniball project and sincere friendship. I have greatly benefited from co-workers, Drs Betty Tsang, Tatsuya Murakami, and David Bowman in learning the computer programs for data acquisition and experimental skills. I want to thank very much Dr. Scott Pratt at University of Wisconsin for his support with a nice classical formula calculating correlation functions. I also appreciate Professor William Friedman for valuable discussions about time scales and IMF emission. It has been a great pleasure working with the excellent N SCL staff; Dr. John Yurkon and Dennis Swan at detector lab. informed me on the techniques used in making Miniball and other detectors. Jack Ottarson was extremely helpful with Miniball design. Without his clever design work Miniball would be in a different shape. I appreciate greatly the assistance of Reg Ronningen, Dave Sanderson, iv Ron Fox during the preparation for Miniball experiment. I want to thank Fred Denovich at Machine Shop, Mike Noe at Purchasing Dept, Orilla McI-Iarris at Graphics, Jim Vincent at Electronics Shop and many other people I may forget to refer to here for their help. I also acknowledge my fellow graduate students. I have had many valuable discussions with Wen—Guang Gong about correlation functions. Ziping Chen, Tapan Nayak, Hongming Xu, Fan Zhu, Larry Phair, Michael Lisa, Wen Chien Hsi have helped me and shared their knowledge and friendship with me, making my work much easier than it would have been without them. Finally I sincerely thank my family in Korea; my parents, my brother(Yeongsoo), and my sister for their encouragement and support. When I felt very tired, I was always comforted by just thinking of them. It is a little sad that I have been away so long from my friends in Korea still struggling against the military government. I can never thank enough my wife, Junghee Cho, who has been my perfect companion with criticism, encouragement, and love even though she has been so busy with her own research and our daughter. Errie, who was born during my thesis experiment, deserves my greatest thanks for her husky laughs. TABLE OF CONTENTS Chapter 1 Introduction ............................................................................................... I 1.1 Overview ........................................................................................................... 1 1.2 Theories of nuclear fragmentation ................................................................. 2 1.2.A Nuclear equation of state : liquid-gas phase transition ...................... 2 1.2.3 Microscopic Models .................................................................................. 7 1.3 Detection of multi-fragment emission ........................................................... 8 1.4 Time scales of multi-fragment emission ....................................................... 11 1.5 Organization of the thesis ............................................................................... 12 Chapter 2 Observation of Multi—fragment Emission with Dwarf Ball. ............... 15 2.1 Experimental setup .......................................................................................... 15 2.2 Reaction filter: Impact parameter selection .................................................. 18 2.3 Multi-fragment emission ................................................................................. 24 2.4 Summary ........................................................................................................... 31 Chapter 3 MSU Miniball - 41c fragment detection array ........................................ 32 3.1 Mechanical Construction of the Array .......................................................... 33 3.2 Detector Design ................................................................................................ 40 3.3 Detection Principle ........................................................................................... 44 3.4 Uniformity of Scintillation Efficiency of CsI(Tl) .......................................... 48 3.5 Fabrication of Scintillator Foils ....................................................................... 55 3.6 Data Acquisition Electronics ........................................................................... 60 3.7 Particle Identification ....................................................................................... 62 3.8 Light Pulsing System ....................................................................................... 67 vi Chapter 4 36Ar + 19'7Au reaction at E/ A = 35 MeV with Miniball .................... 73 4.1 Experimental setup .......................................................................................... 73 4.2 Data reduction and Analysis .......................................................................... 74 4.2.A. Particle Identification .............................................................................. 74 4.2.8 Gain drift correction ................................................................................. 8O 4.2.C Energy Calibration ................................................................................... 82 Chapter 5 General reaction characteristics ............................................................. 88 5.1 Multiplicity distributions : impact parameter selection ............................. 88 5.2 IMF elemental distributions ............................................................................ 94 5.3 Angular distributions ...................................................................................... 99 5.4 Energy spectra .................................................................................................. 99 5.5 Charge correlation function ............................................................................ 110 Chapter 6 Intensity interferometry : A tool to study time scales of multi- fragment emission ..................................................................................... 114 6.1 Overview ........................................................................................................... 114 6.2 Koonin-Pratt formalism ................................................................................... 115 6.3 Classical treatment of two-fragment Coulomb interaction ........................ 119 6.4 Validity of classical approximation ............................................................... 122 6.5 Trajectory calculation ....................................................................................... 126 6.6 Conclusion ......................................................................................................... 136 . . . 36 1 7 Chapter 7 Time scales of mum-fragment emissmn : Ar + 9 Au at E ._ /A__35Me V ............................................................................................... 138 7.1 Equal-charge two fragment correlation functions ....................................... 138 vii 7.2 Combinations of different fragment pairs .................................................... 145 7.3 Gated correlation functions ............................................................................ 148 Chapter 8 Summary .................................................................................................... 159 LIST OF REFERENCES .............................................................................................. 163 viii LIST OF TABLES Table 3.1 Coverage in solid angle, polar and azimuthal angles for individual detectors of the Miniball. Ring 1 was not used in this experiment. ......................................................................... 36 Table 5.1 Parameters of three-source fits (Eqs. 5.2-4, using n=3) to inclusive fragment spectra. The normalization constants Ni are given in units of 104/(5r0MeV3/2). ...................... 104 Table 5.2 Parameters of two-source fits (Eqs. 5.2-4, using n=2) to fragment spectra observed in central collisions (N C2 12). The normalization constants Ni are given in 3/2 units Of 10-4/(SI'MBV ). ................................................................... 104 ix LIST OF FIGURES Figure 1.1 Equation of state for nuclear matter with the Skyrme force. The solid lines show the pressure of nuclear matter as a function of the density for various fixed temperaturesfisotherms). The unstable states of the homogeneous system are indicated by the dashed lines [from Sau76]. ............................................................................................ 4 Figure 1.2 Instabilities of nuclear matter. Left gratingzregion of hydrodynamic instability,(%13 <0. Right grating: n)S initial conditions leading to breakup after expansion [from Ber83]. ............................................................................................. 6 Figure 1.3 IMF multiplicity distributions for Au + Au at E/A=200MeV. Five participant charge multiplicity bins increase from MULI to MULS [from Jac87]. ................................................................ 10 Figure 1.4 Two-fragment correlation functions measured with reactions on Au(left) and Ag(right) [from Tr087]. .................................................... 13 Figure 2.1 Schematic of the experimental set-up using the Dwarf Ball/ Wall and two PPACs; bottom part is the geometrical front View of the Dwarf Ball / Wall system consisting of hexagons and pentagons. ....................................................................... 16 Figure 2.2 Measured correlation between charged particle multiplicity and folding angle for the ‘ ‘N+ 2 ’ ‘ U reaction at E/A=50 MeV (left hand side) and for the 3 ° Ar+ 2 ’ 3 U reaction at E/A=35 MeV (right hand side). The angular range over which the charged particles are detected is indicated for each panel. The numbers of events (divided by a factor 1,000) for individual contours are given in the figure. ........................................................................................................ 20 Figure 2.3 Gated multiplicity (left hand panels) and folding angle (right hand panels) distributions measured for the reactions 3 a ' ‘N+2 ’ 8U at E/A=50 MeV (upper part) and ’ ‘Ar+2 U at E / A=35 MeV (lower part). The distributions for each panel are normalized to the same integrated yield. The gates on folding angle, 8f , are 1 ° wide; their centers are indicated in the left hand panels. The gates on multiplicity, M, are indicated in the right hand panels. ............................................................................................. 23 'igure 2.4 Folding angle distributions between coincident fission fragments. The upper scale gives the linear momentum transfer, AP/ P, in units of the projectile momentum to the heavy reaction residue assuming symmetric fission. The various gating conditions are explained in the text. For comparison, the distributions are normalized to give the same integrated yields, with the inclusive distribution multiplied by an additional factor of 2.0. The measured yields of the inclusive, N 1, (b ) (w) (w) NLMF—3’ NIMF—l, and NIMF 3.1x10’, 2.6x10’, 3.8x10’, 2.1x103 counts, respectively. ............................................................................................. 25 IMF: =3 spectra are 2.2x10 ‘ , figure 2.5 Associated charged-particle multiplicities, N C' The various gating conditions are explained in the text. For comparison, the distributions are normalized to give the same integrated yields, with the inclusive distribution multiplied by an additional factor of 2.0. ............................................................................................................. 27 iigure 2.6 Probabilities per fission trigger for the detection of single and multiple fragments with Z 2 4. Open circles: no gate on Off; open squares: fo>160° ; solid points: Offs 133°. Solid curve: Poisson distribution, P(N,v) = e-va/Nl, with v=0.25; dashed curve: prediction of the percolation model [Bau86] for pb=0.55. .............................................. 28 figure 27 probabilities per fission trigger for the detection of N IMF xi fragments with 224 gated on central collision(AP/P>0.5). Curves are Poisson distributions. ...................... 30 Figure 3.1 Artist’s perspective of the assembly structure of the Miniball 4n: fragment detection array. For clarity, electrical connections, the light pulsing system, the cooling system, and portions of the target insertion mechanism have been omitted. .................................................................................. 34 figure 3.2 Half-plane section of the Miniball array. Individual detector rings are labelled 1 through 11; numbers of detectors per ring are given in parentheses; the polar angles for the centers of the rings are indicated. The dashed horizontal line indicates the beam axis. ............................................... 35 "igure 3.3 Front views of different detector shapes. The detectors are labelled by their ring number; numbers of detectors per ring are given in parentheses. ......................................................... 38 "igure 3.4 Isometric View of the target insertion mechanism. .............................. 39 iigure 3.5 Schematic of phoswich assembly of individual detector elements. The u-metal shield covering the photomultiplier is not included. ................................................................................................... 41 iigure 3.6 Photograph of photomultiplier assembly. The scintillator and the first matching light guide are removed. The ring glued to the u-metal shield defines the alignment of the can housing the voltage divider. The can has been removed to expose the voltage divider. ............................................... 43 2igure 3.7 Schematic of the active voltage divider used for the Miniball detectors. ................................................................................................... iigure 3.8 Schematic of photomultiplier sygnal with three time gates applied. ..................................................................................................... 46 2igure 3.9 Relative variation of scintillation efficiency measured for two parallel surfaces of a CsI(Tl) crystal by using a collirnated oc-source. The axes of the coordinate system are parallel to the sides of the scintillator; the xii coordinates are fixed with respect to the scintillator. ................................................................................................ 50 Figure 3.10 Variations of scintillation efficiency measured with a charge integrating ADC for two different time gates selecting the fast and slow components of scintillation for CsI(Tl). ....................................................................................................... 51 Figure 3.11 Variations of scintillation efficiency detected with collimated oc-particles of 8.785 MeV energy (solid points) and collimated y-rays of 662 keV energy (Open points). The left hand panel shows the measurement for a detector which was rejected. The right hand panel shows the measurement for a detector which was incorporated into the Miniball. ............................................................. 53 Figure 3.12 Maximum detected scintillation nonuniformity for crystals obtained from different suppliers. The number of crystals scanned for each supplier is given in parenthesis. ............................................................................................... 54 Figure 3.13 Relation between scintillator foil thickness and rotational frequency of spinning measured for solutions of Beta- paint of different viscosity. The lines show fits with . -oc the power law expresswn, toc v . ........................................................ 57 Figure 3.14 Position dependent response of phoswich detectors with (upper panel) and without (lower panel) glue bond between plastic and CsI(Tl)scinti11ators. Solid and open points show charges integrated by the "fast" and "slow" time gates, At f =0-50 ns and At =60~400 ast slow ns, respectively. The same foil and CsI(Tl) crystal were used in this test. Various positions sampling the entrance window are labelled 1-11. ...................................................... 59 Figure 3.15 Schematic diagram of the data acquisition electronics of the Miniball. .................................................................................................... 61 Figure 3.16 Timing diagram of analog signal and gates. ....................................... 63 xiii ’igure 3.17 Particle identification obtained from two-dimensional "fast" vs. "slow" matrix for reaction products emitted at elab=35° for the ‘ ° Ar+' ° 7 Au reaction at E / A=35 MeV; a scintillator foil of 4 mg/ cm 2 areal density was used with a thin glue coupling to the CsI(Tl) crystal. (a) Matrix of raw data. (b) Linearized matrix. An intensity threshold of 2 counts per channel has been set. .......................................... 64 ’igure 3.18 Projections of linearized particle identification spectra for I reaction products emitted in the ‘ ° Ar+l ° 7 Au reaction at Olab=35° using phoswich detectors using scintillator foils of 4 mg/ cm 2 thickness, (a) with and (b) without glue coupling between the scintillator foil and the CsI(Tl) crystal. ..................................................................... 66 "igure 3.19 Sensitivity of particle identification to time shift of the "fast" charge integration gate. The solid and dashed lines show the loci of representative particle identification lines in the "fast" vs. "slow" identification matrix obtained for gates displaced by 2 ns with respect to each other. ............................................................. 68 Figure 3.20 Schematics of light pulser assembly and LED trigger circuit. ......... 69 ligure 3.21 Open points: gain variations of a CsI(Tl) photomultiplier assembly determined by measuring the detector response to 8.785 MeV a-particles; solid points: same data corrected for gain shifts in the off-line analysis by using information from the light pulser system. ................................ 72 'igure 4.1 "slow" (upper) and "tail" (bottom) versus fast' spectra of a Bicron detector (det # 2-8) ...................................................................... 75 'igure 4.2 "slow" versus fast ' spectrum of a Hilger detector (det # 2-9). The solid line is drawn to separate the light charged particles and IMFs for each detector. ................................................... 77 igure 4.3 "tail" (upper) and tail ' (bottom) versus "slow" spectra. The transformation is explained in the text. ............................................... 78 xiv 'igure 4.4 "fast" versus time of flight (left) and "time" (right) spectra. Time of flight has been achieved "time"(ns) - "RF"(ns). .................................................................................................... 79 'igure 4.5 Proton punch-through point of one detector in ring 5. Two vertical bars indicate the systematic uncertainty to decide the punch-through points .......................................................... 81 'igure 4.6 The correlation between correction from light pulse information and the correction from punch-through points determined for different detectors. ....................................................... 83 "igure 4.7 Energy calibration of det 2-9 as an example. The points are elasticlly scattered points and the curves are fits with equation 4.4. .................................................................................... 86 ’igure 4.8 Energy calibration of all detectors in ring 2, 3, and 4 to get reference curve. The data points are normalized to detector 2-3 at the 80MeV carbon point. .............................................. 87 figure 5.1 Part a: multiplicity distribution of identified charged- particles detected in this experiment; consistent with the hardware trigger of N hitzz’ a software cut of N C22 was applied. Part b : relation between charged- particle multiplicity and impact parameter obtained from the geometrical prescription of reference [Cav90]. ..................................................................................................... 89 'igure 5.2 Normalized conditional probability distributions dP/ dN IMF of detecting N intermediate mass fragments in IMF collisions preselected by the indicated gates on charged-particle multiplicity N C' Mean values are given in the figure. ............................................................................ 91 igure 5.3 Mean IMF multiplicity (circular points) as a function of MP the charged-particle multiplicity N C' Triangular points show mean multiplicities when lithium nuclei are excluded from the definition of IMFs. ........................................... 93 XV Figure 5.4 Charged-particle multiplicity distributions dP/ dN C measured for collisions preselected by the detection of N =0, 1, IMF and 2 intermediate mass fragments at any angle (left hand panel) and by the detection of a fragment at a given angle (right hand panel). ............................................................. 95 2igure 5.5 Comparison of angle integrated element distributions measured inclusively (solid points), in central collisions (N C2 12: open squares), and in peripheral collisions (25N C s 7: open circles). The lines represent parametrizations with exponential functions; ranges of parameters consistent with charge distributions are indicated. .................................................................................................. 96 3igure 5.6 Parameters of exponential (e-aZ) fits to element distributions selected by cuts on NIMF (left panel) and N C (right panel). ........................................................................................................ 98 iigure 5.7 Comparison of IMF angular distributions measured inclusively (solid points), in central collisions (N C2 12: open squares), and in peripheral collisions (2 sN s7: open C circles). Different panels show distributions for the indicated fragment Z. The distributions represent normalized conditional probability distributions for collisions preselected by the indicated gates on N C' ......................... 100 iigure 5.8 Inclusive energy spectra for intermediate mass fragment of charge Z=4-9. The curves represent moving source fits, Eqs. 5.2-4, with the parameters given in Table 5.1. ............................ 101 'igure 5.9 Energy distributions of carbon fragments produced in peripheral (25NC57: solid circular points), mid-central (8 SN C5.11: open points), and central (N C212: solid triangular points) collisions. The fragment detection angles are indicated in the individual panels. The distributions xvi represent conditional probability distributions per unit energy and solid angle for detecting a given fragment in collisions preselected by the indicated gate on N C' ............................................................................................... 103 .10 Energy spectra for intermediate mass fragments of charge Z=4-9 detected in central collisions (N C212). The curves represent moving source fits, Eqs. 5.2-4, with the parameters given in Table 5.2. The distributions represent conditional probability distributions per unit energy and solid angle. .................................................................. 106 .11 Equilibrium fraction, 0 / extracted from the fit the slow ototal’ data in Figure 5.10. Top panel: dependence of energy integrated yield on laboratory angle; bottom panel: dependence on P / A, the momentum per nucleon, of fragments detected at 16°s Blabs31°. .................................................... 108 .12 Extrapolations of the energy spectra to below the threshold for 2:6 and central collisions. The solid lines show estimates of the cross section below the detection threshold. .................................................................................................. 109 .13 Angular probabilities of 2:6 and 2:10 for central collisions. Solid points and open points are the data points with and without the extrapolation in the energy spectra. Solid and dashed curves are the extrapolations up to 0 ° in 3—1; assuming exponential and flat distribution respectively. .............................................................................................. 111 .14 Charge correlation functions as a function of 21+Z2 constructed according to equation 5.5. ................................................. 113 .1 Comparison of angle-integrated carbon-carbon correlation functions. Calculations with Eq. 6.3 are shown by points; calculations with the classical approximation, Eq. 6.17, are shown by the curves. The emission times xvii are indicated in the figure. ... .................................................................. 124 5.2 Longitudinal and transverse correlation functions for 1:200 fm/c. Longitudinal correlation functions (w=0 ° -30 ° or \y=150 ° -180 °) are shown by the solid curve and circular points; transverse correlation functions (\y=70 ° -110 °) are shown by the dashed curve and square points. Curves show calculations with Eq. 6.3; points show calculations with Eqs. 6.6-9, 6.13-16. ..................................................... 125 5.3 The points represent correlation functions obtained via trajectory calculations assuming emission from an uncharged source; the curves show the results obtained from Eq. 6.17. The parameters used in these calculations are indicated in the figure. ............................................... 128 3.4 Calculations for flat fragment spectra, T=°°, integrated over the indicated momentum intervals. The solid points represent correlation functions obtained via trajectory calculations assuming emission from an uncharged source; the curves show the results obtained from Eq. 6.17; the open points represent three-body Coulomb calculations neglecting recoil effects. The parameters used in these calculations are indicated in the figure. .................................................................................................. 130 5.5 Comparison of two-carbon correlation functions calculated by means of three—body trajectory calculations (points) with correlation functions calculated with Eq. 6.17 (curves). Upper and lower panels show results for total momenta per nucleon of P/ A5110 MeV/c and P/Az 110 MeV/ c, respectively. The parameters used in these calculations are indicated in the figure. ............................................... 131 6.6 Longitudinal and transverse correlation functions between two carbon fragments calculated via trajectory calculations for 1:200 fm/ c. .................................................................. 133 6.7 Two-carbon correlation functions calculated via three-body trajectory calculations for sources of different xviii masses. Solid points: M =226 u, open points M =1000O S S u. The upper panel shows calculations for thermal energy spectra and the lower panel shows calculations for flat energy spectra, T=oo, integrated over the indicated energies. The parameters used in these calculations are indicated in the figure. ............................................... 135 .1 Inclusive (left hand panels) and gated (right hand panels) two- fragment correlation functions at small relative momenta. The calculations are explained in the text. ........................ 139 .2 Estimates of experimental and theoretical uncertainties; a detailed discussion is given in the text. ................................................ 142 ’3 Panel (a): Comparison of inclusive C-C correlation functions to three-body Coulomb trajectory calculations. Panel (b): Correlation functions calculated from Koonin-Pratt formula (curve) and from three-body trajectory calculations (points). ............................................................................... 144 '.4 Dependence of inclusive, energy integrated two-fragment correlation functions on relative momentum q (top panel) and on reduced relative velocrty vrel/JZ1 +22. One fragment was carbon (Z=6); the atomic number Z2 of the other fragment is indicated. ............................................................ 146 '.5 Dependence of inclusive; energy integrated two-fragment correlation functions on relative velocity Vrel (top panel) and on reduced relative velocity Vrel/ 721 +22 (bottom panel). Two-fragment correlation functions are shown for each combination of Z1 and 22, with ................................................................... 147 4521,2239. ............................. 7.6 Two-fragment correlation functions summed over all combinations of Z1 and 22 (with 4sZ1,Zzs9) and selected by the xix indicated gates on charged-particle multiplicity N C (top panel) and on IMF multiplicity N IIVIF (bottom panel). ........................................................................................................ 149 '.7 Two-fragment correlation functions summed over all combinations of Z1 and 22 (with 4le,Zzs9) and selected by central collisions (N C2 12). The correlation functions are , evaluated for the indicated ranges of P/ A, the total momentum per nucleon of the detected fragment pair. .................... 151 ’.8 Inclusive, energy integrated two-fragment correlation functions summed over all combinations of Z1 and 22 (with 4sZ 1,Z 239). The different panels give comparisons with different calculations discussed in the text. ......................................... 152 7.9 Two-fragment correlation functions summed over all combinations of 21 and 22 (with 4sZ 1,Zzs9) and selected by central collisions (N C212). Individual panels show the correlation functions for the indicated cuts on P / A, the total momentum per nucleon of the detected fragment pair. The curves represent calculations with the Koonin-Pratt formula for the indicated emission times. ......................................................................................................... 154 7.10 Two-fragment correlation functions summed over all combinations of Z1 and 22 (with 4sZ1,Zzs9) and selected by central collisions (N C.212). Individual panels show the correlation functions for the indicated cuts on P / A, the total momentum per nucleon of the detected fragment pair. The curves represent the results of trajectory calculations for which the Coulomb interaction with the residual system is turned off. The key for emission times is given in the figure. ........................................................................................................ 155 7.11 Two-fragment correlation functions summed over all XX combinations of Z1 and 22 (with 4sZ 1,2259) and selected by central collisions (N C212). Individual panels show the correlation functions for the indicated cuts on P/ A, the total momentum per nucleon of the detected fragment pair. The curves represent the results of three-body Coulomb trajectory calculations in which the two fragments are assumed to be emitted from a source of initial charge number 25:79. The key for emission times is given in the figure. ........................................................................................................ 156 xxi pter 1 Introduction In this thesis we will present a study of intermediate mass fragment ion in nucleus-nucleus collisions at energies between E/A=20-50MeV. At energies(E/AleMeV) quasi-elastic, deep inelastic, and complete fusion ions dominate the exit channel in proportions which depend on the impact neter of the collision. At these energies mean-field dynamics plays a 3minant role. In contrast, the experimental data at higher energies .2200MeV) indicate a geometrical picture (participant-spectator model), e the role of nucleon-nucleon collisions is more important. At intermediate gies, one observes the interplay between mean-field and nucleon-nucleon ions and a transition between two different reaction mechanisms at lower ugher energies [Ge187,Lyn87]. In addition a new phenomenon may appear is energy regime, namely the liquid-gas phase transition of highly excited ear matter [Cse86]. When the excitation energy is not too large, the decay of an excited nuclear m is believed to be a succession of evaporation steps of neutron and light ged particles (H and He isotopes). As the excitation energy is raised to about 'al MeV per nucleon, however, decays by intermediate mass fragment (IMF : 20) emission are predicted and observed to become important. For :iently high excitation energies, 45 MeV/ nucleon, a number of theoretical lations predict a simultaneous breakup of an excited nuclear system into 'al IMFs (Multi-fragmentation). This process should be qualitatively ent from the conventional decay process like evaporation or fission. Over the pit del fi— 7 fl the last decade there has bee a great deal of interest in multi-fragmentation processes because information about the low-density nuclear equation of state and the liquid-gas phase transition of nuclear matter may be obtainable from detailed studies of multi-fragment emission processes. 1.2 Theories of nuclear fragmentation , 1.2.A Nuclear equation of state : liquid-gas phase transition The thermodynamical properties of nuclear matter at finite temperatures have been studied to investigate the nuclear equation of state at low density(p- IO - (D (Z LU z 2 LL! 0 5 _ 2 El 2 g FRAGMOENgATION — 0: ‘ V I 2&4 ’/// \\\V . w) // —| o -5 .. < z2 / (r E! .2. “'0 ~ ovsasraesseo ~15 ”'6 C J 4 4 L I I I 7— o 5 I |.5 2.0 DENSITY n / "0 Figure 1. 2 Instabilities of nuclear matter. Left gratingzregion of hydrodynamic instabflity,%—( PS) <0. Right grating: initial conditions leading to breakup after expansion [frnom Ber83]. C0115 inst: syst frag inve 1.2 ext frag the pro can rea considerations behind this curve can be found in reference [Ber83]). In this region instablities can grow exponentially from small amplitude, thus can split the system into liquid and gas phases rapidly. The possibility of reaching the fragmentation zone via heavy ion collisions presents a persuasive argument for investigating multi-fragmentation process in such collisions. 1.2.B Microscopic Models The schematic arguments described in the previous section predict the existence of maid-fragmentation processes. A detailed investigations of fragmentation processes requestes theories that can be compared directly with the experimental observables. Such a quantitative comparisons are only provided by microscopic simulations of nucleus-nucleus collisions, in which one can introduce finite size effects, surface and Coulomb energy effects, and a reasonable dissipative dynamics [Sur89]. In most statistical multi-fragmentation models, the basic assumption is that a compound nucleus has been formed with given excitation energy, angular momentum and parity. Once formed, such a nucleus is assumed to decay according to the statistical weights of the various allowed decay configurations. Bondorf et a1. and Gross et al. predict the onset of multifragmentation at temparatures of about 5-6 MeV [Bon85b,Gro86] within canonical and microcanonical treatments respectively. Recently Friedman explored multi- fragmentation with an evaporation model of an expanding nuclei. This calculations predict a multi-fragment rapid massive cluster formation for an excitation energy of about 1.5 GeV in a mass 190 system [Fri90]. However, statistical calculations generally have several limitations : (1) Thermodynamical equilibrium is a priori assumed, while the experimental angular distributions of have ener frag Statt €011; lele: C011 me: 8 Ms indicate nonequilibrium emissions, (2) The formation step of the outgoing fragments is not described due to the lack of dynamical effects. A number of microscopic dynamical calculations indicate that the reaction trajectories of excited nuclear systems produced in heavy-ion collisions may indeed enter the mechanical instablity region [Vic85,Boa86,Sur89], in which density fluctuations can grow exponentially [Boa90a]. The initial conditions leading to evaporation, multi-fragmentation and total vaporization have been studied by mapping reaction trajectories on the phase diagram [Vic85,Boa86,Sur89]. Recent calculations for Ca+Ca collisions at b=0fm estimate a multi-fragmentation threshold at around E/ A = 30-35 MeV [Sur89]. These calculations, however, may not contain sufficient fluctuations to describe the fragmentation process. For this reason, it is also interesting to explore statistical fragmentation approaches. 1.3 Detection of multi-fragment emission A number of experiments on the IMF production in heavy-ion reactions have been performed at intermediate energies for a variety of systems and energies [Chi83, Fie]84, FieJ86, Jac87, Bow87]. To identify IMF emission as a multi- fragmentation clearly, however, one must examine the multi-fragment final states on an event-by-event basis. First experimental evidence of multi-fragment emission was established from Plastic Ball data obtained by the GSI/LBL collaboration. The Plastic Ball/Wall [Bad82] consists of 815 CaFZ-plastic telescope modules covering the angular region from 10 ° 5 9 s 160 °. For Au+Au lab collisions at E/A=200MeV, IMFs(3sZs10) were detected at 330° 9 lab corresponding to the forward hemisphere in the center of mass system. These measurements were performed with an energy threshold of E th/ A=35~40MeV. Figt chat [lac the data cha frag cov det mu COI Figure 1.3 shows IMF multiplicity distributions gated on various participant charge multiplicities thereby representing different regions of impact parameter Uac87]. On average, 3-4 fragments were detected per central collision. Therefore the data clearly established the existence of multi-fragment emission. Also, the data show a smooth increase in the probability of IMF emission as the participant charge multiplicity increases or as the impact parameter decreases. At intermediate energies, a few experimental measurements of multi- fragment emission have been performed with relatively restricted phase space coverage. Bougault et al. studied Kr+Au and Kr+Ag at E/A=43MeV using multi- detector system covering about 50% of 41: [Bou88]. They detected IMF multiplicities as large as N = 6 with non-negligible cross sections. They concluded that even though/[Fthe multi-fragmentation is observed in these measurements, it does not result from a simultaneous explosion of the total system. Recently Trockel et al. reported multi-fragment emission from O + Au at E/ A = 84 MeV with mean IMF multiplicity about 0.5 [Tro89,Fri90]. Comparisons of data with both statistical multi-fragmentation calculations [Bon85b] and calculations with a sequential binary decay model [Cha88] were inconclusive. Earlier a series of experiments have been performed by Berkeley Group using reverse kinematics. From the coincident measurement of fission fragments for La + C at E/ A = 50 MeV, they concluded that sequential binary decay dominates this reaction with an estimated excitation energy of the composite system of about 280 MeV [Bow87]. This conclusion was challenged by Gross [Gr088]. He showed that his microcanonical multi—fragmentation model could reproduce the experimental binary charge correlations reasonably, and concluded that the real cranking should be observable at excitation energy above 600 MeV for nuclei in this mass range. the par lid] 10 Au + Au Ftagment (Z > 2) Multiplicity O < 30° 106 7 T ' I I IEAB r I r I r I 3 1101.1 — j 4 14012 ....... IO MULB —-- 3* MUL4 —.- 3 MUL5 —— 3 o” \ — Events \Il‘ \ 102 \.i \ : _ ._\ \ 1 p a.) \. , 10‘ = .\ i = : \ 1 ’ 10° 4 L n L 4‘ hi ‘_ . 1 r L 2 4 8 8 10 12 14 Multiplicity Figure 1.3 IMF multiplicity distributions for Au + Au at E/A=200MeV. Five participant charge multiplicity bins increase from MULI to MUL5 [from Jac87]. par sysl ons pro eqi '1an det pie aPl 11 It has become clear that more exclusive experimental data are needed for an understanding of multi-fragment emission processes. For example, impact parameter selection could eliminate less interesting peripheral collisions. A systematic study of the excitation function of IMF production could reveal the onset of multi-fragmentation reactions. Dynamical calculations suggest that the probability or onset for multi-fragmentation may be sensitive to the nuclear equation of state at low density [Sur89]. Therefore we can expect to gain information about the equation of state of low density nuclear matter from a detailed studies of the multi-fragment emission processes. The first task of the present work was to measure exclusively multi-fragment emissions with an apparatus of large phase space coverage. 1.4 Time scales of multi-fragment emission At present, it is not clear that the theoretical calculations described in Section 1.2 predict final states which are different from final states predicted by conventional evaporation or fission like processes. In principle one cannot exclude the possibility that sequential multi-step binary decay mechanisms can generate multi-fragment final states [Mor88]. As pointed out by Bertsch [Ber83], the usual liquid-gas phase transition is a first order transition that applies to processes that occur slowly enough for an equilibrium to be established across a phase boundary. A lower limit for the time required for such decays is given by the time required for exponential growth of density fluctuations or the expansion of the nuclear system; this time scales are of the order of 50fm/c in dynamical calculations [Sur89,Boa90]. On the other hand, conventional evaporation-like binary decay assumes a nearly complete relaxation of the excited composite system between each decay. For example, the time scale of ii ordt eva] [De' lll€E 12 of fission at initial nuclear temperatures of about 5 MeV is estimated to be fo the order of 10-205econd or several thousand fm/c [Hi189]. The time scale for evaporative processes is predicted to be of the order of 1-5x10-21(z1000 fm/c) [DeY89]. Therefore key to experimental confirmation can be provided by the measurement of the fragment emission time scales for the breakup. One of the first experimental time scale measurements were obtained by Trockel et. a1. [Tr087]. They constructed correlation functions between heavy residues, fission fragments, and intermediate mass fragments as a function of relative velocity as shown in Figure 1.4. Comparing the IMF-IMF correlation functions with trajectory calculations, they extracted mean IMF emission times of the order of 300-1400fm/ c (the mean emission time, t, and the "half-life" for ) IMF-IMF’ IMF-IMF ' The similar time scales of HR(heavy residue)-IMF and IMF-IMF decay IMF emission, 1: defined by [Tro87], are related by : t = 1.441 processes led them to conclude a sequential nature of multi-fragment emission. However the range of possible time scales extracted is quite large in this measurement. A qualitatively similar result has been obtained by Bougault et. al. who quote a mean emission time 300fm/c without a corresponding estimate of the uncertainty [Bou89]. More experimental data are needed to explore these issues. The second part of this thesis will address this problem. 1.5 Organization of the thesis We first performed an exploratory experiment to begin the investigation of the multi-fragment emission processes. In Chapter 2, we describe this experiment which combined the Dwarf Ball/ Wall 4n phoswhich detector with two parallel plate avalanche counters(PPAC) [Bre77]. After this experiment, we began to develop our own 41c detector with improved characteristics. Chapter 3 explains R12 (vrel) Figure All(]ef 13 180 +197Au 84AMeV 1"0 +"0*Ag 84AMev t r T T r .r r I r I r r uT r r 3 - (a) i “0‘“ v (d) I V‘°‘° ‘ 0 o 2 _ . o. _- .. ’9 1~ o. '.¢++ . - - IIII‘ - -F 0 F-F O—éiFeieii 'iet'fiteie 1.5 -' 5351323 -- ;.;.;.; .. (b) e (c) . ’1 10:. °'. ”W l l "Nut” ii i 2 . _ . ' +0, .. I ++ .. > 4, . 7; 0.5 — 9,. IMF - HR -~ . IMF - HR 4 v- -. ¢ -- .’ <1 O50.0 e++e:+et: r‘tteisle’l,‘ 1.5 IMF - IMF "' (f) I .fi I Y J. ...o° I .......... ' I, --- 10 fmlc . , “m" 100 fmlc 0.5 - ’ --- 1000 fmlc "‘ °' --,- 1000 frnlc " — /// ------- 10000 fmlc ‘- ....... 10000 fmlca o 1" L ‘1 L 1 j 4 L i _J.’ L L L L I L L M 0 1 Z 3 4 0 1 2 3 L S vrel ( cm/ns) Figure 1.4 Two-fragment correlation functions measured with reactions on Au(left) and Ag(right) [from Tr087]. the deta MSU M ChapteI reductiI fragme: conelat T1 the tec . theoret the val experii and th ChaptI 14 the detailed technical aspects of the MSU Miniball. The first experiment with the MSU Miniball was a study of the reaction of 36Ar + 197Au at E/ A = 35MeV. In Chapter 4, we describe the experimental setup and the methods of data reduction. In Chapter 5 we present experimental results; elemental and angular fragment distributions, total and IMF multiplicity, distributions, and charge correlation functions. To obtain informations about the time scale of IMF emission we employed the technique of intensity interferometry. In Chapter 6 we present a simple theoretical expression for the two—fragment correlation functions and investigate the validity of the approximations used in the calculations. In Chapter 7 our experimental two-fragment correlation functions are compared to calculations and the time scales are extracted. Summary and conclusions are provided in Chapter 8. _ I ion 4t we gl'c a p9 coi de Chapteriz Observation of Multi-fragment Emission ' with Dwarf Ball. To study multi-fragment emission processes in intermediate energy heavy- ion collisions we performed an exploratory experiment with a newly developed 41: detector from Washington University [Sar88,Str90]. Two additional PPACs were constructed for this experiment by our group. To clarify the relative importance of single and multi-fragment emission mechanisms, we have measured the IMF multiplicity distributions for fragments emitted beyond the grazing angle for the reactions 36Ar + 238U at E / A = 20 and 35 MeV and 14N + 238U at E/ A = 50MeV. To discriminate between fusion-like central and less violent peripheral collisions, the IMF multiplicity distributions were measured in coincidence with two fission fragments and light charged particles from the decay of the associated heavy reaction residues. 2.1 Experimental setup In the experiment, a 238UF4 target of 400 jig/C1112 areal density was bombarded by 14N and 36Ar beam of three different energies from the K500 cyclotron at Michigan State University. Charged particles were detected with 96 phoswich detectors of the "Dwarf Ball/Wall" array developed at Washington University[Sar88,Str90]. A schematic diagram of the experimental setup is shown in Figure 2.1. The geometrical front view of the Wall and ball detectors is shown at the bottom part of the figure. Each phoswich detector consisted of a thin fast plastic scintillator foil followed by a thick CsI(Tl) scintillator, with 200 um plastic 15 Dwarf Experi Figur PPAI const 16 Dwarf Ball/Wall + PPAC Experimental Setup Q PPAC .‘ Wall T I Beam fl PPAC (a, Figure 2.1 Schematic of experimental set-up using Dwarf Ball/Wall and two PPACs; bottom part is the geometrical front view of Dwarf Ball/ Wall system consisting of hexagons and pentagons. sci to 4 590 he. 105 ele an Wi 17 scintillation foils and 20 mm CsI(Tl) scintillators at the forward angles, reducing to 40 um foils and 4 mm CsI(Tl) scintillators at the backward angles. To suppress secondary electrons and X-rays, the Dwarf Ball detectors (located at 623?) and Dwarf Wall detectors (located at 12 ° 59535 °) were covered by 5 mg/ cm2 thick Au foils and 10 mg/cm2 thick Ta foils, respectively. The most forward detectors at 851?, covering the grazing angle, were shielded by 1 g/cm2 Pb absorbers to stop elastically scattered projectile nuclei. These absorbers also stopped IMF’s, but allowed the detection of energetic light particles. For light particles, the detection array provided an angular coverage corresponding to about 85% of 4n. For heavy fragments, the coverage was reduced to about 78% of 4n; about 1% of the loss in coverage is due to the Pb absorbers with additional losses due to poor elemental resolution for detectors located at 83150? Particle identification was achieved by integrating the photomultiplier anode current over three different time gates (See Section 3.3). This information was combined to obtain the energy of the detected particle as well as elemental identification up to about Z==6 and isotopic identification for light particles (Z32) stopped in the CsI(Tl) scintillators. Fission fragments were detected with two X-Y- position sensitive parallel plate multiwire detectors [Bre84] covering angular ranges of 81:36°-116°, and ®2=-(39°-89°) in the reaction plane. In the off-line analysis, the energy thresholds for the Dwarf Ball detectors were set to 12 and 18 MeV for hydrogen and helium nuclei, respectively; for the Dwarf Wall detectors they were set to 20 MeV for both hydrogen and helium. All heavier particles which passed through the absorber and scintillator foils were analyzed, corresponding to thresholds of E / Az6-9 MeV and E/Az2-3 MeV for the Dwarf Wall and Dwarf Ball detectors, respectively. Unfortunately, double-hits by two a- particles caused significant contaminations in the Z23 particle identification gate. Therefore, IMF multiplicity distributions for fragments with 224 have been C0111 obSI 18 constructed. Contributions from random events were negligible for all observables discussed in this thesis. 2.2 Reaction filter: Impact parameter selection Inclusive experiments determine only impact-parameter averaged quantities, while the detailed reaction mechanism largely depend on the impact parameter. The resulting loss of information can lead to ambiguous interpretations, which could be overcome by employing suitable reaction filters capable of selecting specific classes of collisions. Such filters are particularly useful if they can be used to select well defined narrow ranges of impact parameters. A number of different techniques have been explored. The folding angle, off: 6f1+ efz, between coincident fission fragments provides information about the linear momentum transfer to fissionable heavy reaction residues [Fat85,Che87]. (91:1 and 8 f2 respect to the beam axis.) Folding angle measurements can discriminate between are the polar angles of the two fission fragments with peripheral, quasi-elastic collisions and central, fusion-like reactions. An alternative technique employs neutron multiplicity measurements [Ga185]. For the 20Ne+U reaction at E/A=14.5 MeV, the measured neutron multiplicity was found to exhibit a monotonic and nearly linear dependence on folding angle, indicating a close relation between these two kinds of reaction filters, at least at low incident energies. Existing 41c neutron detectors are mostly sensitive to low-energy neutrons emitted in the statistical decay of fully equilibrated reaction products. Hence, the measured neutron multiplicity is closely related to the amount of energy deposited into equilibrated degrees of freedom. Since energies and angles of the emi 1110 0V6 9116 “GE em prI till“ 19 emitted neutrons are not measured, other details of the reaction remain undetermined, e.g. the orientation of the reaction plane or the transverse momentum of the emitted particles [Dan85]. Charged particle detector arrays can overcome such limitations and provide information about the multiplicities of energetic emissions for which 41: neutron detectors become inefficient. For low incident energies, the multiplicity of charged particles emitted statistically to backward angles will largely reflect the excitation energies of the nearly equilibrated reaction residues. The multiplicities of charged particles emitted to forward angles, on the other hand, are dominated by nonequilibrium processes. With increasing energy, the multiplicities of these nonequilibrium emissions should reflect to an increasing degree the impact parameter dependent geometry or "participant volume" characteristic of the early stages of the collision [Wes76]. At intermediate energies, both equilibrium and nonequilibrium emissions are present. The sensitivity of any charged particle array to either process may depend strongly upon the angular range subtended by the device. Therefore we have performed a cross calibration of charged particle multiplicity filters for intermediate energy collisions, through an experimental investigation of the relationship between charged particle multiplicities and fission fragment folding angles for the reactions 36Ar + 238U at E/ A =35 MeV and 14N + 238U at E/ A = 50 MeV. Figure 2.2 shows two-dimensional contour diagrams of folding angle, Off, versus measured charged particle multiplicities. (These raw multiplicities are not corrected for efficiency losses due to detection thresholds and incomplete solid angle coverage; therefore, they are slightly smaller than the true charged particle multiplicities.) The left and right hand panels show data for the 14N+238U and 36 2 . . . Ar+ 38U reactions, respectively. For orientation, the upper scales give the linear momentum transfer, AP / P, to the heavy reaction residue in units of the projectile 20 “SU ~ 36-175 AP/P 1.0 0.5 0.0 |.O 0.5 0.0 8 L I I r 1 11f f f T 1 I I I 1 r r I y 1 1— T— I 13 1 “N + 238U “A I' ,+ ZGOU '1 l. 2 7 E/’A=50 MeV 4' EM = .35 MéVi 11 sea—175' 9x5 -175 110 6 i :9 «18 5 . -I7 4 ~ 46 «'5 3 F 4 , 2 A 3 >4 >~ : 3:1 lb ..1 U '2: 6 r 9 '5. 9* a .3 -—'O 5 L. -—n :3 7 3 5’ '6 2 4 _ 2 5 3 ' 4 3 2 h 2 l - 1 4 L — 6 a. 5 3 ” 4.4 H \ Wi KWfi 2 1 e 1 L__ l J_ 11_J 120 130 140 150 I60 170 180 IOLO 120 140 16180 911 (deg) Figure 2.2 Measured correlation between charged particle multiplicity and folding angle for the ‘ ‘N+’ ’ 8U reaction at E/A=50 MeV (left hand side) and for the ’ ‘Ar+’ 3 1’U reaction at E/A=35 MeV (right hand side). The angular range over which the charged particles are detected is indicated for each panel. The numbers of events (divided by a factor 1,000) for individual contours are given in the figure. 21 momentum, P, assuming symmetric fission of the compound nucleus. The upper panels show the total multiplicities, M, measured with the entire array (covering the angular range of 8=5°-170°). The center panels show the number of particles detected with the "Dwarf Ball" (covering the angular range of 9=35°-170°) and the lower panels show the number of particles detected with the "Dwarf Wall" (covering the angular range of e=5 °-35 ° ). For both reactions, the average multiplicity of charged particles emitted over the entire angular range ((-)=5°-170°) increases monotonically with increasing linear momentum transfer to the heavy reaction residue. This dependence is qualitatively similar to the monotonic relation between neutron multiplicity and folding angle which has been established for reactions at lower incident energies [Gal85,Mor88]. For the 36Ar+238U reaction, the dependence of the average multiplicity on folding angle is similar to the results obtained for a similar reaction at a slightly lower energy U acq85]. A nearly linear correlation between multiplicity and linear momentum transfer is observed for charged particles emitted at intermediate and large angles (@=35 °'-170°). In contrast, the multiplicity of charged particles emitted at forward angles (6=5°-35°) is nearly independent of the linear momentum transfer to the heavy reaction residue Uacq85]. At least for the present reactions, the multiplicities detected in forward arrays [B1286] cannot be used to select violent projectile-target interactions in which large amounts of energy and / or momentum are dissipated. Charged particle multiplicity as well as folding angle measurements represent reaction filters of finite resolution. Even for the ideal case of formation and decay of a composite nuclear system with unique spin, excitation energy, and recoil momentum, multiplicity and folding angle distributions have finite widths. The widths of folding angle distributions reflect the finite widths of the mass and kinetic energy distributions of the fission fragments as well as smI adr in; ref] me mt cor ml res 1111 im m1 22 smearing from light“ particle evaporation. For more energetic collisions, additional broadening can be caused by the emission of intermediate mass fragments [Fat87]. The widths of charged particle multiplicity distributions reflect the stochastic nature of neutron and charged particle emission mechanisms. Particularly at lower energies, where the charged particle multiplicities are small, statistical fluctuations in the multiplicities can be comparable to the mean values and the selectivity of charged particle multiplicities as reaction filters can be reduced. The widths of various gated distributions are shown in Figure 2.3. The upper and lower sections show data for the 14N+238U and the 36Ar+238U reactions, respectively. The left hand panels show distributions for total charged particle multiplicity gated by folding angle intervals, 1° wide and centered at the indicated angles. The right hand panels show folding angle distributions measured for different total charged particle multiplicities, M = 1, 3, 6 for the 14N +238U reaction and M = 1, 6, 12for the 36Ar +238U reaction. The high multiplicity gates (M=6 for 14[N +238U and M=12for 36Ar +238U) represent cuts in the tails of the charged particle multiplicity distributions, at values approximately 50% larger than the most probable multiplicity observed for fusion-like collisions (see left hand panels, or Figure 2.2). Charged particle multiplicity distributions gated by large folding angles, ®ff=175°, are clearly peaked at Msl. However, they exhibit remarkably long tails extending to high multiplicities. Multiplicity distributions gated by small folding angles (or fusion-like collisions) are peaked at M=4 and 8 for the 14:N+238U and 36 Ar+238U reactions, respectively. They are rather broad indicating significant fluctuations of the number of emitted charged particles. There is considerable overlap between the multiplicity distributions gated by very different folding angles. i — I F WHmCH fl p> T—Q. .Ffitnmw\ Figu pane (upr eaCh angl gate 23 MSU-88-274 4 AP P 8 .. 13N + 233U i 1.0 0.5/ 01.0 :- I I _ _ E/A=50 MeV :— _. 6 I- M=1 " — 0ff=175° '5' : (I) 145° ' '1 4Q - _- _ ‘5 ;. _‘. 2 D b 125° d: : ° I . . . .t. ‘. F0 0 1 1 1 1 O .. AP/P _ :3 - 36Ar+233U -_ 1.0 0.5 0.0 I u 7 v - E/A=35 MeV .. ' .. g " 9151735" {— 33: 4 12 -~ — 145° --~ - >1 2 / . \ _ F /’ ._ —I 2 _. .. J 0 "- 0 0 5 10 100 150 200 M 911 (deg) Figure 2.3 Gated multiplicity (left hand panels) and folding angle (right hand panels) distributions measured for the reactions ‘ ‘N+3 3 3U at E/A=5O MeV (upper part) and 3 ‘ Ar+ 3 3 3 U at E / A=35 MeV (lower part). The distributions for each panel are normalized to the same integrated yield. The gates on folding angle, 6f , are 1° wide; their centers are indicated in the left hand panels. The gates on multiplicity, M, are indicated in the right hand panels. 24 Folding angle distributions gated by low (high) charged particle multiplicities are peaked at large (small) folding angles, respectively. The widths of these distributions are very similar to the widths measured for the out-of- plane distributions of fission fragments selected by the same multiplicity gates, indicating that these gates do not introduce significant broadening beyond the intrinsic resolution of the folding angle technique. In contrast, folding angle distributions gated by intermediate charged particle multiplicities (M=3 for 14N+238U and M=6 for 36Ar+238U) are significantly broader indicating that such multiplicity gates are less selective. 2.3 Multi-fragment emission The solid line in Figure 2.4 shows the inclusive folding angle distribution for fission-fission coincidences. The dashed and dotted-dashed curves show folding angle distributions gated by the detection of at least one and three (b ) IMF 2 1 and 3 are remarkably intermediate mass fragments at backward angles (9 2 35 °) in the Dwarf Ball, N (b ) IMF similar indicating that single and multi~fragment emissions to intermediate and 2 1 and 3, respectively. Distributions gated by N large angles can be associated with violent, fusion-like collisions characterized by large linear momentum transfers [Fat85]. The light and heavy dotted curves show distributions gated by the detection of at least one and three intermediate (w) IMF 2 1 and 3, mass fragments at forward angles—(6 S 35°) in the Dwarf Wall, N respectively. The condition NR2; 2 1 does not select a very specific class of collisions, with contributions from both incomplete fusion reactions and (W) peripheral collisions. The requirement N 2 3, On the other hand, clearly selects IMF more peripheral collisions with smaller linear momentum transfers to the heavy reaction residue. nits) D (arb . u YIEL Figure upper I momer Varioru disiribr iIlclusi Yields ( 25 MSU—ss-obi AP/P 1.0 0.5 0.0 - l I l . I I I I I7 I .I I 4 5? °°Ar + 238U, E/A = 35 MeV i Z — inclusive : ”v? 4— ‘— :‘é ; 1133.: --- 21, 23 (3‘2) ‘ fl _ N871)? . ...... g1, ...... g3 : .0 3.— . 7 3-0 _ ." 3\ - «1 _ . ............. . v ,- .. '-. . Q 2 L— , // \\\’\ _ El I , ' \.,..--;. ------ '=. i H P ’ ..... "'\.\\.'. 1 " >3 _ .i 3‘ ° ‘3‘ \ 1°. - 1 T I" \". \\\ ...: d. F / I "\_ \\\ : r- I ............. -..-.-N‘K'r‘. “ 0‘ l' 1 l 4 4:“ ' 100 200 611 (deg) igure 2.4 Folding angle distributions between coincident fission fragments. The pper scale gives the linear momentum transfer, AP/ P, in units of the projectile tomentum to the heavy reaction residue assuming symmetric fission. The irious gating conditions are explained in the text. For comparison, the stributions are normalized to give the same integrated yields, with the elusive distribution multiplied by an additional factor of 2.0. The measured .. . . (b)_ (b)_ _ (w)_ . ..lds of the mclusrve, NIMF-'1’NlMFm3’N IMF—1, and N IMF-3 spectra are 2.2x10 , 1x103, 2.6x103, 3.8x10’, 2.1x103 countsrrespec‘dvely‘ F fission curves for fls dotted respec partic‘ theNi chargI tollisi classe inter: choice intern C0095 26 Figure 2.5 shows associated charged-particle multiplicities, N = Nto - C t N (g ate), where N is the total observed charged particle multiplicity (excluding IMF tot (3 ate) :15310n fragments) and NIMF :urves are defined as in Figure 2.4: the solid line shows the inclusive distribution denotes the number of fragments in the gate. The for fission-fission coincidences; the dashed and dotted-dashed light and heavy (b ) (w) IMF 2 1 and 3 (NIMF 2 1 and 3), respectively. Consistent with this finding, the observed associated charged dotted) curves show distributions gated by N particle multiplicities are large for IMF’s emitted at e Z 35 ° and rather similar for (b ) . (W) the N IMF 2 1 and 3 gates. The requirement NIMF charged particle multiplicities consistent with a bias towards more peripheral 2 3 leads to lower mean associated collisions. The different gates discussed in Figs. 2.4 and 2.5 clearly select different classes of collisions reflecting impact parameter ambiguities which obscure the interpretation of inclusive data. These ambiguities can be removed by suitable choices of reaction filters. Figure 2.6 shows probabilities for observed multiplicities, NIMF’ of intermediate mass fragments for different gates on folding angle. Open circles correspond to the inclusive distributions for fission-fission coincidences; open squares correspond to small momentum transfer collisions gated by of? 160° (AP/P3 0.5 — __ .. ° I _ .. I \ 4 - . \ i O O L‘l 1 L L l l l l _l _l ML L _L L J l 0 5 10 15 NC ' iigure 2.5 Associated charged-particle multiplicities, N . The various gating C ronditions are explained in the text. For comparison, the distributions are formalized to give the same integrated yields, with the inclusive distribution multiplied by an additional factor of 2.0. 28 MSU—89—003 100 -__;- 3GAI‘ + 238U ‘— E E/A=35MeV f E i Z 1 >5 ”"2 E" “i 4..) Z I a I 1 .0 10-3 5'"- “"5 (t5 5 . . E .Q : O 1nclu81ve 3 O _ r L4 10-4 E— . 6“ g. 133° ‘3 0.. '-' i 5 a 9,, > 160° 3 L . .10—5 g— -— Poisson (u=0.25) “i E -- percolation (pb=0.55) 10"6 =— 6 -= s 1 l l I L l a ‘ O 1 2 3 4 5 6 NIMF gure 2.6 Probabilities per fission trigger for the detection of single and ultiple fragments with Z 2 4. Open circles: no gate on off; open squares: t>160°; solid points: effsl33°. Solid curve: Poisson distribution, P(N,v) = v N v / N I, with v=0.25; dashed curve: prediction of the percolation model [Bau86] ' pb=0.55. by Poi and C( excita‘ statist partic seque Poiss< multi; energ expec incid reacl foldi distr 29 by Poisson probability distributions (an example is shown by the solid curve). Poisson distributions are characteristic of processes which occur with a low md constant probability. Such conditions may prevail for nuclear decays at high excitation energies and with low fragment multiplicities. For example, the statistical weight for fragment emission is small as compared to that for light particle emission when calculated in a statistical rate equation approach such as a sequential evaporation model [Mor75,Fri83]. In this case, deviations from Poisson distributions would be expected to occur mainly for high fragment multiplicities or small systems when emission probabilities are affected by energy or mass conservation. To illustrate that Poisson distributions may also be expected in a static theory which encompasses a phase transition we show in Figure 2.6 a distribution predicted by the percolation model [Bau86]. While the IMF detection efficiency of the present experiment is sufficient to reveal the qualitative trends of the multiplicity distributions, the IMF yields below the detection thresholds of the present apparatus would be required to determine the exact shape of the multiplicity distributions and hence, definitive values for the percolation parameter, pb, or the average v of the Poisson distribution. A model independent correction of the IMF detection efficiency is beyond the scope of the present measurement. To see the dependence of IMF production probability on the system and incident energy Figure 2.7 shows the IMF multiplicity distribution for all three reactions preformed in this experiment with central gate, AP/P>O.5., on the folding angle. For all three cases the data can be described by the Poisson distributions shown by the curves in the figure. k. _ l Ohm NEHZDMQ Wich 30 MSU-89403 O - central collisions AP/P>O.5 : E X + 238U 3 10—1 5— ‘Q -—2 '__ A 3‘ 10 e ‘1; L: E 3 Q _ _ as 10‘3 e— *2 e E 3 L4 - .. E o 36Ar(35 MeV) q) 3 10—5 ___0 36Ar(20 MeV) __ '3' I 1"‘N (50 MeV) ; i 1045 ' l L l n l l ‘ o 1 2 3 4 5 NIMF(Z%4) gure 2.7 Probabilities per fission trigger forthe detection of NIMF fragments lth 224 gated on central collision(AP/P>0.5). Curves are Poisson distributions. 2.4 S a no gran Ball; the c 4r ( dete desi; that 31 2.4 Summary This experiment was the first 41: experiment at N SCL which was performed with nearly 41: solid angle coverage. The detector, "Dwarf Ball/ Wall", was a pioneering device for intermediate energy heavy-ion nuclear collision experiments. It has a larger phase space coverage and dynamic range than that which characterized the previous devices used to study multi-fragment emission at intermediate energies [Bou87]. Nevertheless, this device also diSplay some limitations such as a poor resolution in particle identification specially for Ms, a non-polar geometry, a relatively small dynamic range(1sZle), and a low granularity. Moreover, energy calibrations for the response of the Dwarf Ball/ Wall were not performed for this first experiment, which seriously limited the directions of the data analysis. Therefore our group decided to build a new 41: detector which could overcome some of these limitations. This new 41: detector, the MSU Miniball, was completed in the spring of 1990 after 2 years of design work, test runs, and the mass- production of the detectors. The next chapter describes the technical details of this device. Cha with 1 grant as fol ChapterBMSU Miniball - 41c fragment detection array In order to study multi-fragment disintegrations of hot nuclear systems with nearly 41: coverage, we have constructed a fragment detection array of high granularity and low detection thresholds. The design criteria we considered were as follows : (1) High granularity to handle large multiplicity events. (2) Good particle identification for Z = 1-20. (3) Low energy threshold. (4) Large solid angle coverage. (5) Mass identification for light charged particles (H+He). (6) Good energy resolution. (7) Modular design; compact and portable. (8) Polar coordinate geometry. In its original configuration, the device covers the angular range of elab= 9 0-160 ° with 188 phoswich detectors consisting of 40 pm thick plastic scintillator foils and 2 cm thick CsI(Tl) scintillators. Thresholds for particle identification range from E/Az1.5 MeV for alpha particles to E/Az3 MeV for Ca ions. Individual detectors can resolve elements from Zz1-18, as well as isotopes of hydrogen and helium. Fission fragments can also be identified. Charged particle detector arrays based on similar techniques have also been built by two other groups [Sar88,Dra89]. 32 3.1 l Figui aSSQI mou phol is n tern} plats aftei allm neig bear dett 33 3.1 Mechanical Construction of the Array The Miniball phoswich detector array is designed to operate in vacuum. Figure 3.1 shows an artist’s perspective of the three-dimensional geometrical assembly of the Miniball sitting on the turntable of the 92" scattering chamber. The array consists of 11 independent rings coaxial about the beam axis. For ease of assembly and servicing, the individual rings are mounted on separate base plates which slide on two precision rails. The rings and detector mounts are made of aluminum. Good thermal conductivity between detectors and the mounting structure allows the conduction of heat generated by the photomultiplier voltage divider network into the array superstructure. This heat is removed from the Miniball by cooling the base plates to the desired temperature by circulating ethanol through a copper pipe attached to the base plates. By this means, constant operating temperature in vacuum is achieved after a brief equilibration time. The individual detector mounts are designed to allow the removal of any detector without interfering with the alignment of neighboring detectors. The entire assembly is placed on an adjustable mounting structure which allows for the alignment of the apparatus with respect to the beam axis. Figure 3.2 shows a half-plane section of the array in the vertical plane which contains the beam axis. Individual rings are labelled by the ring numbers 1- 11 which increase from forward to backward angles. For each ring, the number of detectors is given in parentheses. For a given ring, the detectors are identical in shape and have the same polar angle coordinates with respect to the beam axis; these angles are indicated in Figure 3.2. In Table 3.1, we list the solid angles and the ranges of polar and azimuthal angles which are covered by inidvidual detectors of the Miniball. Since the angular distributions of the emitted particles Figure fragme SYStem‘ Omittec 34 ... . WE Figure 3.1 Artist’s perspective of the assembly structure of the.Miniball.41r fragment detection array. For clarity, electrical connections, the light pulsmg system, the cooling system, and the target insertion mechamsm have been omitted. Fig lab pol ind 35 MSU-90-046 Figure 3.2 Half-plane section of the Miniball array. Individual detector rings are labelled 1 through 11; numbers of detectors per ring are given in parentheses; the polar angles for the centers of the rings are indicated. The dashed horizontal line indicates the beam axis. Table 3 36 Table 3 .1 Coverage in solid angle, polar and azimuthal angles for individual detectors of the Miniball. Ring 1 was not used in this experiment. IRggttl e(d_eg) I AQ(msr) I A9(dg) I Ad>(deg) I I 1 I 12.5 I 12.3 I 7 I 30 I J; I 19.5 I 14.7 I 7 I 22; I I 3 | 27.0 I 18.5 I 8 I 18.0 | I 4 | 35.5 I 22.9 I 9 I 15.0 I I 5 I 45.0 I 30.8 I 10 | 15.0 I I 6 I 57.5 I 64.8 I 15 I 18.0 I | 7 I 72.5 I 74.0 I 15 I 18.0 I I 8 I 90.0 I 113.3 | 20 I 20.0 I I 9 | 110.0 I 135.1 I 20 | 25.7 I |10 I 130.0 I 128.3 | 20 I 30.0 I I11 I 150.0 I 125.7 I 20 I 45.0 I are sma achi kee] indI Iabt giv: . ang pla cor bet IOII 3.4 are ele on ml the ex IIII C0 C0 37 are strongly forward peaked, the solid angle subtended by forward detectors is smaller than for detectors at backward angles. Variations in solid angle were achieved largely by placing detectors at different distances from the target while keeping their size approximately constant. The front face geometries of the individual CsI(Tl) crystals are shown in Figure 3.3. Different detector shapes are labelled by the respective ring numbers with the number of detectors per ring given in parentheses (see Figure 3.2 for the definition of the ring numbers). The crystals are tapered such that front and back surfaces subtend the same solid angle with respect to the target location. In order to save cost of fabrication, the curved surfaces corresponding to constant polar angle were approximated by planar surfaces. The resulting loss in solid angle coverage is on the order of 2%, comparable in magnitude to the loss in solid angle coverage resulting from gaps between individual detectors (which must be provided to allow for mechanical tolerances and optical isolation between neighboring crystals). An isometric drawing of the target insertion mechanism is shown in Figure 3.4. The targets are mounted on frames made of flat shim stock of 0.2 mm thickness. Each target frame is attached to an insertion rod. The insertion rods are mounted on a tray which can be moved parallel to the beam axis. An electromagnetic clutch provides the coupling to the insertion and retraction drive once a target rod is located at the appropriate position. A third drive allows rotation of an inserted target about the axis of the insertion rod. This rotation of the target plane is useful for the determination of the shadowing a detector experiences when it is located in the plane of the target frame. All the motions of the target system are remotely controlled by a computer program through controllers located outside of the vacuum chamber. In its present configuration, the detector array covers a solid angle corresponding to about 89% of 41:. The loss in solid angle can be decomposed Figu theii 38 MSU-90-O45 I 2 3 4 . (.2)‘ as) <20) (24) I 5 6 7 8 (24) (20) (20) ('8) 9 IO ll (I4) (I2) (8) I fl I fi 5 cm Figure 3.3 Front views of different detector shapes. The detectors are labelled by their ring number; numbers of detectors per ring are given in parentheses. Izigure 39 Figure 3.4 Isometric View of the target insertion mechanism. into appr med prov 3.2 thic. scin the cho im} via 40 into the following contributions: (i) beam entrance and exit holes (4% of 41:); (ii) approximation of the curved surfaces corresponding to constant polar angle by planar surfaces (2% of 41:); (iii) optical isolation of detectors and allowance for mechanical tolerances (4% of 41:); (iv) removal of one detector at 9=90° to provide space for target insertion mechanism (1% of 41:). 3.2 Detector Design All phoswich detectors of the array are composed of a thin (typical thickness: 4 mg/cm2 or 40 um) plastic scintillator foil, spun from Bicron BC-498X scintillator solution, and a 2 cm thick CsI(Tl) scintillator crystal. A schematic of the detector design is given in Figure 3.5. In order to retain flexibility in the choice of scintillator foil thickness, the scintillator foil is placed on the front face of the CsI(Tl) crystal without bonding material. (Tests indicated that a slightly improved resolution could be achieved when the two scintillators are coupled via a thin layer of optical cement, see Section 3.8 below.) The back face of the CsI(Tl) scintillator is glued with Optical cement (Bicron BC600) to a flat light guide made of UVT plexiglas. This light guide is 12 mm thick and matches the geometrical shape of the back face of the CsI(Tl) crystal. This light guide is glued to a second cylindrical piece of UVT plexiglass (9.5 mm thick and 25 mm diameter) which, in turn, is glued to the front window of the photomultiplier tube (Burle Industries model C83062E). The photomultiplier tube and the cylindrical light guide are surrounded by a cylindrical u-metal shield (not shown in the figure). Front and back faces of the CsI(Tl) crystals are polished; the tapered sides are sanded and wrapped with white teflon tape. The front face of the phoswich assembly is covered by an aluminized mylar foil (0.15 mg/cm2 mylar and 0.02 mg/cm2 aluminum). Figul'l Il-met 41 MSU-90-03l Fast scintillator L‘ght gunde \ \\\\\\\\\\\\§ Q % PMT I I Al-mylar foil I \ Optical cement \\ Figure 3.5 Schematic of phoswich assembly of individual detector elements. The u~metal shield covering the photomultiplier is not included. tema [Km max max scin ma) . ath C51! C51 trar can thic 42 The Bicron Beta-paint (BC-498X) used to make fast scintillator foils is a ternary plastic scintillator consisting of a solvent (PVT:Polyvynyle Toluene), a primary solute, and a secondary solute, wave shifter, dissolved into xylene [Kn089,Hur85]. The primary scintillation of the plastic scintillator foil has its maximum intensity at 370 nm. In bulk material of the scintillator, the intensity maximum is shifted to 420 nm by the addition of a wavelength shifter. Our scintillator foils are, however, too thin for an effective wavelength shift and maximum emission remains in the far blue region of the spectrum (the full width I at half maximum lies between 350 and 450 nm). The absorption of this light in CsI(Tl) places a constraint on the maximum useful thickness of the CsI(Tl) crystals. For example the transmittancy of visible light at 350nm through 20mm CsI crystal is about 79%, while it is about 85% at 450nm and reaches a maximum transmittancy at about 700nm [Har88]. Additional absorption in the light guides can be minimized by using UVT plexiglas rather than standard plexiglas light guides. The transmittancy of the 24mm thick standard plexiglas is almost 0% at 350 nm and about 50% at 380 nm., which can seriously reduce the blue light of Beta-paint [Bic90]. On the other hand the transmittancy of UVT of the same thickness is about 90% over 350nm. The cathode responsivity of our photomultiplier tube has maximum of 70% at between 350 and 450 nm, which fits the emission spectrum of Beta-paint well. Such considerations become particularly important for phoswich detectors utilizing thin scintillator foils in efforts to reduce particle detection thresholds. Figure 3.6 shows a photograph of the basic photomultiplier assembly used for all detectors. The phoswich and matching first light guide have not yet been attached in this photograph. A precision- machined aluminum ring is glued to the u-metal shield surrounding the photomultiplier and the cylindrical light 43 Figure 3.6 Photograph of photomultiplier assembly. The scintillator and the first matching light guide are removed. The ring glued to the u-metal shield defines the alignment of the can housing the voltage divider. The can has been removed to expose the voltage divider. guid whe volt; the lead opei 44 guide. This ring provides the alignment for a precision machined aluminum can which houses the voltage divider and which defines the detector alignment when bolted to the rings of the array support structure. In order to expose the voltage divider chain, this aluminum can has been removed and placed next to the photomultiplier. The active voltage divider chain is soldered to the flying leads of the phototube. To prevent destruction of the FETs by sparking during operation in poor vacuum, the entire divider chain, including the leads to the photomultiplier tube, is encapsulated in silicone rubber (Dow Chemical Sylgard 184). Vacuum accidents occurring with fully biased detectors do not lead to divider chain failures. In fact, the detectors are designed to survive a full pumping cycle from atmospheric pressure to vacuum with bias applied to them. The 10-stage Burle Industries model C83062E photomultiplier tube was chosen because of its good timing characteristics (1: ==2.3 ns), its large nominal gain (z107), and its good linearity for fast signals. Sin: the apparatus is designed to operate in vacuum, active divider chains were chosen to minimize the generation of heat. (The base circuit was designed by M. Maier.) A schematic of the active divider chain is given in Figure 3.7. The final stages of the divider chain are of the "booster" type which provides improved linearity for high peak currents generated by large signals of the fast scintillator. 3.3 Detection Principle With this highly granular geometry the additional requirements for the Miniball telescope are good particle identification, energy resolution, and low energy threshold. The detection principle of the Miniball telescope is based on the plastic-CsI(Tl) phoswich technique employed for example in the Washington University’s Dwarf Wall/ Ball. The anode signal from the photomultiplier tube Figur detec 45 MSU-QO-OSZ Awe 300. COAXIAL CALELE Ikoi: J 1- IOMa DIOO— —V Z$§FT ’ InF t— :I IOMQ. 09H firéVAVAV ”ll-FT ‘ t— 4?“ :3 :E 8.2 M9. 03% '"F fir—w. 'I‘FT 47o v~24osij :: 07c— lnF I— 4“ MOSFET : [#3 9”“ ‘rga mess ”F l— 1:820k015 3.9m: 1» 680m lnF ’ 05E ’1! ‘ I "F t— 5; 680m 04.— \/—_L InF I"‘I 5; 680k0 oar fiz——-—L lnF t-— s IMO. 020— ”F I: ‘ scam |—» <: I. 8M9. PHOTO- lnF ‘ SHV CABLE -HV r —v-——J )— CATHODE L IOka J 1 InF 7" Figure 3.7 Schematic of the active voltage divider used for the Miniball detectors. FiguI'E 46 Photomultiplier tube signal z, 0 150ns 1.5us I_I | | I "faSt " "slow" " tail" 33ns 390m 1.5us Figure 3.8 Schematic of photomultiplier sygnal with three time gates applied. sci. de rel de 0.4 47 has the shape depicted in Figure 3.8. It consists of fast and slow light components. We apply and integrate the charges with three different gates, "faSt", "slow", and "tail". The "fast" and "slow" signals represent approximate AE (energy loss in fast plastic foil) and E (energy loss in CsI(Tl) crystal) signals. The elemental resolution is achieved in the 2-dimensional map of the "fast" and "slow" signals because of the dependence of stopping power dE/dx on the particle charge, mass, and energy. The CsI(Tl) crystal has several good characteristics as a stopping scintillator. First, it has large stopping power because of its relatively large density (4.51g/cm3). Hence a large dynamic range could be achieved with relatively short crystals. Second, its light output has two components of different decay times. One component (dominating the "slow" signal) has decay times of 0.4 - 1 us, depending on various particle type and energy [St058,Ben89,Kes69]. The second component is governed by decay times of the order of 4-7 us [Ben89,St058,Kes69]. This component dominates the "tail" signal. The relative intensity of the "slow" and "tail" components depends on the specific energy loss of the detected particles. From this prOperty the mass resolution of H and He isotopes can be achieved using the pulse shape discrimination [Ala86,Ben89]. An example of the applied time gates is shown in Figure 3.8. Third, CsI(Tl) has good energy resolution for charged particles (about 1% for 100MeV on particle) compared with other scintillator materials such as plastic scintillator [Gon88]. So to achieve the requirements listed earlier we investigated the characteristics and fabrication methods of these scintillators in great detail. 48 3.4 Uniformity of Scintillation Efficiency of CsI(Tl) Previous experience with CsI(Tl) crystals used for the detection of energetic particles had revealed difficulties with the production of scintillators with uniform scintillation response [Gon88,Gon90c]. Therefore, considerable attention was paid to select CsI(Tl) crystals of uniform scintillation efficiency. In previous tests of large cylindrical CsI(Tl) crystals [Gon88,Gon90c], non-uniformities of the scintillation efficiency were detected by measuring the response to collimated y- rays. Such measurements are relatively easy to perform since they can be done in air. However, they are less suitable for small volume crystals, since collimated y- rays sample a relatively large volume of the crystal. Small scale fluctuations of the scintillation efficiency may stay undetected. In addition, measurements for small non-cylindrical crystals are less precise, since the shape of the Compton background depends on the position of the collimated y-ray source. Such dependences lead to additional uncertainties in the extraction of the photopeak position. It was determined, however, that nonuniformities of the scintillation efficiency can be detected very sensitively by scanning the CsI(Tl) crystals with a collimated a-source in vacuum. All crystals ordered from various manufacturers were rectangular in shape with dimensions of 2"x1.5"x1". They were polished at the front and back faces (with dimensions of 2"x1.5") and Sanded at the sides. The back face was optically coupled to a clear acrylic light guide with the same dimensions as the crystal. This light guide, in turn, was optically connected to a photomultiplier tube of 1" diameter. The sides of the CsI(Tl) crystal and of the light guide were wrapped with white teflon tape. By covering the front face of the CsI(Tl) scintillator with an aluminized mylar foil, a uniform light collection efficiency was achieved. (Without a reflective entrance foil, the light collection etfil 8.7I avc an; 49 efficiency decreased by about 5% from the center of the front face to its sides.) The front face of the crystal was scanned in vacuum and the peak location of the 8.785 MeV a-line from a collimated 228Th oc-source was monitored. In order to avoid edge effects, regions within about 2 mm of the side boundaries of the crystal were not scanned. Most tests were performed with a simple multichannel analyzer equipped with a peak sensing ADC; in those instances the anode signal of the photomultiplier was shaped and amplified with standard electronics, using integration and differentiation times of 1 us. Figure 3.9 shows the results of a scan for a crystal exhibiting a large gradient of the scintillation efficiency. The horizontal axis of the plot shows the location of the collimated a-source with respect to the center, along the short symmetry axis of the front face. Different surface treatments of the front face of the scintillator did not affect the measured variation of the scintillation efficiency. In order to demonstrate that such variations were related to the bulk material of the scintillator we exchanged the role of front and back faces of this scintillator and performed an equivalent scan of the parallel surface (i.e. the previous back face). The results of the two scans are compared by the solid and open points in Figure 3.9. (The coordinate system was kept fixed with respect to the CsI(Tl) crystal.) Nearly identical variations of the scintillation efficiency are observed across the two parallel scintillator surfaces indicating that the measured large gradient of the scintillation efficiency persists through the bulk material of the sample. The measurements shown in Figure 3.10 were performed by integrating the anode current of the photomultiplier with a charge integrating ADC using time gates of At = 0.1-0.5 us and At = 1.1-4.1 us which select the fast and slow scintillation components of CsI(Tl). The fast component exhibits a larger variation of the scintillation efficiency than the slow component. Since the 50 MSU- 90-036 10TIrI—IIIfIIrrTIIITIIIIIIIIrTIIIITI l —— front side )9 - -- back side g’ lllllJlLfilLlUlll1114‘ll_ll_LLLl Relative Light Output (7.) N TIIIr—TITIIITII—IFII—IIIIjITIflT—r LLngJlLll—IIIIILLLLLl—llIlJllJzLi -1o -5 o 5 10 15 ‘ X(mm) t I ...; 01 Figure 3.9 Relative variation of scintillation efficiency measured for two parallel surfaces of a CsI(Tl) crystal by using a collimated a-source. The axes of the coordinate system are parallel to the sides of the scintillator; the coordinates are fixed with respect to the scintillator. N! I mufifind. H:.Q.-..— 01? O DH: 8.: nwhy Figu integ com 51 MSU-eo-oas 6 - I I I I r r I I r I I I I I I r I r I I r : E3 5 E" —- ADC gate : 0.1-0.5 [LS I V t ---- . _ : ...: 4 _____ ADC gate. 1.1 4.1 [1.8 _: 1'3 Z 2 3' 3 :— __: g 3 j ~03 2 l— —;I £3 E 3 53° 1 3— -a 0 E ,’§"'§§ 0 ‘- _. .2 : j +3 - I .4 .94 -1; , -— ’ __‘ g 5%” § _§ 3 “3.3-£74? —; _3: I l 1_ 1 J I l L l__l J J_l i _L L. L4 L L: ~20 -—10 0 10 20 X(mm) Figure 3.10 Variations of scintillation efficiency measured with a charge integrating ADC for two different time gates selecting the fast and slow components of scintillation for CsI(Tl). 52 relative intensity of fast and slow scintillation components depends strongly on the T1 concentration [Bir64,Man62], the observed variations of scintillation efficiency are most likely due to gradients in the T1 concentration. Crystals incorporated into the Miniball were preselected by scanning the 1.5"x2.0" rectangular surface of original crystals along two perpendicular axes and requiring a uniformity of scintillation response better than 3%. The preselected crystals were then milled into their final shapes and scanned a second time, requiring uniformity of response within 2.5%. The preselection process avoided expensive machining of poor quality crystals; it was about 90% efficient for the selection of crystals of the desired quality. Figure 3.11 compares variations of scintillation efficiency detected with collimated a-particles of 8.785 MeV energy (source: 228Th) and y-rays of 662 keV energy (source: 137Cs). The left and right hand panels give examples for a rejected and an accepted crystal, respectively. The enhanced sensitivity of the a-particle scan is obvious. It is probably caused by the fact that a-particles sample a much smaller volume than y-rays and that the two kinds of radiation exhibit different sensitivities to the T1 concentration [Bir64,Man62]. The results of our standardized prescans are diSplayed in Figure 3.12. The histogram shows the number of scanned crystals, obtained from different companies, as a function of maximum detected scintillation uniformity. Crystals obtained from one supplier (a) were grown with the Czochralski technique [C2018,Bric65]; crystals obtained from three other suppliers (b-d) were grown with the Stockbarger-Bridgman technique [Bric65,Brid25]. Crystals grown with the Czochralski technique exhibited, on average, more uniform scintillation efficiencies than crystals grown with the Stockbarger-Bridgman technique. Quality differences between different suppliers using the Stockbarger-Bridgman technique may not be significant since only a limited number of crystals were PNV ..VFFCJLFC ..FC—KQTh QKFW+GFQQ FFEVV Relative Light Output (7.) 53 15 MSU-90-034 IIIFIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIITIIrrrI — Rejected —- Accepted '- __ __ . a __ IIILLLIIIIIILILLIIJJILJIIJ LJIJJIilLLIIJlLILllLIIJlLI -10 -5 0 5 10 —10 -5 0 5 10 X(mm) Figure 3.11 Variations of scintillation efficiency detected with collimated 0t- particles of 8.785 MeV energy (solid points) and collimated y-rays of 662 keV energy (open points). The left hand panel shows the measurement for a detector which was rejected. The right hand panel shows the measurement for a detector which was incorporated into the Miniball. {M Figure from ( given i 54 MSU-90-051 6O - Companies E? $3 - a (127) ‘5 b (75) g 50 — :35 E c (41) 3‘ _:_: d (23) . U =_'-: "c 40 t- \3', 33. §:-: [_‘ \f-E . \‘ == tr: s... O \5‘3 \:-- ‘7' §-:-'- §.: L: §:=:I § s3; §§ §':: \: 10 - $33? $3: §3.='-: § s3? §§ §§2 \: \E‘ \E \3; RE >3. ta ks; 0-1 1-2 2-3 ' > 3- Scintillation nonuniformity (%) Figure 3.12 Maximum detected scintillation nonuniformity for crystals obtained from different suppliers. The number of crystals scanned for each supplier is given in parenthesis. scanm given to res since direct reveai side V some with i relati respo US. 3.5 I [Mey ratio to ab meas bottl Optic befOJ ball J Law the \ 55 scanned for each supplier. (The number of crystals scanned from each supplier is given in parentheses in Figure 3.12.) Furthermore, some differences are expected to result from different ways of cutting the 2"x1.5"x1" crystals from the ingot, since Tl concentration gradients build up largely in the vertical crystal growth direction. Our standard scan of the 1.5"x2.0" face of each crystal should then reveal an especially large gradient of the scintillation efficiency when the 2" long side was cut parallel to the vertical growth direction of the ingot. Selective cuts of some crystals from various locations of the ingot and with different orientations with respect to the crystal growth direction provided supporting evidence for the relationship between the T1 concentration and the uniformity of scintillation response. For most of the crystals, such detailed information was not available to US. 3.5 Fabrication of Scintillator Foils Scintillator foils were made by using the so called "spin-coating method" [Mey78,Nor87]. The original Beta-paint solution was ordered with a 40% weight ratio of scintillator material to xylene. It was then diluted by adding more xylene to about 30% until the solution had the desired viscosity of 20-30 poise. After we measured the viscosity of the solution, we stirred this solution inside a brown bottle using a stainless steel rod for about 5-10 minutes to mix it well and achieve optical uniformity. Then we let the solution in this bottle setle for about half day before spinning to remove all the bubbles generated during stirring. The viscosity was determined by measuring the terminal speed, v, of a steel ball sinking in a glass tube filled with a sample of Beta-paint. Correcting Stoke’s Law for the finite diameter of the glass tube gives the following expression for the viscosity [Din62]: Her( mot SW 5110 56 2 = W (1-2.104(§)+2.09(§)3-0.95(§)5 ) . (3.1) Ti 9v Here, n denotes the viscosity, r and R are the diametersof the steel ball and the glass tube, g is the gravitational acceleration, and p o and p are the densities of the steel ball and the Beta-paint, respectively. For the fabrication of scintillator foils, a glass plate of 23 cm diameter was mounted horizontally on a small platform connected to the drive of an electrical motor which allowed spinning of the plate about its center at a preselected speed. To facilitate the removal of spun foils, the glass plate was covered successively with metasilicate solution and Teepol 610 and then wiped to leave only a thin film of the releasing agents on the glass substrate. An appropriate amount of beta paint was poured on the center of a glass plate. In order to provide rapid spreading of the initial solution, the plate was spun at an enhanced speed for the first few seconds until the entire plate was covered with Beta-paint. Following this rapid startup, the glass plate was spun at the preset rotational frequency for a duration of approximately 4 min until a solid foil had formed. After spinning, the glass plate was stored in a flow of dry nitrogen for a duration of about eight hours. The foil was then peeled from the glass plate, mounted on a frame, and placed in a dry nitrogen atmOSphere for another 24 hours to allow further evaporation of residual xylene. We obtained good and reproducible results by using more dilute solutions and spinning at lower rotational frequencies than described in refence [N or87]. A number of measurements were performed to determine the relation between rotational frequency and foil thickness in this Operating range. The results of these measurements are shown in Figure 3.13. For each foil, thickness and homogeneity were determined by scanning the foil in vacuum with a collimated 228 . . . Th a-source and measuring the energy of the transmitted a— particles in a o\mEc 9. GE Figu of 3} show 57 MSU-90-037 8 I T I l Irlrlrrrl—rIIIrI—rti ‘7— _ t (mg/cmz) '7 3 .— .— 2 1 J— I L J l l L L l l__ 1__ I I l l L [J 1_ L4 200 300 400 500 600 V (rpm) Figure 3.13 Relation between scintillator foil thickness and rotational frequency of spinning measured for solutions of Beta-paint of different viscosity. The lines show fits with the power law expression, toc v calit area hit] for i ath wit] Cor opt foil ion dis an M pl; va “F sci Cs 58 calibrated silicon detector. The energy loss in the foil was then converted to an areal density according to reference [Lit80]. The spun foils were uniform in thickness to within typically 1-2% over an area of 7x7 cmz. Scintillator foils used for instrumenting the Miniball in its present configuration were selected to have a thickness of 4.0 :t 0.12 mg/cmz. We have explored the light collection efficiency for scintillator foils placed with and without optical cement on the front faces of the CsI(Tl) scintillators. Considerable effort was spent on developing a technique which provides an optically clear glue bond of minimum thickness between the plastic scintillator foil and CsI(Tl) crystal. Best results were obtained with Epo-tek 301 which has a low viscosity of 1 poise and a curing time of 1 day. A thin layer of the epoxy was distributed on the scintillator foil by spin coating. By means of a thin rubber pad and a weight, the epoxy coated foil was pressed onto the CsI(Tl) crystal in vacuum and cured in a clean and dry nitrogen atmosphere. The pressure applied by the weight was typically 13 kPa. Glue layers of 300-500 ug/cm2 areal density were achieved, with nonuniformities of the order of 50%. We investigated the position dependence of the light collection efficiency for phoswich detectors prepared with and without glue bonds between the plastic scintillator foil and the CsI(Tl) crystal, by measuring the response at various locations on the front face (labelled 1 through 11 in Figure 3.14). The upper and lower panels in Figure 3.14 show the relative variations of the plastic scintillator response (integrated over the time interval At st: 0-50 ns) and the fast w: 60-400 ns) for or- fa CsI(Tl) component (integrated over the time interval Ats 10 particles sampling different locations of a phoswich fabricated with (upper panel) and without (lower panel) a glue bond between the two scintillators. Both detectors have sufficient uniformity of response to provide good particle identification (see also Section 3.7). In this bench test, a better uniformity of m 070 tbs 0 aflms flaw: H5 ARV Figu Pan. 30in "810' and em 59 nsu-eo-oaz e h- .1 ~ 4 L— GLUED 0 fast plastic 9 4 I 2 _ [I \ " "' - CSI // ‘0 4 o _ _. -2— \ _. _ ‘d \ ’ \ ’ . _4__ 0’ U __ 6 ' l i 1 L L i i l J J J ‘ g - .. E 4 r- WRAPPED fast plastic -i 2 1. - - - CsI — E o —- 4 F —2'— L . - 4 _4—_ —.i _at L n I L | L L 1 t i i ‘ o 1 z 3 4 5 e 7 a 9 10 11 12 position e 3.14 Position dependent response of phoswich detectors with (upper nel) and without (lower panel) glue bond between plastic and CsI(Tl) ntillators. Solid and open points show charges integrated by the "fast" and ow" time gates, At St=0-50 ns and Ats W=60-4:00 ns, respectively. The same foil fa 10 d CsI(Tl) crystal were used in this test. Various positions sampling the trance window are labelled 1-11. reS] 6O esponse was obtained for the phoswich without glue bond. Somewhat larger but still tolerable) fluctuations are observed when a glue bond is applied. The bserved fluctuations can be attributed to thickness variations of the glue layer hich cause variations in energy deposition in the stopping CsI(Tl) scintillator see open points in the top panel). Since some of the CsI(Tl) signal overlaps in ’me with the fast plastic scintillator signal, a correlated modulation appears in e charge integrated by the fast time gate. For more energetic particles, this ffect should be less important as the energy loss in the glue layer decreases. The fabrication of phoswich detectors without glue bond has the advantage f allowing changes in scintillator foil thickness to be made relatively easily. lued phoswich detectors, on the other hand, are mechanically more rugged. In ur test runs, improved particle identification was obtained with glued phoswich etecto'rs (see Section 3.7). However, these improvements do not appear to be :ompelling enough to justify the initial use of glue bonds in the Miniball array. 30 all the experiments up to the present time (September of 1991) have been performed without glue layer between fast scintillator and CsI(Tl) crystal. 3.6 Data Acquisition Electronics Figure 3.15 shows a block diagram of the data acquisition electronics. The mode current from the photomultipliers is split via passive splitters into the 'fast", "slow", "tail", and "trigger" branches of relative amplitudes I fas t : I510W : Itail: trig z 0.82 : 0.04 : 0.04 : 0.1. The "slow" and "tail branches are connected directly ’mm the Splitter to their respective fast encoding readout analog—to-digital :onverters (FERAs). The gates for the "slow" and "tail" FERAs are 390 ns and 1.5 is wide and open 150 ns and 1.5 us after the leading edge of the linear signal, 'espectively. For the "fast" branch, a linear gate(Phillips 7145) is inserted between 61 Multiplicity Selector disc. AND veto F.I.F.0. — FERA J r___A . 0.1 180m DGG Splitter . . . a» disc TFC E/N OR \ rog. ‘69 -° {5'3 sealer i 2 a ... lStlns fast lin. ate FERA E/N OR sum but re ister Oscill- Linear g slow oscope F.1.F.0. FERA gm @ [E] \ tail 1 [- VIEERA E/N @ DGG LED ECL/NIM converter PMT Photomultiplier Tube dual gate generator Fan in Fan out Fast Encoding and Readout ADC Time-to-FERA Converter gure 3.15 Schematic diagram of the data acquisition electronics of the Miniball. he di 62 e passive splitter and the "fast" FERA. This linear gate allows the individual gating of each "fast" channel which cannot be achieved with the common gate FERAs (see also the discussion in Section 3.7). The linear gate is opened 10ns prior to the leading edge of the linear signal and for a duration of 33 ns. The "fast" FERA is gated by a common gate of 140 ns width which begins approximately 60 ns prior to the leading edge of the linear input signal. The trigger branch, I trig’ is re-amplified by a fast amplifier and fed into a leading edge discriminator module (Phillips 7106), the output of which provides the stop signal for the time-to-FERA-converter and opens the linear gate for the ”fast" channel. The detailed timing diagram is shown in Figure 3.16. Each discriminator module provides a sum output for its 16 channels. The amplitude of this sum signal is proportional to the number of channels which have triggered. By setting a discriminator level on the linear addition of all discriminator sum outputs, a simple multiplicity trigger is obtained. Each discriminator channel has 1.5 us internal dead time to prevent double triggering. In order to reduce dispersive losses for the fast anode current pulse representing the response of the plastic scintillator, the data acquisition electronics is located close to the measurement station. Discriminator thresholds and photomultiplier gains are adjusted via remote computer control. Remote inspection of each detector signal is also possible by using the sum output of the linear gate modules and selectively masking the discriminators. 3.7 Particle Identification The particle identification resolution of various phoswich detectors was 4 1 7 tested for fragments emitted at about 91ab~35° in the 0Ar+ 9 Au reaction at E/A=35 MeV. All data were taken with the standard electronics setup described Fig 63 TimingDiagram t=0 d . . Vf Signal Input 33 l output of 16 ch. Disc. \/ input of linear gate t=l70ns V output of linear gate output of multiplicity selector t=100ns w=140ns Ir input of slow FERA t=150ns 390ns ‘ gate for slow FERA (( 1V “V/ input of tail FERA t=1500ns (L .....___...._.._ ” 1500ns gate for tail FERA —— F ERA fast gate gure 3.16 Timing diagram of analog signal and gates. (a) ~ .-....,. .‘ 7V. . . (b) SLOW igure 3.17 Particle identification obtained from two-dimensional "fast" vs. :low" matrix for reaction products emitted at 81ab=35° for the ‘ "Ar+l ° 7Au :action at E/A=35 MeV; a scintillator foil of 4 mg/cm2 areal density was used 'ith a thin glue coupling to the CsI(Tl) crystal. (a) Matrix of raw data. (b) inearized matrix. An intensity threshold of 2 counts per channel has been set. 65 n the previous section. Part (a) of Figure 3.17 shows a two-dimensional plot of e "fast" versus the "slow" charge integration parameters for a phoswich onsisting of a 4 mg/cm2 scintillator foil glued to the CsI(Tl) crystal. Elemental 'dentification up to Z=18 is achieved over a considerable dynamical range of article energies. Part (b) of the figure shows a linearized presentation of these ata which is more suitable to diSplay the resolution of the device. From such a inearized presentation, projections on the particle identification axis can be generated which show the particle identification resolution in a more quantitative form. Spectra projected on the particle identification axis are shown in Figure 3.18. Part (a) of the figure shows the projection of the data displayed in Figure 3.17; part (b) show the result for a phoswich using a 4 mg/cm2 scintillator foil without glue bond. Better resolution is obtained by using a glue bond between the scintillator foil and the CsI(Tl) crystal. Additional improvements in particle identification resolution can be achieved by increasing the thickness of the scintillator foil. For a specific experiment, the benefits of improved particle identification resolution due to an increase in scintillator foil thickness must be weighed against the ensuing higher energy threshold. We have also explored the use of thinner foils and found that the resolution deteriorates rapidly for scintillator foils thinner than 3 mg/ cmz. For most purposes, particle identification provided by foils of 4 mg/cm2 thickness is satisfactory. The use of thin scintillator foils in phoswich detectors for particle identification is complicated by the fact that the fast plastic scintillator signal is superimposed on the rising signal from the CsI(Tl) scintillator. Good particle identification via direct charge integration depends critically on well defined integration times. Electronic walk introduced by leading edge discriminators changes the detailed shape of a particle identification line in the "fast" versus 66 — ‘ ~1— d)- +— ”F 1- “r ..l. .1)- cut-FF- q llll]llllJLlllellIJLllljjll llllljlllljljlllllJJllllJlJl 1.5x105 err.,....l....l. 40pmglued A D v 5 1:100 1:500 1.0x10 0.51110‘5 IIIIIIIIIITIIFIIITIIIIIIIlll “- 0.01:10‘5 Counts 1.01:10“ _ 0.51:105 IIIlIIIIlIIIIIIIIIIIIIIIIIII OE— l. t l' L l—. t 0.01:10‘5 U! H O ...; 01 N C Figure 3.18 Projections of linearized particle identification spectra for reaction products emitted in the ‘ °Ar+‘ ° 7Au reaction at Blah—135° using phoswich detectors using scintillator foils of 4 mg/cm2 thickness, (a) with and (b) without glue coupling between the scintillator foil and the CsI(Tl) crystal. 67 "slow" matrix, but it has only minor effects on the separation between adjacent particle identification lines. Time jitter in the integration gate, however, does have an adverse effect on the particle identification resolution, since it produces fluctuations in the amount of CsI(Tl) scintillation integrated by the fast time gate. If the fast component is integrated by ADCs with a common gate mode, loss in resolution will occur whenever there is a time jitter between gates provided by different channels. Loss in resolution will be inevitable when more than two detected particles with different flight times are converted in separate channels of a common gate ADC. For example, the difference of flight times between 70MeV proton and 70MeV carbon detected in ring 2 is approximately 2ns. The problem can be avoided by introducing linear gates into the "fast" channel which are individually opened (see Figure 3.15). Figure 3.19 illustrates the sensitivity of the particle identification to time jitter in the integration gate for the "fast" channel. The solid curves correspond to the centers of selected particle identification lines in the "fast" versus "slow" matrix (see, e.g. Figure 3.17). The dashed curves show how these particle identification lines are shifted when the gate of the "fast" ADC arrives two 2 ns later in time. (In these two measurements, the gate width was kept constant.) Even a 2 ns time jitter in the “fast" gate is sufficient to mix the particle identification lines of neighboring elements in the "fast" versus "slow" matrix. 3.8 Light Pulsing System Gain drifts of the photomultiplier tubes are monitored by a simple and compact light pulser system which operates in vacuum. In order to preserve the modularity of the device and avoid unnecessary removal of optical fibers during transport, each detector ring is provided with its own light pulser system. Figure 68 MSU-90-O38 I 400 300 — \ —( ;“= i . 7) _ U 200 a ,._. _ 100 —- 100 200 300 Slow (ch#) igure 3.19 Sensitivity of particle identification to time shift of the "fast" charge itegration gate. The solid and dashed lines show the loci of representative article identification lines in the "fast" vs. "slow" identification matrix obtained )r gates displaced by 2 ns with respect to each other. 69 MSU- 90-049 Fiber optics bundle PIN diodes ' ......... ......... ......... ..... LEDs LED dnver cucurt +24 V ---------------- ................ ........... -------------- .............. .............. uuuuuuuuuuuu .......... ......... ......... gure 3.20 Schematics of light pulser assembly and LED trigger circuit. 7O .20 shows schematics Of the mechanical assembly Of the light pulser system and Of the driving circuit for the light emitting diodes (LEDs) which is triggered by an external N IM logic signal. During experiments, the light pulser is triggered at a rate of about 1 Hz. Light is generated by simultaneously pulsing an array Of eight LEDs (Hewlett Packard HLMP-3950) which generate light at wavelengths around 565 nm. The emitted light is diffused by reflection from an inclined teflon surface. Light fibers which only view scattered light transport the light to the individual photomultipliers. Because of temperature fluctuations and aging effects, operation of light emitting diodes may not be stable over long periods of time. Therefore, the intensity of each light pulse is monitored by two PIN diodes (Hamamatsu 81223) read out by standard solid state detector electronics. The ratio of the signals of the two PIN diodes can be used to monitor their stability. The ratio of pin diode and photomultiplier signals can then be used to monitor the gain of the individual photomultiplier tubes according to the relation: Ch’ = Chx(Pin1+Pin2)/2Pmt . (3.2) Here, Ch denotes the ADC conversion measured for a given event, Ch’ is the conversion corrected for gain shifts, and Pmt, Pinl, and Pin2, are the channel numbers for LED generated light pulses detected by the individual photomultiplier tube and the two PIN diodes, respectively. Better than 1% gain stabilization is achieved if the temperature of the CsI(Tl) crystals is kept constant. (Variations of the scintillation efficiency of CsI(Tl) caused by temperature fluctuations cannot be detected with the light pulser.) It was verified, however, that active cooling of the base plate ensures rapid achievement Of a stable Operating temperature for the Miniball. 71 Figure 3.21 illustrates the gain stabilization achieved with the light pulser tem. The gain variations Of a photomultiplier (enhanced by variations Of the >ply voltage) were directly measured by irradiating a CsI(Tl) crystal with or- 'ticles emitted from a collimated 228Th source and monitoring the peak location the 8.785 MeV Ot-line; they are shown by the open points in the figure. The id points in the figure show the peak positions obtained in the Off-line analysis er correcting the gain variations according to information from obtained by light pulser system. Gain stability to better than 1% was achieved. 72 . MSU-90—O48 1-4 I. l I | | I I I I I I 3 I o o I 1.3 :- O uncorrected ‘j I O corrected 1 E 1.2 Er ” 0 "j KS : 1 5 1.1 L— ‘2 a) - _ Z : 3 E; 1.0;— --0- +- o- +-- -—+-—.--+-—-. 1 0.9 _— 0 a “i 0.8 :— J - o “ _ O 7 ' I I I I I L I L I I 3 0 1 2 3 4 5 6 7 8 9 10 11 Run # igure 3.21 Open points: gain variations of a CsI(Tl) photomultiplier assembly etermined by measuring the detector response to 8.785 MeV a-particles; solid oints: same data corrected for gain shifts in the Off-line analysis by using tformation from the light pulser system. 1 7 . . napter 4 36Ar + 9 Au reaction at E/A = 35 MeV With Minib all 1 EXperimental setup The experiment was performed using the MSU Miniball in the 92" diameter :attering chamber of the National Superconducting Cyclotron Laboratory at lichigan State University. An 36Ar beam of energy E/A=35 MeV and intensity =108 particles per second (~0.3nA) was extracted from the K500 cyclotron. ypical beam spot diameters were Of the order Of 2-3 mm. The target foil onsisted of 1 mg / cm2 197Au. During the experiment, a vacuum of better than 10-5 m was maintained in the scattering chamber. Water and hydrocarbon vapor omponents in the residual gas were strongly reduced by a large cold trap filled ith liquid nitrogen. By this means, carbon deposits on the target were reduced a negligible level eliminating contamination from auxiliary reactions. Light particles and complex fragments were detected using rings 2-11 of the iniball covering scattering angles Of (9 =16°-160.° and subtending a solid angle orresponding to 85% of 41c. As a precaljtlion against secondary electrons, rings 2 nd 3 were covered by aluminum foils of 0.81 mg/cm2 areal density. All the etectors were checked with a Th228 01 source located at the target position looking t "fast" and "slow" 2-dimensional spectra to see the resolution of the 8.785 MeV signals. The high voltage supplied to each detectors was adjusted to ompensate the variations in the gain and give similar dynamic ranges (Z=1-20) the "fast" and "slow" 2-dimensional spectra obtained with the beam on target uring the early stage of the experiment. Moderately suitable adjustments in the 73 74 ain of the photomultipliers could be made by using the approximate relation, \_l = (V ) . (4.1) this empirical relation was obtained from tests with an a source in which the relation between photomultiplier tube gain and cathode voltage was explored. The temperature was monitored with temperature sensors located at various positions, and the reservoir of the cooling system has been kept at constant temperature of about 15 °C over the duration of the experiment. The discriminator thresholds are common for sets Of 16 channels. They were set slightly above the signal level by electrons emitted in large numbers from the target. Most data were taken with a hardware trigger of N t2 2, where hi Nhit denotes the number of detectors firing; a run with reduced statistics was taken with Nhitzl' Due to the low beam intensity, random coincidences were negligible. 4.2 Data reduction and Analysis 4.2.A. Particle Identification For each detector three gated charge signals were recorded : "fast", "slow", and "tail". Hence, one can generate three combinations of 2-dimensional spectra for particle identifications. This information is combined to establish the logic to get optimal particle identification. As explained in Section 3.3, elemental resolution for intermediate mass fragments has been achieved from a 2- dirnensional "fast" versus "slow" map. To get better channel resolution we transformed the "fast" signal to ; 75 “Slow” u 3.11” Figure 4.1 "slow"(upper) and "tail"(bottom) versus fast' spectra of a Bicron detector(det # 2-8) 76 fast' = "fast" -c"slow", (4.2) where c is a constant less than 1. However a number of detectors supplied by one company (Bicron) had better elemental resolution for IMFs (3sZs6) on the "fast" versus "tail" spectra, possibly due to different thallium doping. Figure 4.1 shows "slow" vs fast' (upper section) and "tail" vs fast' 2-dimensional spectra for a detector using a Bicron crystal (compare this with Figure 4.2 which shows a spectrum from a Hilger crystal). Especially Li, Be, and B have better separations in "tail" vs. fast' spectrum than "slow" vs. fast'. Overall Hilger crystals had better particle identifications than Bicron crystals. We have made individual particle identification gates for each detector to keep the best resolution. At first one line was drawn to separate light charged particles (252) and intermediate mass fragments (Z 23) on the fast’ versus "slow" or "tail" spectra (Figure 4.2), and the particles identified as light charged particles were identified on the "slow" versus "tail" spectra by mass. Figure 4.3 (upper section) shows this "slow" vs. "tail" spectrum, and H and He isotopes are cleary separated on this plot except in the very low energy region where all particles merge together. To get better resolution in the low energy region we transformed the 2D-matrix putting upper and lower bounds as shown in the upper section of Figure 4.3 and expanding the useful dynamic region with full channel resolution. SO tailLupper bound) - "tail" tall. = C A(tail) , (4.3) where C is constant and A(tail) is the difference between upper and lower bounds for given "slow" channel. The transformed 2—dimensional spectrum "slow" versus tail' is shown in the Figure 4.3 at bottom section. Hydrogen .Houomuoc 6mm 18% mug/m can 33$qu Umwumfi Em: o5 mumuemmm 8 55ch m“ as: 6:8 9E. .8-N a $3583an Emmi m we Edbummm L93 mam; :33». New 33mm imam “M018” 78 “Tail” Tail! “Slow” Figure 4.3 "tail"(upper) and tail'(bottom) versus "slow" spectra. The transformation is explained in the text. 79 Fission Fragments LP + IMF “Fast” Time of Flight “Time” Figure 4.4 "fast" versus time Of flight (left) and "time" (right) spectra. Time of flight has been achieved "time"(ns) - "RF"(ns). 80 isotopes are well separated over the entire energy region up to the punch- through points, and 3He and 4He are well separated except at low energy. The double on lines are well separated from Li, and not included as the intermediate mass fragments. Also there is some separation in the Li and Be isotopes, but as discussed in reference [Ben89] particle identification by pulse shape discrimination using CsI(Tl) at these energies seems to be limited to about Zs4. Low energy particles and fission fragments are stopped in the 4mg/ cm thick fast plastic and could not be identified by charge because Of the lack of a "slow" signal. However fission fragments which have distinctly slower velocities could be identified in the "fast" versus ”time" Spectra. The right section in Figure 4.4 shows such "fast" versus "time" spectra for one detector in ring 2. Improved resolution has been achieved using RF timing information. The left section shows the separation of fission fragments from intermediate mass fragments and light charged particles, where x axis is the absolute time of flight in arbitrary units reconstructed from RF time and "time" signal of a detector. Unfortunately RF information was not recorded for all events. Therefore raw detector timing had to be used for identification Of fission fragments with slightly worse resolution. 4.2.B Gain drift correction As discussed in Section 3.8, each ring had a light pulsing system to monitor the gain drifts of the photomultiplier tubes. Unfortunately, for the current experiment, most of the information Of the light pulsing system had been lost (except for two runs) because Of an error in the data acquisition software. Thererefore we had to monitor gain drifts of the photomultiplier tubes in an alternative way. For this purpose we determined the channel variations Of punch- through points which are well defined by the detector geometry. Figure 4.5 81 197Au(36Ar,p), E/A=35MeV, 01ab=45° ' I '1 I . I ' 10000 I . [ punch—through point 1000 L M ~ VJ . *5 g 100 -+ -* O U 10 a J ‘ r h l a J m l n l n . L 100 200 300 400 500 600 Channel # Figure 4.5 Proton punch-through point of one detector in ring 5. Two vertical bars indicate the systematic uncertainty to decide the punch-through points 82 shows a typical punch-through point of proton spectra. The two vertical bars represent the systematic uncertainty in deciding this point, i.e. start and end points of the rapid fall-Off of the cross sectioin. Figure 4.6 shows the correlation of gain drifts determined from both light pulser and punch-through points obtained with the data of two runs for which the light pulser information had been recorded. Both methods Of extracting gain drifts agree within 1-2% indicating the validity of the more cumbersome gain drift correction via the punch-through points. For backward detectors, punch-through points could not be determined. Here we assummed that drifts were linear with time and used light pulser data at the beginning and at the end Of the experiment to determine the rate of change. The maximum gain drift of all detectors over whole run was Of the order Of 4%. Therefore the gain correction was not very significant. This constancy was probably achieved because we allowed for a "warm-up" time before taking data, and because we stabilized the temperature of the apparatus during the experiment. 4.2.C Energy Calibration (1) method Energy calibrations were Obtained by measuring the elastic scattering of 4He, 6Li, 10B, 12C, 160 and 35C1 beams from a 197Au target at incident energies of E(4He)/ A = 4.5, 9.4, 12.9, 16, and 20 MeV; E(6Li)/ A = 8.9 MeV; E(IOB)/ A = 15 MeV; E(12C)/ A = 6, 8, 13, and 20 MeV; E(160)/ A = 6, 8, 16, and 20 MeV; and E(35Cl) / A = 8.8, 12.3, and 15 MeV. The limitation of this method is that elastic scattering cross section decreases dramatically for scattering angles larger than the grazing angle Of the reaction. Therefore backward detectors did not have significant statistics to allow the extraction of the elastic scattering peaks. To 83 Gain drifts from punch—through and light pulse I I I I I T I I I I I I I I I I I I I I I I I I I_ 1.04 — _ _ o _ 1.02 —' 0 a I— oo ‘ Q) - o 080 °°o .1" O .L” - og 6'; o a O a - 5‘0 000 o -i 1.00 — 89 o —-— +3 7 00 00 Q, o "I FED *- o o 0 a) 000% — -:-t o 0 GD 0 o r—4 7 80 00° 8 0 T 7 o 00 0° 0° 0 o .. 0 ° o 0.98 — O O o .— — O O — ._ 0 O o _I 0.96 — —- "'1 I_|n_I LI 1 I Iml l min #14 LLJiLl‘ 0.96 0.98 1.00 1.02 1.04 punch—through Figure 4.6 The correlation between correction from light pulse information and the correction from punch-through points determined for different detectors. 84 compensate this limitation we rotated the whole miniball by 180° to expose backward detectors to elastically scattered 4He. For each elastic scattering peak, we calculated the net energy loss inside the CsI(Tl) crystal by subtracting the energy losses in the absorbers and fast scintillator foils. Then we compared this energy loss with the "slow" ADC conversion which is not contaminated by fast plastic light output and was a measure of the energy loss inside of CsI(Tl) crystal. (2) Energy response of CsI(Tl) The light response of CsI(Tl) crystals to various impinging particles has been investigated by several experiments for light charged particles, and heavy ions [QuiS9,Gon88,St058,Ala86]. The nonlinearity of the light output Of CsI(Tl) crystal seems prominent for particles of lower energy and increasing atomic number. At energies above 56 MeV / A, the light output seems to be quite linear. As an example, Figure 4.7 shows a calibration achieved by a typical Miniball detector. The x-axis is the net energy loss inside the CsI(Tl) crystal and y-axis is the channel number of "slow" signal. The pedestal had been subtracted. The curves in the Figure 4.7 are fits to the data points with the function, -cE L(E) = aE + b(e -1), (4.4) where a,b, and c are adjustible parameters, and E is the energy loss in the CsI(Tl) crystal. This functional form is consistent with previous measurements [Qui59,Val90]. It reproduces the nonlinear behavior of the CsI(Tl) light putput in the low energy region and the linear behavior at higher energies. For the forward rings, ring# 2 and 3, we had enough data points to determine the fit parameters for the individual detectors. We constructed a 85 reference calibration representing the average calibration points Of the detectors in ring 2,3, and 4, and used this average calibration for the more backward detectors. Figure 4.8 shows these reference lines and data points of the detectors in ring 2,3, and 4 normalized at the 80 MeV Carbon point. The variations in the channel numbers for the same energy are largely attributed to the slightly different response functions between detectors due to different thallium dopping. For detectors of ring 4-7, we used these reference lines and one normalization factor to match 1 or 2 data points for each Z line. The spread of the data points in Figure 4.8 would be the main source Of uncertainty in the energy calibration of detectors back of ring 4. For the calibration of Z=4,7,9 we interpolated) from adjacent low and high Z lines to obtain the calibrations shown in Figure 4.8 as dashed lines. The overall accuracy Of this energy calibration of intermediate mass fragments was estimated about 5%. For hydrogen isotopes we used the punch-through energies calculated with the program of refence [Zie85] assuming a linear relation between light output and energy [Gon88]. 86 Energy Calibration 1200 — IIIIIIIIIrII TITT Illl—F Detector 2—9 1000 N II N OJ 0‘ 0') (I) H 0 L41 l I II 800 17 600 Ch# 400 200 lllllllllIllllllllla llIlll_l LLLLngl l l 1 L41 ll 100 200 300 400 500 ECsl(MeV) O OlllIIllllIllIIIIIIIIIIIIIIIII I Figure 4.7 Energy calibration of det 2-9 as an example. The points are elasticlly scattered points and the curves are fits with equation 4.4. iflli- ‘ 87 Energy Calibration 1200 l—fil l Tl'fil F‘I—fil II l/fl l/Tl Ring 2,3,4. / ’ 1000 800 600 Ch# 400 200 IIIIJIJILIIJJIIJIILJIJJ 500 Figure 4.8 Energy calibration Of all detectors in ring 2, 3, and 4 tO get reference curve. The data points are normalized to detector 2-3 at the 80MeV carbon point. Chapter 5 General reaction characteristics 5.1 Multiplicity distributions : impact parameter selection A qualitative perspective of the reaction is depicted by the multiplicity distribution of identified charged particles in Figure 5.1. The threshold at N C: 2 is due to the hardware trigger employed during most of the experiment. To a rough approximation, the charged particle multiplicity is determined by the deposition of energy from relative motion into internal degrees of freedom. Within a simple geometric picture of the collision, this energy deposition depends on the impact parameter Of the collision - more central collisions being associated with greater energy deposition and hence larger charged particle multiplicities. At higher bombarding energies [Cav90], cuts on the charged- particle multiplicity are commonly used for the selection of different impact parameter ranges. While a purely geometric interpretation of the measured charged-particle multiplicities may not be strictly applicable at this low energy, it can nevertheless be utilized to generate a rough impact parameter scale. In the bottom part of Figure 5.1, such a scale is constructed by using the geometric prescription of reference [Cav90] in which a monotonic relation between charged- particle multiplicity and impact parameter is assumed: 2 °° , , , (b/bmax) = l (dP(NC)/dNC)dNC, (5.1) NC here dP(N C)/ dN C is the normalized probability distribution for the measured charged-particle multiplicity and bmax is the maximum impact parameter for 88 89 MSU-9l~167 'r"'T""|'"TFI"TI"'_ 10“1 (a) o —2 z 10 3 3 m 10 "d 10‘4“ : 3”Ar+197Au,1:1"/A=35MeV : .l_l L l Ij#1_l_ L1 ELI—l Pl L__l l : “fi'frrr' (Iv' ' tr '“' r. FT" 1.0-0 (b) - . o - o I . oo 2’— .. d _ 00 .I E. o ...O 0.5— o - \ p O .0 . 00 4 . o . OO 4 0.0” 0000004 ..laiLLmeinJLmelniL 0 5 10 15 20 NC Figure 5.1 Part a: multiplicity distribution Of identified charged-particles detected in this experiment; consistent with the hardware trigger Of Nhi software cut of N C22 was applied. Part b : relation between charged-particle multiplicity and impact parameter Obtained from the geometrical prescription Of reference [Cav90]. 22, t a 90 which particles were detected in the Miniball (N C21). This relative scale between impact parameter and charged-particle multiplicity must not be over-interpreted since considerable fluctuations of the charged-particle multiplicity must be expected even for collisions of well defined impact parameter. Nevertheless, it appears reasonable to distinguish between "central", "mid-central", and "peripheral" collisions by applying multiplicity cuts corresponding to N 212, 8sN $11, and 2sN C C definition of peripheral collisions. C s7, respectively. Events with N C=1 were not included in this Experimental multiplicity distributions for intermediate mass fragments detected in peripheral, mid-central, and central collisions are shown in Figure 5.2. The Poisson-like shape Of the IMF distribution is consistent with our previous measurement described in Chapter 2. The "raw" IMF multiplicity is larger in the current measurement with Miniball than the previous measurement by factor of about three, due to lower energy threshold, larger solid angle coverage, and no fission requirement. For peripheral collisions, the IMF multiplicity distribution is peaked at N =0 indicating that IMF emission is an IMF improbable outcome. For mid-central collisions, the peak of the IMF multiplicity distribution is at N IMF: 1. For these collisions, IMF emission is a common process and multiple IMF emission occurs fairly frequently. For central collisions, the IMF multiplicity distribution does not change dramatically from the mid-central distribution. The IMF distribution is still peaked at N =1, but the probabilities IMF for multiple IMF emission (NI = 3-7) increase in probability by factors Of 2-10. MF The dependence of the mean IMF multiplicity on the charged- particle multiplicity N is depicted in Figure 5.3. Over a broad range of charged- . C particle multiplicities N the mean IMF multiplicity exhibits an CI approximately linear rise as a function of N C' For very high multiplicities, N C312, the mean IMF multiplicity levels off as a function of NC and reaches an 91 36Ar+ 1971111, E /A=35MeV z I I I I I I I I . 10O __ E? 52 10-1 1 Cl: i 1 10‘2 _ a I z“ 10’3 ‘i "d _ E \ *4 — -i "C3 E i 10_5 5— NO 55. \\ __ F \ f 6 1' 0 2—7 0285 \ ~ 10" \ j 0 8—11 0.916 '.th (5 3 10:7 0 a12 1.139 I | ~ I I L I I L n I 0 1 2 3 4 5 6 7 a N IMF Figure 5.2 Normalized conditional probability distributions dP/dNIMF Of detecting N intermediate mass fragments in collisions preselected by the IMF indicated gates on charged-particle multiplicity N C' Mean values are given in the figure. 92 asymptotic value of ~1.2. To determine whether this behavior is dominated by the inclusion of Li fragments in our definition of IMFs, we have explored the dependence of the mean IMF multiplicity on the charged- particle multiplicity N when an IMF is redefined to exclude Li fragments (452320). Qualitatively,Cthe dependence Of mean IMF multiplicity on charged particle multiplicity exhibits a similar trend, irrespective of the inclusion of Li fragments in the definition of IMF. The relationship between the IMF and total charged particle multiplicities can be qualitatively understood by assuming that the charged particle multiplicity is strongly correlated with energy deposition and that the production of intermediate mass fragments depends primarily upon this energy deposition. The Observed rise Of as a function of N C can thus be viewed as due to the selection of interactions involving progressively larger amounts Of internal energy deposition. In the extreme tails Of the N distributions, however, C the correlation between internal energy and N becomes dominated by C fluctuations of the charged particle multiplicity. Hence, very large values of NC become ineffective in selecting nuclei of increasing internal energy, thus causing the Observed saturation Of at large values of N . This loss Of selectivity for IMF C N 312 is also expected from the impact parameter scale provided in Figure 5.1b. C In order to explore whether the detection Of intermediate mass fragments can provide impact parameter selectivity, we investigated the dependence Of the charged particle multiplicity distribution on the IMF multiplicity and on the detection angle Of the IMF. The left panel Of Figure 5.4 shows the correlation between IMF multiplicity and associated charged particle multiplicity. Such a relationship is qualitatively similar to the data discussed in Chapter 2 for reactions contributing to the fission channel. Events without an IMF are dominated by peripheral (low charged particle multiplicity) reactions. 93 MSU—91—073 TTl—FITFlI—IFIfiTIIIfiII 1.5 -- — — 36Ar+197Au, E/A=35MeV — Figure 5.3 Mean IMF multiplicity (circular points) as a function Of the charged-particle multiplicity N C' Triangular points show mean multiplicities when lithium nuclei are excluded from the definition of IMFs. 94 Requirement Of one intermediate mass fragment substantially suppresses contributions from peripheral processes. The dependence Of the charged particle multiplicity on the detection angle of the IMF is shown in the right hand panel of Figure 5.4. In all cases, IMF emission selects violent collisions characterized by large charged-particle multiplicities. While a slight shift in the charged-particle multiplicity distribution to lower multiplicities is observed when fragments are detected at 16° selabs23°, the angular sensitivity of the charged-particle multiplicity distributions to the fragment detection angle is insignificant at larger angles. 5.2 IMF elemental distributions Examples of elemental distributions, integrated over the angular range of 16 ° :9 $120 °, are shown in Figure 5.5. Solid points show the inclusive (N 22) lab distributions; Open squares and open circles show distributions measured fofthe peripheral (2sNCs7) and central (N C212) gates on charged-particle multiplicity, respectively. (To facilitate the comparison of relative shapes, the distributions were renormalized). All three elemental distributions exhibit near exponential shapes. Examples Of fits with exponential functions, e-az, are shown by the solid and dashed curves for the parameters given in the figure; the indicated errors are estimates of the systematic uncertainties arising from the fact that the elemental distributions do not strictly follow exponential shapes. The elemental distribution measured for the peripheral gate exhibits a slightly steeper slope than that measured for the central gate. However, the inclusive distribution is very similar in shape to that Observed in central collisions. This similarity is related to the fact that IMF emission is strongly suppressed for peripheral collisions which, hence, make only minor ' ‘ dP/dN Figure 5.4 Charged-particle multiplicity distributions dP/ dN C collisions preselected by the detection Of N 95 O 25 MSU-91-178 . rIIIIIIIIIIIrrIfiTILrII—FIIIIIIIIIIIIII A - I am: -— 16°-23° : (xz) 1 — — 23°-31° - . - - - 31°-40° L—.. — —— -I 0.20 _ I Inclusrve q_ __ 4Oo_500 _ - I N1111- I ----- 100°-120° ‘ .. ‘ .". ' I "' ’ 0 “I C‘\ I 0.15 —- I -—"' \\-\" d ' \ ‘ ' 1 ‘i- .- “. . ’ I i_ ,’ \Iit' : '..~ 2 :- III II. . -l _— 'I \ _- I \ ... O. 1 0 - \ . I ‘ .4 I; A .i e \\ .’ 5 ‘ ‘5 I/I' \. ‘ _ .1 \ 'i- [II 'I \ 3 d I- .I .\ \ \ .I“. ///{.I. X‘ —t 0.05 r a .- \\ -; ,,,- (I... 1 .- I \ . .J- [.7..' \x. .1 _ I. \ ‘. .." ”7..” a L \ ‘.\ I- Ill”. \.. d 000 ' 'L_1 L I I L 1 1 I I l L l__I L m 0 5 10 15 20 5 10 15 20 NC No IMF 0.20 0.15 0.10 3Np/dp 0.05 0.00 measured for =0, 1, and 2 intermediate mass fragments at any angle (left hand panel) and by the detection of a fragment at a given angle (right hand panel). 96 0 MSU-91-173 10 E fi r l I I I r 1 r r f T I r I r I r r I f F i E 1"71111831032=.-3—20), E/A=35MeV f : 9,,b=16-120° : \ —1 _. —aZ __ A 10 E e 2 m b- : ...; — a u-O - .I I: - . :3 . -I rd ”.2 ._ -.—I LI 10 E : c0 . : v t .. A .- j N _ _ 10“3 5— . inclusive 026110.016 o —: E o 2-7 034110.017 3 ; 0 gm 024110.018 j J 4 L l_ L P Pl 1 J 4 L J— 1 LL 1 J A l J— l O 5 10 15 20 Z Figure 5.5 Comparison of angle integrated element distributions measured inclusively (solid points), in central collisions (N 212: open squares), and in peripheral collisions (ZSN s7: open circles). The lines represent parametrizations with exponential functions; ranges of parameters consistent with charge distributions are indicated. 97 contributions to the inclusive cross section. Inclusive mass or charge distributions may not always be quite as meaningless as suggested previously [Aic88a,Aic88b,Pei89]. The parameters oc describing elemental distributions gated by different IMF multiplicities are shown in the left hand panel of Figure 5.6. While the shapes of the elemental distributions are not very sensitive to the IMF multiplicity, there is a discernible trend for the elemental distributions to become slightly steeper as more fragments are emitted. We cannot offer a quantitative explanation of the dependence of on on N C' However some of the qualitative trends are consistent with statistical considerations. Within a statistical picture of fragment emission, the slope of the charge distribution is expected to become less steep when the temperature of the emitting system is raised and Coulomb barrier effects are reduced. As increasing values of N C are related to higher internal energies of the emitting system, the qualitative trend of reduced slopes in the elemental distributions is expected. However, the increase at very large values of N is not, and its origin remains less C clear. It could be related to self-correlations imposed by energy conservation. Gates on the extreme tails of the charged-particle multiplicity distributions select events in which statistical fluctuations have lead to the emission of more charged particles than average. If intermediate mass fragments are emitted at the later stages of the reaction, the fragment emitting system will then have a slightly reduced excitation energy, and the slope of the resulting fragment distribution should be slightly steeper. 98 MSU-91—179 Illfl'lIllrIIlllllllll 0.5—I l—I I l I [TI I] I I 36Ar+197Au, E/A=35MeV Blah = 16"1200 — 0.4 —- 0.3 — 0.2 — __ OlLl J liLL;[ I l 1 LLluh-LIJIILLMIJlllI—LLJPIII 2 3 6 '70 5 10 15 20 NIMF NC Figure 5.6 Parameters of exponential (eflz) fits to element distributions selected by cuts on NIMF (left panel) and N C (right panel). 99 5.3 Angular distributions Angular distributions of representative intermediate mass fragments, gated by peripheral (ZsN 57: open circular points) and central (N C212: open square shaped points) collisions, are shown in Figure 5.7. For comparison, inclusive angular distributions are depicted by solid points. The angular distributions shown in the figure are normalized as conditional probability distributions, i.e. as the probability per unit solid angle to detect a given fragment in collisions preselected by the indicated gate on charged-particle multiplicity. In general, the angular distributions become more forward peaked for increasing fragment charge. The slopes of the angular distributions are steeper for peripheral collisions than for central collisions which could indicate increasing degrees of equilibration and possibly diminishing contributions from projectile-like sources for collisions with larger charged-particle multiplicities. 5.4 Energy spectra Examples of inclusive energy spectra are presented in Figure 5.8 for intermediate mass fragments of charge Z=4-9. The energy spectra exhibit qualitative characteristics already observed in other heavy-ion induced reactions at comparable energies [Lyn87,FieJ86,FieE89]. In order to provide a reasonable analytic parametrization of these cross sections we have fitted them with a simple parametrization allowing for contributions from three sources of different velocities. Each source was assumed to emit particles with a 1/sin9 angular distribution in its respective rest frame. The explicit form of the adopted parametrization is taken from reference [Ch186]: 100 MSU-9I-l65 197 36 Au( Ar,Z), E/A235MeV FfTTTTYIITTTTI‘rVY‘ II I [VIVT'IVIIYV—fiT‘IIT il' ITT V I I fr 7 l h; L 1 I 1 L l l l 1 l V I T I Y I I I I I r A ‘1 P T 10 .r 5_, : 10‘2 3; 5‘ % 10‘3 E a 1M- "U ' ' . 10-1 E 10‘2 .r . 10-3 :_ 'inclusive 1 E oNciZ“? ‘ 10"4 E‘ och12 1: :.a.l.1.1I1.‘3~. ..L111L.: 0 50 100 150 0 50 100 150 0 50 100 150 elab (deg) Figure 5.7 Comparison of IMF angular distributions measured inclusively (solid points), in central collisions (N 212: open squares), and in peripheral collisions (2sN s7: open circles). Different panels show distributions for the indicated fragment Z. The distributions represent normalized conditional probability distributions for collisions preselected by the indicated gates on N C' dZP/deE (sf1 MeV—'1) 10“3 10‘4 10-5 10—6 10"7 L 10"3 10—4 10—5 10'"6 10—7 10“3 10—4 10*5 10‘6 10"7 101 36Ar+197Au, E/A=35MeV . . , . . . . . z, . ’B I I 1— T r I I Be I“ I I ll" lilil‘l‘ Inlfijii l1 rll"“1 I II 0 200 00‘ ‘ E (MeV) Figure 5.8 Inclusive energy spectra for intermediate mass fragment of charge Z=4-9. The curves represent moving source fits, Eqs. 5.2-4, with the parameters given in Table 5.1. 102 d2P n _ —— = ' - 5.2 dEdfl ilei(‘/ESi/51n9)exp( Hsi/Ti) , ( ) where ' E . = E - V + E. - 2JE.(E-V )cost) (5.3) 51 C 1 1 C and E. = l mv.2 . (5.4) 1 2 1 Here, Ti is a kinetic temperature parameter characterizing the slope of the energy spectrum for the i-th source, and vi is the source velocity. The parameter V C is introduced to account for Coulomb repulsion from a heavy charge assumed, for simplicity, at rest in the laboratory system. To simulate the kinematics of fragment emission from equilibrated residues formed in an incomplete fusion reaction, the velocity of one source was fixed at v2=0.035c. This value corresponds to a linear momentum transfer of 80%, consistent with the systematics of references [Fat85,Cas89]. The fits are shown by the solid curves in Figure 5.8, and the parameters are listed in Table 5.1. The fitted parameters may not be unique, since they depend on the specific parametrization adopted and, in addition, on the energy and angular range included in the fit [Wil91]. Nevertheless, the fits strongly suggest contributions from fusion-like and projectile-like sources (i=2 and 3, respectively) as well as from an intermediate velocity source (i=1) representing nonequilibrium processes. Rather large differences are observed in the shapes of energy spectra gated by different ranges of impact parameters. For illustration, Figure 5.9 presents the energy distributions measured for carbon fragments produced in peripheral (ZsN s7: solid circular points), mid-central (8sNCs11: open points), and central C (N 212: solid triangular points) collisions. As before, the distributions represent C 103 197Au (36Ar,C), E/A=35MeV ff r r f , f, r f g f r l l -3 _ m 10 :' 19.5° 27° f4 5 6 58 I- 3 d I A 10-4 5L3 2- ‘a s I : . \ : 00°00 i O q, . % -5 b t 3’ i a"? T 2 10 5— 4. is ‘i 59 ‘1 H ‘t 0:. fiuk 00 It... “ °.' 4 ‘ l m _3 _ _ V 10 355° 45° 5 c I a. a "U - 3° to ‘ 53 10“" :‘e "22. 2° - Na 4. \ : {\fiq’o 5°52: . 2—7 3 Q-I : . ‘0 . .0.” .4 N °o° .p U 1.0—5 :- ‘o 5:200 D 8‘11 —5 1%. to. . g 12 o ‘00 . 4.Lingfigel. _11LJ§LJ_1_LL 0 200 400 o 200 400 E (MeV) Figure 5.9 Energy distributions of carbon fragments produced in peripheral (ZsNCs7: solid circular points), mid-central (8 sN Cs11: open points), and central (N C212: solid triangular points) collisions. The fragment detection angles are indicated in the individual panels. The distributions represent conditional probability distributions per unit energy and solid angle for detecting a given fragment in collisions preselected by the indicated gate on N C' 104 Table5.1 Parameters of three-source fits (Eqs. 5.2-4, using n=3) to inclusive fragment spectra. The normalization constants Ni are given in units of 10’4/(sr-Mev3/2). IZI N1 lvl/c IT1(MeV)I N2 Ivz/c |T2(MeV)I N IVS/C IT3(MeV)I 3 I I I I I I I I . I I I I4 |0.748 I0.107 I 15.3 I0.516I 0.035 I 11.4 I 0.734 I 0.196 I 9.2 I I5 |0.793 I0.108 I 14.8 I0.526 I 0.035 I 11.4 I 0.896 I 0.200 I 9.4 I I6 I0.712 I0.096 I 14.9 I0.378I 0.035 I 12.6 I 0.739 I 0.194 I 10.4 I I7 I0.521 I0.085 I 14.1 I0.223I 0.035 I 9.9 I 0.375 I 0.183 I 11.8 I I8 I0.349 I0.078 I 14.1 I0.231 I 0.035 I 8.8 I 0.223 I 0.170 I 12.6 I I9 I0.231 I0.078 I 14.8 I0.123 I 0.035 I 10.6 I 0.148 I 0.170 I 12.3 I Table5.2 Parameters of two-source fits (Eqs. 5.2-4, using n=2) to fragment spectra observed in central collisions (N C212). The normalization constants N i are given in units of 10-4/ (erMeV3/2). I Z I N1 I Vl/C | T1(MeV) I N2 I vz/c IT2(MeV) I I I I I I I I I I 4 I 1.195 I 0.111 I 13.8 I 1.333 I 0.035 I 11.4 I I 5 I 1.306 I 0.109 I 13.3 I 1.226 I 0.035 I 11.4 I I 6 I 1.207 I 0.096 I 13.6 I 0.923 I 0.035 I 12.6 I I 7 I 0.900 I 0.085 I 13.3 I 0.591 I 0.035 I 10.1 I I 8 I 0.620 I 0.077 I 13.2 I 0.533 I 0.035 I 9.1 I I 9 I 0.437 I 0.077 I 13.6 I 0.276 I 0.035 I 10.7 I 105 conditional probability distributions per unit energy and solid angle for detecting a given fragment in collisions preselected by the indicated gate on N C' Consistent with Figure 5.2, the fragment emission probabilities are smaller in magnitude for peripheral than for central collisions. For central collisions, the energy spectra exhibit rather featureless exponential slopes which become steeper at larger emission angles. For peripheral collisions, the energy spectrum at elab=195° exhibits a large high-energy shoulder, indicating possible contributions from the decay of projectile-like residues or from strongly damped collisions. Mid-central collisions exhibit an intermediate behavior. At larger angles, this shoulder vanishes, and the energy spectra attain structureless exponential shapes. However, the slopes of the energy spectra observed in peripheral collisions are less steep than the slopes of the energy spectra measured for central collisions. The exponential shapes of the energy spectra observed in central collisions suggest emission from a system characterized by a relatively high degree of equilibration. In order to explore whether fragment production in central collisions is consistent with emission from fully equilibrated reaction residues formed in fusion-like processes, we have fitted the differential emission probabilities with a two-source parametrization allowing for contributions from a fusion-like source and an intermediate velocity source, each parameterized according to Equations 5.2-5.4 (using n=2). Figure 5.10 shows energy spectra and two-source fits for fragments with 4sZs9 observed in central collisions (N 212); C parameters are listed in Table 5.3. The fits indicate significant contributions from nonequilibrium emission processes. To give an overall qualitative perspective of the ratio of equilibrium to non— equilibrium emission in central collisions, we show in Figure 5.11 the / equilibrium fraction, 0 indicated by the fits in Figure 5.10. Here, 0 / total slow 106 36Ar+197Au, E/A=35MeV,- 5102.12 10_3 F f I I fie 'I’ I 10-4 : A 10-5 g 10‘7 5- 10‘3 5‘ . Is... 104 5' 1: U) : : V 1.0—5 g- to“! : 0.: C: 10—7 5 A. 1 L 13 *3 t r r r I . I r . éL 10—4 5 g 13 _5 E i 10 E— .551 10"6 f E 10—7 5 .51 . . l . 0 100 200 300 0 100 200 300 E (MeV) Figure 5.10 Energy spectra for intermediate mass fragments of charge Z=4-9 detected in central collisions (N C212). The curves represent moving source fits, Eqs. 5.2-4, with the parameters given in Table 5.2. The distributions represent conditional probability distributions per unit energy and solid angle. 107 Oslow represents the cross section represented by the slow fusion—like source (i=2) and a total represents the summed cross section from the two sources (i=1 +2). The top panel depicts the equilibrium fraction as a function of laboratory angle with the individual source contributions integrated over the energy spectrum above the detection threshold. The bottom panel shows this fraction as a function of fragment momentum per nucleon with the individual source contributions integrated over the angular range of 16 ° 5 9 531 °. While the decomposition into equilibrium and nonequilibrium processgzbmay not be unique, the extracted ratios should, nevertheless, provide a qualitative estimate of the fraction which may be emitted by an equilibrated heavy residue. Nonequilibrium contributions are most important at forward angles and for fragments emitted with high kinetic energy. At larger angles, the fragment cross sections are increasingly consistent with emission from equilibrium decays. Because of the finite energy threshold, the true IMF multiplicity should be higher than the observed multiplicity. In order to estimate the order of magnitude of this differences we have proceeded as follows. Figure 5.12 shows the extrapolations made on the energy spectra for central collisions at three different angles in the laboratory coordinate. As shown in the figure we extrapolated the energy spectra to zero energy by fitting the lower part of the Coulomb bump when the energy spectra show explicit Coulomb bumps or by assuming flat energy distributions when the energy spectra do not show explicit Coulomb bumps. This way we get the approximate uncertainties in cross section below the detection threshold. This rough estimate shows about 10% increase in mean IMF multiplicity for peripheral collision, 15% increase for mid-central collision, and 19% increase for central collision. The uncertainty caused by the finite angle coverage is estimated by extrapolating the angular distribution it: to 108 MSU-9l-I72 9151b (deg) 20 40 60 80 1'0 'fiv ‘TTT' 'Tv' 'fi*_ 08;. 36Ar+lwAu, E/A=35MeV hr» _‘1 _ ' __I I l I J I Uslow/ Utotal 100 150 200 P/A (MeV/C) Figure 5.1 1 Equilibrium fraction, 0 / 6 total’ extracted from the fit the data in Fig. slow 9. Top panel: dependence of energy integrated yield on laboratory angle; bottom panel: dependence on P/ A, the momentum per nucleon, of fragments detected at 16° °. selabs31 I¥ 109 IOOEW'T"iv'rvl‘s'rTr'wr‘ 1 ' Z=6, NC_2_12 I 10“ 5- 1 A I T : I > 10—2 :5: \ 01ab=19'5° i (D E a.“ 1 -3 ._ .5 a i . ' I ”I; 1 -4, 5- —J V 10 . m.elab=52.5° I 5 “I. o 3 £3 E '-. “to ‘ 10"5 r 0. "a £5 5 to 91.1,: 1 10° 0?“ ‘ O E 10“6 L '1 III ‘ 1 NT? *5 10"7 5— 1{I ‘1 10—8 L l 1 J J 1 J 1 1 1 4 I J A 1 1 I 1 +4 4 I n 11 o 100 200 300 400 E (MeV) Figure 5.12 Extrapolations of the energy spectra to below the threshold for 2:6 and central collisions. The solid lines show estimates of the cross section below the detection threshold. 110 0 ° and 180 ° as shown in the Figure 5.13 for Z=6 and 10 cases. The solid and open points are the probabilities with and without the extrapolations in the energy spectra which are used as higher and lower bounds respectively. (gig at less than 15° has been estimated by extrapolating as exponential(solid line) and flat(dashes line) distributions for higher and lower bounds respectively. The estimated true multiplicities(upper bounds) are 0.62 for peripheral collisions(114% increase), 1.68 for mid-central collisions(87% increase), and 1.96 for central collisions(76% increase). To get more accurate true IMF multiplicities measurements with low-threshold gas detector are necessary. Multiplicity gated energy spectra and angular distributions provide clear evidence for significant non-equilibrium IMF emission even in central collisions selected by large charged particle multiplicities. To better characterize the multifragment final state and distinguish between different theoretical scenarios, knowledge of the IMF emission time scale is necessary. This issue will be taken up in Chapters 6 and 7. 5.5 Charge correlation function To test the assumption of statistically independent emission of intermediate mass fragments, with a reduced sensitivity to the detection inefficiency, we have examined the "charge-correlation function" as suggested in reference [Boa88], C(Z1,Zz)=NO {.2 Yik(Zl’ZZ) } / {.22 Yi(Zl)-Yk(Zz) }. (5.5) lik 1¢k Here, Yik(Zl,Z2) denotes the coincidence yield for particles of charge number Z1 and 22 detected in detectors i and k, respectively, and Yi(Z) denotes the singles 111 1.0—2 ¥TTTfTTITfiTiTT IT—FT‘TTI I I—rTr—FTIT§ [ Ncgiz - o Z=6 I I a 2:10 I co m "d v 8 \ 10—4 .— L _. 1 CL E ' e I 3 "d 5 l \ j - I ._ .e- __ _ ...1 ' I I. ‘- ‘-CI—1 I z 5 I' L l L l J. L4 L4 L4 J_ l__L [ L :18:- LJ L l_I_ L J. l l .1 o 30 60 90 120 150 180 91ab (deg) Figure 5.13 Angular probabilities of Z=6 and 2:10 for central collisions. Solid points and open points are the data points with and without the extrapolation in the energy spectra. Solid and dashed curves are the extrapolations up to 0 ° in d 9 assuming exponential and flat distribution respectively. 112 yield for particles of charge number Z in detector i. The charge correlation functions obtained from the data of 36Ar + 238U, at E/ A = 35 MeV with Dwarf Wall/ Ball Were constant within about :I:10% for 3521,2256 [Kim89]. Figure 5.14 shows charge correlation functions for 36Ar+197Au at E/A=35MeV with a broad range of $21,2st0 as a function of total charge Z1 +Z2. Except for the case that one IMF is Li, the correlation functions are again constant within about i 10% when 21 +Z2520, supporting the assumption of statistically independent emission. For Z1 +ZZ>20 correlation functions decrease monotonically as a function of Z1+ZZ reaching about 0.6 for heaviest combinations. This effect may be due to finite size of the emitting composite system. C(szz) 1.2 1.0 — 0.8 0.6 113 197Au(“iAruz122), E/A=35MeV T I I l I I I I I r T I I I I I I I d :3 _ mr§§ Z1 - mmmm .1 ss-aasmmmwg a §§§§ 21:4 __ 9° Bfi§m§§§::§ 21:5 J wég rags : l l I I l l l I I l l l I l l I _ 10 20 30 4o Z1+22 Figure 5.14 Charge correlation functions as a function of Z 1+Z2 constructed according to equation 5.5. Chapter6 Intensity interferometry : A tool to study time scales of multi-fragment emission 6.1 Overview When two particles are emitted in close proximity and with small relative momenta, their relative wave-function is affected by final state interactions and, for identical particles, by quantum statistics. Therefore, two-particle correlation functions at small relative momenta are sensitive to the space-time evolution of the emitting system [Boa90b,Koo77,Gon91a]. When interactions with the residual system can be neglected, the two-particle correlation functions depend essentially on the single-particle phase-space density of the emitted particles [Pra87,Gon91a]. Up to now, quantitative comparisons of experimental correlation functions with predictions from specific reaction models have largely been performed for the case of two-proton correlation functions. Rather gratifying agreement between theory and experiment was found for emission processes occurring on rather different time scales. Two-proton correlation functions measured for evaporative emission from equilibrated reaction residues could be understood in terms of slow emission time scales predicted by compound nucleus decay models [DeY89,Qui89,Ard89,Gon90a,Gon91b]. Two-proton correlation functions measured for non-compound emission processes were found to be in good agreement with the space-time evolution of the reaction zone as predicted by the Boltzmann-Uehling-Uhlenbeck equation [Gon91b,Gon90b]. 114 115 While the time scales of nucleon emission processes appear to be reasonably well understood, those governing the emission of intermediate mass fragments in energetic nucleus-nucleus collisions are much less certain and subject to controversial interpretations. In this chapter, we explore the sensitivity of correlations between two intermediate mass fragments to the emission time scales. For this purpose, we apply the formalism of refs. [Koo77,Pra87,Gon91a] to the calculation of two-fragment correlation functions resulting from the final- state Coulomb interaction between the two detected fragments. To a high degree of accuracy, such correlations can be treated classically. In many cases of practical interest, the fragments are emitted in the vicinity of a heavy reaction residue. In such cases, the neglect of Coulomb interactions with the residual system may not be justified. In order to assess the effects of distortions in the Coulomb field of the residual system, we will also present correlation functions calculated by means of three-body trajectory calculations. This chapter is organized as follows. In Section 6.2, we summarize the Koonin~Pratt formalism of reference [Gon91a], and give a brief derivation of the classical approximation to this formalism treating only the Coulomb interaction between the two detected fragments in Section 6.3. In Section 6.4, we compare longitudinal, transverse, and angle-averaged correlation functions, calculated with the generalized Koonin-Pratt formula by treating the Coulomb interaction between the two detected particles classically and quantum mechanically. Results of three-body trajectory calculations are presented in Section 6.5. 6.2 Koonin-Pratt formalism In intensity interferometry the two fragment correlation function 15126323) is defined by the expression : 116 1165. ,5.) = (1+R’ . >116.) , (6.1) whereP=p . +5 2 3:16; , /m . :5 2 / m 2 )= “crel and uare the totalmomentum, relative momentum and reduced mass,respectively, which are quantities constructed from the individual momenta i5, and E, and individual masses ml and m2 of fragments 1 and 2, respectively, and 1161,52) and l'I(p) denote the two-particle and single-particle emission probabilities. The experimental correlation function, 1+R, is constructed from the measured coincidenceandsingle-particleyields,Y1 2631,52) anin(pi),respecfively, according to the expression £Y12(p1,p2) = C(1+R(C)) 2Y1(p1)Y2(p2) , (6.2) where C is relative momentum q or "reduced" relative velocity Vre d (see eq. 6.19 below). The normalization constant C is determined by the requirement that =0 for values of C sufficiently large that the final state interaction between the emitted fragments can be neglected. For each gating condition (on P and/ or other quantities), the correlation function is evaluated by summing both sides of Eq. 6.2 over all momentum combinations corresponding to given values of C. Since the yields differ from the emission probabilities only by overall normalization constants, Eq. 6.2 is equivalent to Eq. 6.1 The usual treatment of two-proton correlations at higher energies is based upon the Koonin-Pratt formula. The derivation of this formula is based upon the assumptions 1) that the final-state interaction between the two detected protons dominates, 2) that final-state interactions with all remaining particles can be neglected, 3) that single particle phase space distribution varies slowly over characteristic momenta, and 4) that the correlation functions are determined by 117 the two-body density of states as corrected by the interactions between the two particles. The density of states for two particles separated by the relative coordinate 1:, and emitted in the asymptotic state characterized by the relative momentum 21’ is the square of the relative wave function. If the two fragments are emitted simultaneously with thermal distributions or if the momentum distributions are very broad, as is the case at high energy, this approach provides a near-exact answer provided interactions with third bodies may be neglected. For emission from long-lived compound nuclei, the emission is thermal but far from simultaneous. Furthermore, distortions in the long-range Coulomb field of the residual nucleus might not be negligible, especially for particles emitted with energies close to the exit channel Coulomb barrier. While the Koonin-Pratt formula is expected to be reliable for higher energy collisions which are characterized by rapid disintegrations, its range of validity is less clear when it is applied to less energetic regimes. The Koonin-Pratt formula allows the calculation of the two-particle correlation function in terms of the single particle phase space distribution and the relative wave function of the emitted particle pair [Gon91a]: —§ —) 3 ——) -) —-) 2 1+R(P,q) = I d rFfi(r) I ¢(q,r) I . (6.3) Here,l3)=p’1+p2 is the total momentum of the proton pair and 6 (3,?) is the relative two- particle wave function. Here fir-11d; / dt is the relative momentum between the two particles, ; is the relative coordinate, and u=m m / (m 1 2 1 The relative Wigner function F136,) is defined by: +m2) is the reduced mass. 3 -) --) -) -> —-1 ——> Id Xf1(Pu/m2, X+r u/m1,t>)f2(Pu/m1,X-ru/mz,t>) —) 3 —> -—) (64) X1f1(Pu /m2, X1, t >)0Id X2f2(P u/ml, X2, t>)| 12.4?) = P I Id3 118 Here, i is the coordinate of the center-of-mass of the fragment pair, 15132,? i,t>) is the phase-space distribution of particles of type i with momentum {ii at position 1:) i at some time t; after the emission process. If the particles cease to interact at a time earlier than t), then the relative Wigner function is independent of the particular choice of t> and the function fi(p)i,?i,t>) can be expressed in terms of the emission function, gi(pi,?i,t), i.e. the probability of emitting a particle of type i with momentum Bi at location 5 i and time t [Gon91 a]: t > fi(pi,r i,t>) = .idt gi [ pi; i-pi(t>-t)/mi,t1 . (6.5) The relative wave function M3,?) is influenced by three different effects : identical particle interference, short range nuclear interaction, and the long range Coulomb interaction. It has been shown that all these three effects are important for proton-proton correlation function. For the calculation of the correlation functions between two IMF’s, we neglect the identical particle interference and nuclear interaction because average separations between two fragments are expected to be larger than the range of the nuclear force and Coulomb effects are expected to be predominant. Furthermore, only phase space points are integrated over for which the relative separation between the two fragments is larger than the sum of their radii, r>1.2(Ai/3+A:/3) fm. 119 6.3 Classical treatment of two-fragment Coulomb interaction Calculations based upon Eq. 6.2 using Coulomb wave function are rather tedious and time consuming. It is, therefore, of interest to derive a classical expression. Again, we assume that the correlation function is the ratio of available states with and without interactions. If {1’0 and E denote the initial and final relative momentum vectors, we have: 2 . .94 q dq Sine d9 do C(q,r0)== I 0 0 0 0 0 I. (6.6) q dq sinede do 0 and 9 denote the angles between the initial relative position vector 1?p and A A A the initial and final momentum vectors, respectively: cose 0=r 0 0 q 0 and cos 9 =r 0 0 q. From angular momentum conservation, we have for azimuthal rotations about A the ax1s parallel to r Here, 9 0: dq> = dcpo. (6.7) Energy conservation gives the relation 2 2 = _ 6.8 qo q ZuK/ro, ( ) where I<=ZIZZe2 is the product of the charges of the two emitted fragments. From Eq. 6.8 one immediately obtains 2 2 1 2 2 qodq0=(l-2III—>—) -> —> —) A A e = qu/Iuc + r = qx(r xq)/u1< + r , (6.10) is a constant of motion and equating its two components in the scattering plane at the time of emission and at very large times, one obtains the following relations: 2 . 2 . . Inc + quOsm 60 — chose + quqosmGsme (6.11) 0 I 2 . . . roqocose 051110 0 — roqqocosesmt)0 - Iixsme . (6.12) Here, we made use of angular momentum conservation and of the fact that the two vectors f, and E become parallel for t-no. Equation 6.10 can be cast into the form: sinGO = (q/2q0){sin0 i [sin20 - 4uK(1-cosa)/r0q2]1/2} . (6.13) .9 Equation 6.12 has two solutions: for given final relative momentum q, there are two different trajectories which pass through the relative coordinate 50. Both trajectories must be considered when calculating the correlation function. Equation 6.11 can be written as: 2 z _ ' - .14 c0590 (q/q0)cose (uK/r0q0)sme/sm6 (6 ) 0' Differentiation of Eq. 6.13 yields: 121 . . 2 cos esme - ZuKSI ne/qu 0 [sin e ~4IIIc(1-cose) /r0q2]1/2 deO/da = ———q——{ case 1 } . (6.15) Zchose Using Eqs. 6.8, 6.13-6.15, one can express sinGOdGO/sinede in terms of the of the initial relative coordinate rO and the asymptotic values of 9 and q. Hence, Eqs. 6.7- 6.9, 6.13—6.15 allow the expression of Eq. 6.6 in terms of the final relative momentum and the initial relative position of the emitted particles. Integration over all relative positions of the emitted particles gives the final expression for the correlation function: 145120—33): Id3r F—>(? )CG? ) (616) ’ 0 P 0 ’ 0 ' ' If all directions of {j are integrated over, Eq. 6.16 reduces to a particularly simple expression, ->-) 3 -> 2 1/2 1+R(P,q) -— Id rOFI—3(r0)[1-2III function,r andp are unit vectors parallel tor and p, and E=p2/ 2m. The parameter VC is introduced to account for Coulomb repulsion when emission occurs from the surface of a heavy reaction residue assumed to be at rest at the origin of our coordinate system. The relative wave function can be calculated by using a partial wave expansion of the Coulomb scattering amplitude [Mes61] : 0° 2 ion i2: 0(22 +1)i e F1 (n;qr)PJ1 (cose) (6.21) El“ ¢C= Here, a! is Coulomb phase shift and, n=ZIZZauc/ q is the Sommerfeld parameter and a=21re2/hc=1 / 137 is the fine structure constant. In our quantum mechanical treatment of the two-fragment Coulomb interaction, the number, 9. max’ of partial waves included in the calculations was scaled according to the classical relation It max: 20(h/21r)+(q2r2-2n qr)1 /2. We used computer code "COULFG" published by AR. Barnett [Bar81] for numerical calculation of Coulomb function F I (mqr). As a specific example, we calculated correlation functions for the emission of two distinguishable carbon nuclei with the parameters T=15 MeV, R =12 fm, S V = 63 MeV and various mean lifetimes 1:. Figure 6.1 gives a comparison of C correlation functions integrated over all relative orientations between the total and relative momenta of the two emitted fragments. The points show calculations with Eq. 6.3 and the curves show results obtained with the classical expression, Eq. 6.17. The two calculations produce virtually identical results indicating that the classical approximation is very well justified. (Some small differences exist at large relative velocities for the calculations with 1:500 fm/ c; these are due to the limitation in the number of partial waves (<1000) used in our calculations with Eq. 6.3.) I 124 MSU-91438 fir T I I I 7 r—Ii T—Y j—I fit] —T I I I T I I Ifi I—I' T _Ii I I f 1.25 Koonin—Pratt formula — VC=68MeV, Rs=12fm, T=15MeV Z ZI'ZZ=6 : 1.00 oc' .... IITTIITII—fileTIflllllfillr 0 Quantum 1 .I D: 0.75 z -I- o; 1 ‘-I ,' j Classical, ‘T’ frn c - 0.50 ( / > 4 ......... 5O .1 ,9 — — — - 100 ; . ' ————- 200 1 0.25 ~’ -1.-. 500 —~ 1 . 4 000 L w 2.3:9'4IIiiuiunmgIIIILLIIIIInui 0 5 .10 15 20 25 30 Vrel/ V ZI+ZZ (10730) Figure 6.1 Comparison of angle-integrated carbon-carbon correlation functions. Calculations with Eq. 6.3 are shown by points; calculations with the classical approximation, Eq. 6.17, are shown by the curves. The emission times are indicated in the figure. 125 MSU-9M3? ’- fl T—r T—r F I l I' If I _l——]' hi I I l C? I T I r I I T ‘l’ ['17 .- 125 - Koonin—Pratt formula ‘ _ VC=63MeV. R3=12fm, T=15MeV _ : 1- : 200fm/c : 1-00 :— Zl-Zz=6 Qaga-B'O‘O’D'B'Qa‘a'a‘a‘G-‘jb .. 52’9“ I Q: 0.75 :— 1 V—I - .. 0.50 — m — - Quantum Classical :1 t w: O—30° __ o - 0.25 :_ ISO-180° :I ._ V5 70-110° - - ‘ D d I 8’ I 0.00 Q41 L l_| I L l_ lLl J L l I l L L_l J L I Ll J. l LILJLL 0 5 10 15 20 25 30 Vrel/ VZ1+22 (10730) Figure 6.2 Longitudinal and transverse correlation functions for 1:200 fm/c. Longitudinal correlation functions (w=0°-30° or w=150°-180°) are shown by the solid curve and circular points; transverse correlation functions (III=70°~110°) are shown by the dashed curve and square points. Curves show calculations with Eq. 6.3; points show calculations with Eqs. 6.6-9, 6.13-16. 126 In Figure 6.2, calculations for longitudinal and transverse correlation functions are compared for 1:200 fm/c. These correlation functions are defined in terms of the angle wbetween l5 and ti, w=cos-1(l3° a/Pq). Longitudinal correlation functions correspond to the angular interval w=0°-30 ° or w=150°~180 ° ; transverse correlation functions correspond to the angular interval w=80°-100 °. The curves show the results obtained with Eq. 6.3 and the points show the results obtained with the classical relations, Eqs. 6.6-6.9, 6.13-6.16. Again, the two calculations produce virtually identical results. We conclude that classical treatments of correlation functions between intermediate mass fragments provide an excellent approximation. The validity of the classical approximation is, of course, expected for emission from sources with dimensions comparable to or larger than the Bohr radius. For the two-carbon system the Bohr radius is 0.13 fm which is, indeed, much smaller than the size of the emitting system. For emission from long~lived systems, longitudinal correlation functions are predicted to exhibit a wider minimum at qu than transverse correlation functions. This directional dependence is due to the fact that the Coulomb force is parallel to the relative displacement of the particles. Therefore, the Coulomb hole in the correlation function will be strongest when the relative momentum is parallel to the longest dimension of the pair’s separation. For long-lived sources this is the longitudinal direction [Pra87,Gon91a]. 6.5 Trajectory calculation The derivation of Eq. 6.3 is based upon the assumptions that the phase space distribution does not vary significantly over the range 1/$.P:i:q and, furthermore, that dynamical correlations and distortions in the Coulomb field of other reaction residues can be neglected. In this section we assess the validity of 127 these assumptions by means of trajectory calculations. The use of trajectory calculations is justified by the validity of the classical approximation. In our trajectory calculations we consider the sequential emission of two carbon nuclei from the surface of a source of radius R which is initially at rest . S and which has a total charge of Q :2 e=93e and a total mass of M =A u. In order S S S S to isolate the effect of three body Coulomb distortions from dynamic distortions resulting from the recoil of the emitting system, we use an artificially large mass number for the emitting source, A =10000. Further below, we evaluate recoil S effects for a more realistic heavy source of mass number AS=226. To be consistent with the calculations presented in Section 6.4, we assumed an emission function of the form: _, __, A A A A __ _ _ g(p,r ,t) °c (r °p)®(r op)6(r-RS)@(E)JE e E/T e t/T . (6.22) This emission function was sampled by Monte-Carlo techniques. Upon emission, the particle trajectories were calculated by integrating Newton’s equations, and the asymptotic particle momenta were stored as coincidence events and as single- particle spectra. Correlation functions, 1+R, were constructed in the same way as experimental data using Eq. 6.2. First, we explore the validity of using the approximate form of the phase space distribution, Eq. 6.4. For this purpose we have performed trajectory calculations in which the Coulomb interaction with the emitting source was turned off. In this case, the relation between initial and final two-fragment phase space points can be calculated analytically. In Figure 6.3, the results of these calculations are compared to calculations based upon Eq. 6.17 (these latter calculations are identical with results obtained from Eq. 6.3). The open points 128 MSU-9I-I42 I I I I Fr I I I—r I I I I I TI I I r I 1717 I I : 1.25 As=10000, RS=12fm, T=15MeV —5 CI 1.00 2,, 22:6 %W 0.75 IllllIll 0.50 LLLI O Trajectory, ZS=O 0'25 Koonin—Pratt {IIITIIITTIIIrl—rrlllllll r j. + IlllIllllIllllIllll ll_l_lIll L11 m L 1L1 1L1 1L: + 0-00 fljl'jltrj TITITITIWITITITITllI—r "‘ 1.25 1.00 E— 0375 Z— 0.50%— o.25 :2— ; . 0.00 '- iJLII iJLiiaLIJLiIlLIiIlLILLIIL 0 5 10 15 20 25 30 Vrel/VZfiZZ (x10“‘3 0) Figure 6.3 The points represent correlation functions obtained via trajectory calculations assuming emission from an uncharged source; the curves show the results obtained from Eq. 6.17. The parameters used in these calculations are indicated in the figure. 129 represent the results obtained from these trajectory calculations and the curves show the results obtained from Eqs. 6.17. The difference between the two calculations is significant, but not necessarily surprising. The emission function varies over characteristic momenta of the order of ApzflmT .... 0.6 GeV/c. The Coulomb minimum of the two-fragment correlation function has a width of the order of Aq==0.2-0.3 GeV/c. Hence, the condition Aq<<10"3 c) 0.00 Figure 6.5 Comparison of two-carbon correlation functions calculated by means of three-body trajectory calculations (points) with correlation functions calculated with Eq. 6.17 (curves). Upper and lower panels show results for total momenta per nucleon of P/ As110 MeV/c and P/A2110 MeV/c, respectively. The parameters used in these calculations are indicated in the figure. 132 for 1:500 fm/c. However, for shorter lifetimes (15200 frn/c) distortions in the Coulomb field of the emitting source become so large that Eqs. 6.17 become virtually useless. As may be expected, distortions by the Coulomb field of the emitting system are less important for fragment pairs emitted with larger initial velocities. This qualitative expectation is borne out by the calculations for more energetic fragment pairs with P/A2110 MeV/c. For this momentum gate, Eq. 6.17 provide reasonable approximations for emission times 13200 fm/c. However, for shorter emission times, the disagreement is still substantial, though much smaller than for the low-momentum gate. For the emission of rather energetic particles, three-body Coulomb ‘ distortions of the angle averaged correlation functions become small, even for rather short emission times. As an example, the open points in Figure 6.4 show the results of three-body Coulomb trajectory calculations performed for flat energy spectra, T=oo, and for Z =93. For energies well above the Coulomb barrier, P/ A2 150 fm/c, the distortionsS in the field of the emitting source become small and trajectory calculations (for ZS==O as well as 25:93) agree nearly quantitatively with calculations performed with Eq. 6.17 even for emission time scales as small as 1:50 fm/c. However for low-energy emissions, P/As140 MeV/c, the discrepancies are appreciable. It is interesting to note that three-body Coulomb distortions and inaccuracies of the Koonin-Pratt formula for ZS=O appear to become small at comparable energies. Distortions in the Coulomb field of the emitting system can be particularly large for longitudinal and transverse correlation functions. As an example, Figure 6.6 shows longitudinal and transverse correlation functions for a relatively long lifetime, 1:200 fm/c. For this lifetime, the deviations from the angle integrated correlation functions calculated with Eq. 6.3 or 6.17 are still relatively modest (Figure 6.5). In order to increase statistics, we had to widen the 133 HSU-Dl-ISB T I l l I I—T r 1 r I F l [Tl t I vj’ I I 1 ‘l—r l I I F r I f .1 - zs=93. R3=12frn. T=15MeV — 1.25 — __ - T=200£rn/c j? . a Z Zivzz=6 j j .1 % - 1 + l ”W “was r- D U 4 E i o ° {1 l’: D: P— U -—i + 0.75 _ + an 1 _ .4 L D .. - D -1 ~ + c. . 0.50 r" o “l : o : - D a L 45 0 o ¢=(o-5o,1so-180)° - 0.25 - o o o .1 : o D a 41470-110) . L D _ 0 a D 0.00 LPIJAL L J_ l_l I I J. L I I Pl J I L1. l_l_l J J L L L LI__L l 0 5 10 15 20 25 30 val/«21w, (x10’3 0) Figure 6.6 Longitudinal and transverse correlation functions between two carbon fragments calculated via trajectory calculations for 1:200 fm/c. 134 angular acceptance for the longitudinal correlation function, xy=O°-SO° or 130°— 180°. (In the present simulation, the energy spectra are relatively strongly peaked at the Coulomb barrier. As a consequence, only relatively few events in the longitudinal gate fall into the region of relative velocities of interest here.) The longitudinal (circular points) and transverse (square shaped points) correlation functions deduced from the trajectory calculations were normalized with the same normalization constant determined from the asymptotic behavior of the angle integrated correlation function. Although the statistical errors for the longitudinal correlation function are large, it is clear that it exhibits a narrower minimum at Vredzo than the transverse correlation function, in complete disagreement with the qualitative trends predicted from Eqs. 6.3 or 6.17, see Figure 6.2. Hence, the detailed shape of longitudinal and transverse two- fragment correlation functions can be strongly affected by interactions with the Coulomb field of the residual system. It would be of interest to investigate whether such sensitivities to the charge distribution of the emitting system could be exploited to differentiate between various reaction models for multifragment disintegrations. Since intermediate mass fragments are emitted with larger average momenta than light particles, one may expect that two-fragment correlation functions can also be affected by dynamical correlations due to momentum conservation effects. In Figure 6.7, we assess the magnitude of distortions resulting from the recoil of heavy reaction residues. The solid points in the figure show three-body trajectory calculations for emission from a realistic source of mass M =226 u and the open points show the results for emission from an S artificially heavy source of mass MS=IOOOO u for which recoil effects are negligible. The upper panel shows calculations for thermal energy spectra with T215 MeV and the lower panel shows calculations for flat energy spectra, T=<>o, 135 MSU-9|-|4| .- I r‘r—r I I__r I I FT I I I—rI I fiI I I I I I I I r I I I I—r I A _ 23:93, RS=12fm, T=200fm/c , 1’0 F— ZI'ZZ = 6 . q: ,.;-’;o“o'°"" "2;...“- : T = 15MeV ”by...” i 1 r- m j 0'5 F— ..." Koonin—Pratt i _ ." O Trajectory, As: 10000 i L ." 0 Trajectory, As = 226 1 D: - _." J + 0.0 ~ *1 ~ ‘—4 - l. 1.0 r- r- 0.5 m 0.0 41 I__ll_l LILLLIILILLQLLl—l L141_1L1_LL1 0 5 10 15 20 25 30 Vrei/Vzi+zz (X104 C) Figure 6.7 Two-carbon correlation functions calculated via three-body trajectory calculations for sources of different masses. Solid points: MS=226 u, open points MS=10000 u. The upper panel shows calculations for thermal energy spectra and the lower panel shows calculations for flat energy spectra, T=°°, integrated over the indicated energies. The parameters used in these calculations are indicated in the figure. 136 = 63-563 MeV; other parameters of =93, 1:200 fm/c). For the averaged over a broad range of energies E1, E 2 the calculation were kept fixed (RS=12 frn, ZS calculations with T=15 MeV, recoil effects introduce only minor distortions, largely because most particles are emitted with energies close to the Coulomb barrier. For the emission of more energetic fragments, recoil effects become slightly more important as is illustrated by the calculations for T=°o, shown in the bottom panel of Figure 6.7. Our schematic investigation is aimed at emission from rather heavy nuclear systems. For lighter sources, of course, recoil effects become more important. One must stress, however, that reasonably unambiguous assessments of recoil effects can only be made within the framework of specific reaction models, particularly for fast noncompound emission processes expected to set in at higher energies. This is because the effects will depend on the details of size and time evolution of the source. 6.6 Conclusion The schematic model calculations presented in this chapter indicate that two-fragment correlation functions are, indeed, sensitive to the space-time evolution of the emitting system. For large emission times (13200 fm/c) and fragment kinetic energies well above the Coulomb barrier, calculations incorporating the mutual Coulomb interaction between the two emitted fragments allow the extraction of emission time scales with reasonable accuracy. For emission from heavy and highly charged systems, distortions from interactions with the Coulomb field of the residual system can be significant thus rendering the analysis more model dependent. Such distortions are particularly important for fragment emission with low kinetic energies or on fast emission 137 time scales. Additional model dependences can arise from dynamical correlations resulting, for example, from momentum conservation. Directional dependences of two-fragment correlation functions are particularly sensitive to final-state Coulomb interactions with the residual system. Indeed, the shapes of longitudinal and transverse correlation functions can be strongly altered by interactions with the Coulomb field of the emitting system, even when the angle-integrated correlation functions appear to suffer only minor distortions. Such enhanced sensitivities of two-fragment correlation functions to properties of the emitting system may contain useful additional information on the emission process and thus allow more stringent tests for various models for multifragment disintegrations. Thus it appears promising to pursue more detailed calculations of two-fragment correlation functions for specific reaction models capable of making predictions of the space-time evolution of nuclear disintegrations by multi-fragment emission. Chapter7 Time scales of multi-fragment emission : 36Ar + 197Au at E/A=35MeV 7.1 Equal-charge two fragment correlation functions Experimental two-fragment correlation functions are constructed from fragments detected at forward angles 16 ° s 9 $31 ° corresponding to ring 2 and 3 of the Miniball, where the statistics of the eiajferimental data and the granularity of the array are optimal for extracting good-quality correlation functions. First, for simplicity of the discussion, we study correlation functions of two equal- charge fragments. Figure 7.1 shows experimental two-fragment correlation functions constructed according to Eq. 6.2. The top, center, and bottom panels show Be-Be, B-B, and CC correlation functions respectively. (Double hits by a—particles resulting, for example, from 8Be decays were identified by pulse shape discrimination and excluded from the data presented in this thesis, see Chapter 4.) The left panels present inclusive correlation functions and the right panels show correlation functions gated by the requirement N 29 and P/A>150 MeV/c, where N denotes the total number of charged particles detected in the Miniball and P/ A is the total momentum per nucleon of the coincident fragment pair. All correlation functions exhibit pronounced minima at q=0 MeV/c which can be attributed to the repulsive final state Coulomb interaction between the emitted fragments. The correlation functions gated by N 29 and P/A>150 MeV/c are similar in shape to the inclusive correlation functions, indicating that this gate does not select significantly different emission times. 138 77‘1 ' ‘- 1+R(q) I” = ‘ .= O . .: 1.0 L X Be 0 0 1L X Be. .O .Q .‘ i ------ T= 50fm/c 1 05 L — - 1'= 100fm/c j; I -——- r= ZOOhn/c jf ' - -~“ r= 500fln/c 1t ' O 0 A —+ L J— L k L + J— + ‘ . _ l T I 'fi '_T E t r l 197I’lu(361311",XX),E/A=35MeV, 61ab=16°—31° 139 MSU—91-007 f r I I I fl ”—F j * r fin fl _r Irr—Yi‘l’I I «l : X = C i 1.0 —- - : T I I 0.5 _- ‘- . i 0.0 , ‘ . o 100 200 300 400 500 o 100 200 300 400 500 600 q (MeV/c) Figure 7.1 Inclusive (left hand panels) and gated (right hand panels) two- fragment correlation functions at small relative momenta. The calculations are explained in the text. 140 The curves shown in the figure represent calculations with the Koonin-Pratt formula, Eq. 6.17, for an emission function g(p,?,t) similar to Eq. 6.20: dN -t/ r 94 A A A A g(p,r,t ) ... (r °p)€-)(r 'p)6(r-R )— s dEd o 9 (7'1) The energy and angular distributions dN/dEdQ were taken to be consistent with the experimental single fragment distributions. The emitting source was assumed to move with a velocity of vS=0.035c consistent with emission from the surface of heavy reaction residues formed in incomplete fusion reactions in which 80% of the projectile momentum was transferred to the fusion-like residue (taken from the systematics of linear momentum transfer for reactions producing heavy reaction residues [Fat85]). For the emission of the second fragment, we 1/ 3 1/3 required a minimum initial separation of rmin= 1.2 (A 1 +A2 ) fm from the first fragment. In all calculations we assumed a fixed source radius of RS=12 fm. Choosing a smaller source radius leads to more significant Coulomb final state interactions; this, to first order, has the same effect on the correlation function as reducing the emission time scale. Likewise, an increase in the source radius cannot be easily distinguished from an increase in the emission time scale. The comparison between measured and calculated correlation functions in Figure 7.1 indicates that the inclusive correlation functions are reasonably consistent with mean emission times of about t-~100~200 fm/c. The correlation functions gated by N 29 and P/A>150 MeV/c appear to indicate slightly shorter emission time scales. In our calculations, we neglect the possibility that highly excited primary fragments could undergo sequential decays by particle emission. Such decays may be expected to attenuate the minimum at qu MeV/c [Bar90]. It is, therefore, possible that our analysis yields mean emission times which are slightly larger 141 than those of the primary fragments. Our data seem to indicate slightly larger emission times for Be fragments than for B or C fragments. Furthermore, the calculations reproduce the Be-Be correlation function less well than the B-B or C- C correlation functions. It is interesting to speculate that these observations could be related to enhanced feeding of Be nuclei. Additional experimental information concerning the magnitude of sequential feeding would help resolve this issue. Figure 7.2 illustrates systematic uncertainties introduced by our apparatus and data analysis. The curves in panels (a) and (b) show the theoretical B-B correlation functions already presented in the left center panel of Figure 7.1. The solid and open points in panel (a) show the correlation functions for 1:100 and 500 fm/c, respectively, filtered by the response of our apparatus. The resulting distortions are of minor importance. (The instrumental distortions at very small values of q are mainly due to the finite angular resolution of the apparatus; uncertainties in energy calibration (25%) are small in comparison.) Panel (b) illustrates errors which could arise from the unknown mass of the detected fragments. The solid and open points show experimental B-B correlation functions constructed by assuming A=10 and A211, respectively, for the mass of the detected boron fragments. The sensitivity of the calculation to uncertainties of the velocity and size of the emitting source is illustrated in panels (c) and (d). None of these uncertainties changes the extracted lifetimes by more than 50%. For the present reaction, the charged particle multiplicities are sufficiently low to virtually guarantee the survival of a heavy reaction residue (which might subsequently fission). To assess the influence of the Coulomb field of the heavy reaction residue, we have performed three-body Coulomb trajectory calculations as described in Section 6.5, assuming a residual nucleus initially with charge number Z =93, mass number A =226, and velocity V =0.035c. The emission S S S function was the same as in Eq. 7.1. The response of the experimental apparatus 1+R(q) 1.0 0.5 0.0 1.0 0.5 0.0 142 MSU—91—008 197Au(36Ar,BB), E/A=35Mev, 61ab=16°—31° I I I I I I fij—h ‘ r If I (a) Monte Carlo Simulation of 1L I I Apparatus Resolution L r L I _ - L l L t I No) C C— ; Figure 7.2 Estimates of experimental and theoretical uncertainties; a detailed discussion is given in the text. 143 was included. In Figure 7.3a, the results of these calculations (curves) are compared to the inclusive C—C correlation function (points). The three-body Coulomb trajectory calculations predict wider minima in the correlation functions than the calculations based upon Eq. 6.17, and the agreement with the data is slightly worse. Nevertheless, the calculations are in fair agreement with the experimental correlation functions for emission times of the order of 100-200 fm/c. In Figure 7.3b, we compare calculations for 1:200 fm/ c. The curve shows the calculation with Eq. 6.17 and the solid points show the results of the three- body trajectory calculation. The discrepancy between the two model calculations is largely caused by dynamical correlations caused by the recoil of the heavy reaction residue which is included in the trajectory calculation, but not in Eq. 6.17. Indeed, the calculation with Eq. 6.17 is in rather good agreement with three- body calculations when the recoil of the heavy target residue is artificially reduced by increasing its mass to A =SOOO while keeping its charge and radius S constant (open points). Shorter emission times appear to be indicated for the higher momentum cut (P/A> ISOMeV/c). However, this observation is not conclusive due to poor statistics. By combining different fragment pairs of 4sZs9, we can gain an increase in statistical accuracy by a factor of about 20. This makes it possible to study the time scales in more detail by applying gates on the charged particle multiplicity and fragment energies. 144 MSU—91—009 197 6 Au(3 Ar,CC), E/AzBSMeV _ r5 1‘3 l ‘ I TC T‘ l ‘ I’T _ l- (a) B—body calculations .9 1.0— _: : °data : 0.5"— """ ._‘ g ' é V. _ E: ”119.114L.I.LLL.‘ 0-0 T 1 I T I l Ii—l T I T I f + " l H 1 O C. (b) T=100fm/C . I 1 _ J - a asr— ° 3-body.As= 225__ : U 3—body,AS=5000 : * —- 2--body a 0,00 l L 4 l l L I L 4 ‘1 o 200 400 600 <1 (MeV/c) Figure 7.3 Panel (a): Comparison of inclusive C-C correlation functions to three— body Coulomb trajectory calculations. Panel (b): Correlation functions calculated from Eq. 2 (curve) and from three-body trajectory calculations (points). 145 7.2 Combinations of different fragment pairs In Figure 7.4, we compare correlation functions for representative pairs of intermediate mass fragments alternatively as functions of relative momentum q (top panel) or as a functions of reduced relative velocity vred = gel/$21122 (bottom panel). Different symbols in the figure denote correlation functions evaluated for different fragment pairs (Z1=6, and 452 s9). Since the Coulomb 2 repulsion is greater between fragments of greater charge, the correlation functions, 1+R(q), exhibit wider minima at q=0 for increased charge of the second fragment. When plotted as a function of v however, the correlation functions red’ 1+R(vred) are very similar. This suggests that correlation functions may be summed over different pair combinations and evaluated as a function of the reduced relative velocity with little loss in resolution. This "mixed-fragment" analysis permits the exploration of emission timescales with significantly improved statistical precision. In the following, we construct "mixed-fragment" correlation functions according to Eq. 6.2, where the sum is extended over all 1’22 summation to allow the exploration of emission timescales as functions of the charge combinations with 432 $9. Sufficient statistics is achieved via this charged-particle multiplicity and the velocity of the emitted fragments. Our definition of the mixed-fragment correlation function differs slightly from that adopted in reference [Tro87] where two—fragment correlation functions were evaluated as a function of the relative fragment velocity. Figure 7.5 illustrates the difference between the two prescriptions. The top and bottom panels of the figure show correlation functions evaluated, for different fragment combinations with 4le,22 velocities, Vrel and Vred’ respectively. For this range of fragment charges, mixed- 39, as functions of relative and reduced relative L% q (MeV/c) o 100 200 300 400 500 600 m““’TT' WTTF'T'T“T'TT “' 5 19711106111 2122), E/A= 35MeV. 0m=16— -31° ‘5 CO 1.0 — NIM} ' Z1:6 ¢° °°¢:2';¢ €135“ g I fit: ow.“ “‘1‘ . 13¢ o'¢¢‘ M 22 ‘ 0.5 :' (3° g¢¢.. ‘4 U 4 . 7 '71. A ' DO ..q’tl ' 5 A 8 . 0" ‘00-'3o‘fif‘ 06 A9 4 E ()0 -T fiaakaiirek‘fiesst 1.‘ a T a. L+lrt 1. + . _ n T T T T I + H . ‘41“ 3 __ I 1'0 - éjjIIIIIII'IIu I! 1 II . I. T 0.5 — * L £333.! 00 4111 LLLllniLiLllisLLLllli.LLLI 10 Val/«21mg (><10“3 15 20 25 C) 30 Figure 7.4 Dependence of inclusive, energy integrated two-fragment correlation functions on relative momentum q (top panel) and on reduced relative velocity vrel/ JZI +2 2 of the other 2. One fragment was carbon (Z=6); the atomic number Z fragment is indicated. 147 Vrel (XlO-a C) 0 20 4O 6O 80 100 ''TIH'‘IFII'IIH‘IHHI'Iz - m . 1°"Au(""1\r,z,zz), E/A=35MeV, alab=16—31° CE . j 3 L 5 I 1.0 - 1 o - on . o: 0.5 - A ' U-I .- v t D: 00 ‘ + I H u- l . 1.0 r— 0.5 - [ I: .. l l L J l :1 0.0 11L ILL! L14L41—14 LLLI_1 LILJLJ 0 5 10 15 20 25 30 Vrel/VZ.+22 (x10“3 c) Figure 7.5 Dependence of inclusive, energy integrated two-fragment correlation functions on relative velocity vrel (top panel) and on reduced relative velocity V /\/Z1 +Z2 (bottom panel). Two-fragment correlation functions are shown for rel each combination of Z and Z2, with 4521,2259. 1 148 fragment correlation functions display superior resolution when they are evaluated as a function of the reduced relative velocity. Of course, the loss of resolution incurred for 1+R(vrel) can be incorporated into model calculations by performing corresponding averages [Tr087]. 7.3 Gated correlation functions Measured energy integrated two-fragment correlation functions, gated by various conditions on charged-particle and intermediate mass fragment multiplicities, are shown in Figure 7.6. The top and bottom panels show correlation functions gated by various conditions on charged-particle and IMF multiplicity, N C and N IMF’ respectively. For orientation, we include calculations with the Koonin-Pratt formula, Eq. 6.17, for different emission times 1:. Correlation functions measured for peripheral collisions or for events in which only two intermediate mass fragments were detected exhibit considerable distortions at larger reduced relative velocities. Since these distortions are not yet understood, they introduce slight uncertainties in the asymptotic normalization of the correlation functions. They become less significant for central collisions or for events in which at least four fragments are detected. The shapes of the energy integrated correlation functions depend only slightly on these multiplicity gates. The width of the minimum at v zO appears to decrease slightly as more central red collisions (larger charged~particle multiplicities) are selected. This observation is qualitatively consistent with slightly longer time scales (or larger source dimensions) for fragments emitted in central as compared to peripheral collisions. Central collisions are most important because they are better suited to study the properties of hot nuclear matter. Also they have the largest degree of 149 MSU- 9l- WI :1 I I I I I I I I I I I I I I I I I I I I l I I I I I l I I I I 1 1.25 :- 197151u(3"Ar,Z 22), E/A= 35MeV al,b=16— 3133:: 5 4s ._ 21,22s _ 9 0 00 i 1'00 _ all energies ‘3 0.75 :— ~j 0.50 E- -I 2 a Z 1 m 0 00 P 1 r’. 1 1 I 1 1 L I 1 L 1_l l 1 J 1 L 1 4 1 I L 1 d i . : rI I I I T I I If r l I I r I l—I'I' Iir I I—r T FI I I I I: 1.25 E- . ng4 00-0: 0 .1 1.00 :— ° NM=3 E 0.75 :— Koonin—Pratt -: E 1(fm/c) : 0.50 I..— oooooooo 50 _: ’ - - - 100 2 0.25 E- —— 200 1 : . _"" - 500 '1 0.00 _ 1 I 4:1{L' {L1 1 1 L 1i 14 1 1 L1 L11 1 1 LL1I I 1 1 O 5 10 15 20 25 30 Vrel/VZ1+ZZ (Xqu3 C) Figure 7.6 Two-fragment correlation functions summed over all combinations of 21 and Z (with 4sZ ,2 s9) and selected by the indicated gates on charged- particle multiplicity N C (top panel) and on IMF multiplicity N IMF (bottom panel). 150 equilibration according to energy spectra discussed in Section 6.4. For the remainder of our investigation, we will focus on correlation functions gated by central collisions. Figure 7.7 depicts two-fragment correlation functions gated by central collisions (N 212). In order to be able to study fragment emission time scales for different regions of the kinetic energy spectrum, we have evaluated the correlation functions for three different ranges of P/ A, the total momentum per nucleon of the coincident fragment pair. The low momentum gate (P/AsllO MeV/ c: solid circular points) selects fragment kinetic energies at and below the exit channel Coulomb barrier. Kinetic energies slightly above the Coulomb barrier are selected by the intermediate momentum gate (P/A=110-120 MeV/c: open circular points). For these two gates, a considerable part of the emission cross section is consistent with equilibrium emission, see also Fig. 5.12. Kinetic energies significantly above the Coulomb barrier are selected by the high momentum gate (P/A2140 MeV/ c: solid square-shaped points). In this domain, fragment emission for central collisions is dominated by nonequilibrium emission processes different from projectile fragmentation. The experimental correlation functions exhibit a rather pronounced dependence on the total momentum per nucleon of the emitted fragment pairs. The minimum at Vredz 0 becomes considerably wider as the gate on P/ A is raised from below 110 MeV/ c to above 140 MeV/c. Since wider minima are indicative of smaller space—time dimensions, this observation is qualitatively consistent with the expectation that mean emission times should become shorter as the kinetic energy of the emitted fragment is raised from close to the Coulomb barrier to much higher values. In Figure 7.8, the measured inclusive two-fragment correlation functions are compared to correlation functions calculated for the indicated emission times 1:. Calculations with the Koonin-Pratt formula, Eq. 6.17, are presented in the top 151 MSU-9l-I74 fI I I I7 I I I I—T I I I I ITI I I I I I II I I I I I I '4 125 _ 197Au(:“3’Ar.Z122), E/A=35MeV, 61ab=16—31° _: ' ; 4szl,zzs9,NC212 4; 1.00 L Huh .1 331113; I i ¢¢¢ ¢¢§6¢$$QSI + : : ++ ¢¢¢++# I : 0: 0.75 — __ + _ + ¢ ++ d H : ¢ : _ I ¢ +* _ 0'50 .— 7 O ¢ P/A(MeV/c) : Z . $110 : ' * 0|“ 0 110—120 - 0.25 —— ¢ 1. __ ” '1‘ 02 I 2140 : : ..sgg .. 0.00 "'1 1 LA $1 Pl Ll L141 114 14L 1‘1 l_l L l 1 l 1 LL— 0 5 10 15 20 25 30 Vre/VZI+ZZ (><1O'3 c) Figure 7.7 Two-fragment correlation functions summed over all combinations of Z1 and Z (with 4sZ ,2 s9) and selected by central collisions (N 212). The correlation functions are evaluated for the indicated ranges of P/ A, the total momentum per nucleon of the detected fragment pair. 152 MSU-9l- I70 197Au(36Ar,ZIZZ), E/A=35MeV lIIIIrTIIrrIIrrTIIII—fiTIIIITIIIITT I. 0m=16-31°, 4§Zl.23§9, NCEIZ ' all ener ies 1.0 f- g 0.5 :- I ' f l 00L- ITI IT—TII—T—TT IIIFWI 10 E ...J m . + : "‘ 0.5 Trajectory,Zs=0 0.0 1.0 _— 0.5 f' . -1 f/ Trajectory.Zs-=79 00 111: L‘.J._.i_LllLl_11_lJl_Jl_1LlllJLILIL‘ O 5 10 15 20 25 30 vm/x/zpuz2 (x10‘3 c) Figure 7.8 Inclusive, energy integrated two-fragment correlation functions summed over all combinations of Z1 and 22 (with 4sZ1,Zzs9). The different panels give comparisons with different calculations discussed in the text. 153 panel; the center and bottom panels show the results of numerical trajectory calculations for 28:0 and 79, respectively. Calculations neglecting Coulomb interactions with the residual system (upper and center panels) predict shapes of correlation functions which are in slightly better agreement with the experimental correlation functions than the three-body calculations incorporating distortions in the field of a heavy reaction residue (bottom panel). Slightly smaller emission times are indicated by calculations with the Koonin- Pratt formula and by the 25:0 calculations than by the three-charged-body trajectory-calculations. Nevertheless, all calculations are consistent with emission times between 1:100-200 fm/c, in agreement with the results from equal-charge correlation functions. Calculations for correlation functions measured for central collisions and for different cuts on P/ A, the total momenta per nucleon of the coincident fragments pairs, are presented in Figures 7.9-11. Individual panels of these figures depict results for the indicated cuts on P/ A. Figures 7.9, 7.10, and 7.11 present calculations for different emission times using the Koonin-Pratt formula, and trajectory calculations for Z =0 and Z :79, respectively. For the case 2 S S S three-body trajectory calculations could only be performed for the two higher =79, momentum cuts, P/A=110-120 MeV/c and P/A2140 MeV/c, as the cut P/As110 MeV/ c selects mostly energies below the Coulomb barrier. For clarity and consistency, we refrained from lowering the exit channel Coulomb barrier which would require a significant increase in source radius or a significant decrease of the source charge. Either of these parameter modifications would reduce Coulomb distortions and, hence, differences with the Koonin—Pratt and 28:0 calculations. 154 MSU- 91- I75 197A11|(36'11r,Z 7.2),. E/A= 35MeV r- F1 I WI III—FT I] I_r—I I I I T I . P/ASIIOMeV/c 1.0 — 0.5 E- I 0.0 _+ . t P/A:110—120MeV/c I - . .1 1.0 r .. - m _‘ /./. 4 + L ." /.°/° : H 0.5 _ ./ 9.: 91¢=16"31° —. : ./ ... NCZIZ : /./ /...°. 4§Z1,ZZ.S.9 0.0 - ' . I P/Ag 140MeV/c ...‘1 1.0 '— _ . 0.5 :- 0 O 1 /. 2122.11 14 1_Ll LJ.LJL_I 1 1 L14111_.| LLL.‘ 0 5 1015 20 25 30 Vrel/VZI+ZZ (XIO—a C) Figure 7.9 Two-fragment correlation functions summed over all combinations of Z andZ (with 4sZ ,2259) and selected by central collisions (N 212).Individual panels show the correlation functions for the indicated cuts on P/ A, the total momentum per nucleon of the detected fragment pair. The curves represent calculations with the Koonin-Pratt formula for the indicated emission times. 155 MSU-Sl-IGB 7 3 19 Au( 6Ar,z.zz), E/A=35MeV T7 II—I—r ITII IIIITI—rTrIIIrTIIIIr .. P/ASIIOMeV/c 1.0 " 009..” O : / Trajector ,Zs=0 r 1' fm/c) O5 — . ..... 50 d “ — - 100 I -. -—-———200 I 0". fl 1 L_ ';500 0,0 _.:::15H:fitstlthjH—mfitqui “ P/A:110-120MeV/c . ' o 10 L- . . 1 - 00.? 03 - . ° . + ; _ .. _ "" 0.5 — -’ Owls—31° I ‘ ./.' chiz ‘ L x' . ° 43.21.2259 - 0,0 +1-1-++1+>+-j-+-1+1-++++1-I-4—+-1—1—+—1-1—H+1—+—1— - P/A214011eV/c ... ' 0 ..: 1.0 f . , . .' O .1 0.5 '- 00 is. .1.JUL.LL.L.LL.L1-.A.J..E O 5 10 15 20 25 30 val/721mg (x10‘3 0) Figure 7.10 Two-fragment correlation functions summed over all combinations of 21 and 22 (with 4sZ1,Zzs9) and selected by central collisions (NC212). Individual panels show the correlation functions for the indicated cuts on P/ A, the total momentum per nucleon of the detected fragment pair. The curves represent the results of trajectory calculations for which the Coulomb interaction with the residual system is turned off. The key for emission times is given in the figure. 156 MSU-9l-l76 197Au(36Ar,2122), E/A=35MeV "l'—"'I‘T"I"*'T""I""I"' P/A:110—120MeV/o 1 0 ; .” . Z .7 Trajectory,Zs=79 - 4' T(fm/C) 0,5 — /.’ / ........ 50 _ ' ,' '° ““ 100 “ I. #200 j _ f. v ----- 500 .J.Q...1°'1L...L¥L.... ....l..1 Di. oofil+j+j22,+ .—4 L P/A2140MeV/c .j ' ‘9')“. 1.0 *- O.5 — /./l/ 01ab=16"31° : . Nc212 L (.9. 4§ZI,ZZ§9 * 00...“ 1 00 1411#ULJLAQMQLLPLPL.1.AJM1 O 5 10 15 20 25 30 val/«21%, (><10"3 c) Figure 7.11 Two-fragment correlation functions summed over all combinations of Z and 22 (with 4sZ ,Zzs9) and selected by central collisions (N 212). Indivrdual panels show the correlation functions for the indicated cuts on P/ A, the total momentum per nucleon of the detected fragment pair. The curves represent the results of three-body Coulomb trajectory calculations in which the two fragments are assumed to be emitted from a source of initial charge number ZS=79. The key for emission times is given in the figure. 157 For all three approximations investigated, comparisons between theoretical and experimental correlation functions lead to qualitatively similar conclusions: The emission of energetic fragments is governed by significantly smaller time scales than the emission of low-energy fragments with energies close to the exit channel Coulomb barrier. Such a dependence is consistent with the predominance of nonequilibrium emission processes for energetic fragment emissions and the increasing importance of emission from more equilibrated systems for particles emitted with energies close to the Coulomb barrier, see also Fig. 5.11. As was already observed for the inclusive correlation functions, calculations with the Koonin-Pratt formula and those for ZS=0 predict shapes of correlation functions which are in better agreement with the experimental data than those predicted by three-body trajectory calculations. These former calculations indicate emission times of 1:550 fm/c for the emission of energetic fragments selected by P/A2140 MeV/c and considerably larger emission times, T=500 frn/ c, for subbarier emission selected by P/A.<.110 MeV/c. Emission times extracted for P/A=110—120 MeV/c are of the order of t=150 fm/c. Qualitatively similar conclusions are drawn from a comparison with the three-body trajectory calculations for ZS=79. Emission time scales of the order of several hundred fm/c are consistent with time scales expected from statistical models of compound nuclear decays. For example, the model of reference [Fri90] predicts average time intervals of 1:300 fm/c between the emission of two carbon fragments from equilibrated heavy nuclei (A=226, 2:93) of 700 MeV excitation energy. Much shorter time scales, 'czSO-100 fm/c, extracted for the emission of energetic fragments in central collisions, are incompatible with statistical emission from fully equilibrated heavy reaction residues. These emission times are of comparable 158 magnitude as those predicted [Aic88a,Aic88b,Pei89] by dynamical models of fragment production. Chapter 8 Summary In this thesis we addressed the questions related to multi-fragment emission. Theoretical studies show the sensitivity of the probability of multi- fragment breakup to the nuclear equation of state at low density. We tried to measure the multi-fragment emission at intermediate energy nucleus-nucleus collisions and characterize the reaction mechanism by studing the global observables and time scales of the fragment emission processes. We have measured the relation between fission fragment folding angles and total charged particle multiplicities for the reactions 14N+238U at E / A=50 MeV and 36Ar+-238U at E/A= 20 and 35 MeV. The mean values of these two quantities exhibit a monotonic relation. Both reaction filters have finite resolution due to the statistical nature of the decay processes, yet is is possible to distinguish central, large momentum transfer collisions from peripheral, small momentum transfer collisions by gates on very high (M212) or very low (M=1) multiplicity, respectively. Less extreme multiplicity gates, however, are less selective and do not result in cleanly separated momentum transfer distributions. Rather broad momentum transfer distributions are observed, for example, at intermediate multiplicities. In contrast to the total multiplicity or the multiplicity of charged particles emitted to backward angles, practically no selectivity was observed for a reaction filter utilizing the number of charged particles emitted to forward angles, 9<35°. At least for the present reactions, the multiplicities in such forward arrays cannot be used to select violent projectile-target interactions in which large amounts of energy and / or momentum are dissipated. 159 160 We also established the occurrence of multi—fragment emission processes. Single and multiple emissions of intermediate mass fragments to large angles (8235°) occur essentially in reactions characterized by large associated charged particle multiplicities and large linear momentum transfers to the heavy reaction residues. The measured fragment multiplicity distributions are well described by Poisson distributions consistent with a stochastic IMF production process. Final states with only one intermediate mass fragment in the exit channel represent only one of a family of final states. We successfully constructed a low-threshold charged—particle detector array with high granularity and good particle identification, the MSU Miniball. With this device, we studied the emission of intermediate mass fragments in collisions between 36Ar projectiles and 197Au target nuclei at E/A=35 MeV. Intermediate mass fragments are preferentially emitted in violent central collisions characterized by large charged-particle multiplicities N C' In peripheral collisions, fragment emission is a fairly unlikely process; the average IMF multiplicity increases from f 0.1 for N C = 2 to ...... 1.2 for N C 3 15. The elemental distributions observed for various cuts on charged-particle multiplicity exhibit a nearly exponential fall-off as a function of Z. In this reaction, the inclusive element distributions are rather similar to those observed in central collisions. The exponential fall-off is only slightly steeper for fragments produced in peripheral collisions than for fragments produced in central collisions. The angular distributions of fragments produced in peripheral collisions are steeper than those of fragments produced in central collisions. At forward angles, the energy spectra of fragments produced in peripheral collisions exhibit a high-energy shoulder which could be attributed to emission from a projectile-like source. Such a shoulder is not observed in energy spectra 161 gated on central collisions, for which the spectra exhibit nearly exponential shapes. A detailed analysis of the energy and angular distributions of fragments emitted in central collisions (gated by N 212) reveals important contributions C from processes incompatible with emission from equilibrated sources. These contributions dominate for fragments emitted at forward angles and with kinetic energies well above the exit channel Coulomb barrier. Emission with kinetic energies close to the Coglomb barrier and at larger angles is consistent with increasing contributions from the decay of equilibrated heavy reaction residues. The granularity of the apparatus and the statistics of two-fragment coincidences allowed the generation of good quality two-fragment correlation functions in the angular range of 16° s9 331 °. The two-fragment correlation lab function was found to depend mainly on the reduced relative velocity of the fragment pairs. This scaling allows the construction of mixed-fragment correlation functions with little loss in resolution and significantly improved statistics thus allowing a detailed investigation of two-fragment correlation functions for different multiplicity cuts and different ranges of fragment velocities. Average emission time scales of 1:100-200 fm/c were extracted from the Be-Be, B-B, and CC correlation functions and inclusive, mixed-fragment correlation functions. These mean emission times are shorter than those extracted previously [Tro87,Bou89] for reactions induced by lighter projectiles. Only modest dependences were observed for various cuts on charged-particle or IMF multiplicity. These dependences were qualitatively consistent with slightly larger space-time dimensions for central than for peripheral collisions. In order to suppress contributions from the decay of projectile-like residues, we constructed two-fragment correlation functions for central collisions (gated by N C 212). We explored various cuts on P / A, the total momentum per nucleon of 162 the emitted fragment pair, which select different ranges of fragment velocities and, hence, different relative contributions from equilibrium and nonequilibrium emission. The observed dependence on P/ A indicated significant differences in emission time scales. In order to extract fragment emission times, we assumed, for simplicity, emission from the surface of a spherical source of 12 fm radius. For the most energetic fragments, the correlation functions were consistent with mean emission times of 13<100 fm/c, possibly as short as 1250 fm/c. For fragments emitted with kinetic energies at or below the exit channel Coulomb barrier, the correlation functions indicate much longer emission time scales, 13300 fm/c, possibly as large as 1:500 fm/c. For comparison, average emission times predicted for the decay of equilibrated fusion residues are of the order of 300 fm/ c; dynamical models of fragment production predict emission time scales on the order of 50 fm/ c. The shape of the fragment energy spectra and the emission time scales extracted from two-fragment correlation functions indicates that fragment emission in central collisions begins at the very early stages of the reaction and continues throughout the later equilibrated stages. 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