”F3935" Bi. 1' . . .9 33%? :z. :n ‘5.» 25.1. .491. .0 . .52.. fl.» zigzag). .2. 3.3.1.. .fyuii. : . 3. .033 .2: ‘11-‘30 .. viii.- .Ia :- .I;A MICHIGAN STATE UNIVERSITY LIBRARIES IHIIIIIIIHHIIIHIlllllHlHllllllllllllllllllllllllllllllll 7 C ”CO 3 1293 02048 4030 LIBRARY Michigan State University PLACE IN RETURN BOX to remov TO AVOID FlNES return MAY BE RECAU£D with ea e this checkout from your record. on or before date due. rlier due date if requeSted. DATE DUE DATE DUE DATE DUE woo chlRCDaWpGS—p 14 TRANSK TRANSITIONS OF COLLECTIVE FLow OBSERVABLES AT INTERMEDIATE ENERGIES By Daniel Joseph Magestro 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 2000 TRAxsm The cornpari suggested variou clear mean field tions in eolleetirr are mostly imlé‘p! torheor}: Tlrree' occur at NSC L or disappearance of The disappea Measured for array. Previous c but a recent qua cannot be measu Batzmann~l'elrli the Coulomb im e not Slippregs the (r .. ‘Epefldpnce of tl ABSTRACT TRANSITIONS OF COLLECTIVE FLow OBSERVABLES AT INTERMEDIATE ENERGIES By Daniel Joseph Magestro The comparison of experimental collective flow studies to model calculations has suggested various qualities of nuclear matter, such as the momentum dependent nu- clear mean field and the reduced in-medium nucleon-nucleon cross section. Transi- tions in collective flow observables are particularly beneficial, since these transitions are mostly independent of experimental biases and therefore better suited for relating to theory. Three known transitions in collective flow observables exist, two of which occur at NSCL energies: the disappearance of directed flow (balance energy) and the disappearance of elliptic flow (transition energy). The disappearance of directed flow in intermediate energy heavy-ion collisions is measured for the 197Au+197Au system using the Michigan State University 47r Array. Previous experiments have extrapolated values of the Au+Au balance energy, but a recent quantum molecular dynamics study suggested that the balance energy cannot be measured directly for Au+Au due to a prohibitively large Coulomb force. Boltzmann-Uehling-Uhlenbeck (BUU) model calculations are performed to show that ’ the Coulomb interaction lowers the experimental balance energy significantly but does not suppress the observable completely. Also, BUU calculations show that the dual dependence of the balance energy on nuclear compressibility and in—medium cross section seen for light- and medium-sized systems reduces to a single dependence on the nuclear compressibility, allowing for the first time a characterization of the nuclear equation of state using the balance energy. The (lhappt and 197.lu+197.l eter dependent. independent of . pith increasing balance energy. * culations of ”t ‘ may for the l and Ar+Sc ellip equation of state The disappearance of elliptic flow is measured for 40Ar+45Sc, 58Ni+58N i, 86Kr+93N b and 197Au+197Au. The transition energy for Au+Au exhibits a strong impact param- eter dependence, in contrast to the balance energy for the same system which is nearly independent of impact parameter. However, the transition energy appears to decrease with increasing system mass, which is in accordance with the mass dependence of the balance energy. Comparison of experimental 40Ar+45Sc data to published model cal- culations of 48Ca+48Ca allows one to study the nuclear EOS using the transition energy for the first time. For the two independent cases of Au+Au directed flow and Ar+Sc elliptic flow, comparison of data with BUU calculations indicates a soft equation of state and a reduced in-medium cross section. for those who never doubted my intentions iv It is sorncu. with highs and and s'n' months as the depart In commitments it ellorts, NSC L :‘ drew necessary coursework for secretary. The biggest and Astrononu ll'Orlif‘d in one. and llle Comp enthusiasm ax have sud] filler it'lillOut 1.91110] lWOuld “k a ACKNOWLEDGEMENTS It is somewhat of a cliché to say that one’s time in graduate school was filled with highs and lows, but for me the adage is not an exaggeration. In my four years and six months at MSU, I managed to garner attention through great successes, such as the departmental T.A. award, the Research Recognition Day award, and strong commitments to the REU program, graduate student recruitment, NSCL outreach efforts, NSCL student seminars, and teaching scholarship. But my letdowns also drew necessary attention: three attempts at the old Candidacy Exam, below average coursework for the first year, and the deserved ire of more than one departmental secretary. The biggest consequence of all these events is my interaction with the Physics and Astronomy faculty in more ways than I ever imagined possible. I especially have worked in one way or another with nearly every faculty member at the Cyclotron, and the completion of my Ph.D. degree is as much a credit to their guidance and enthusiasm as it is to my own perseverance. It is partly because of them that I have such affection for the NSCL and for Michigan State University, and I proclaim without remorse that the village certainly raised the boy. I would like to thank Gary Westfall, whose strong commitment as an advisor began at a conference in Berkeley when be skipped some talks to convince me to apply to MSU. He has supported me unwaveringly through the tribulations of graduate school in ways too numerous to remember fully, and he always made me believe I was fully capable of trouncing any challenge (data-related or not) I faced. His most appreciated act, but also just a typical example, was the day I expressed my passion for teaching to him. Within a couple hours, I had e-mails from the associate chair of the department and a meeting with Dr. Benenson to discuss ways I could get involved in teaching oithin our dep to his persiste: Bauer. Bernart hastily-formed German chocoi. I also woult loverlapped or Stone. Stefan (A the years. Some been among the| in acknowledger shifts during pm What makes way of being th other unit'ersitir lust as dynamic 0f Past graduate I Would like lde ~ '35 and enthi both valuable at a. ”“9 0f the tlrir 0d FOl'I‘l ll or 5 S‘Wlu- ate Sluder «l ail 0f the Ihln 0. P5. lhe .re when the dg‘rgn , dlld IOgrxp within our department. Even the difficult decision to move to Germany owes greatly to his persistent encouragement! Special thanks goes to my committee, Wolfgang Bauer, Bernard Pope, Phil Duxbury, and Thomas Glasmacher, for serving on my hastily-formed committee without blinking an eye. I owe you all much more than German chocolate! I also would like to thank the other members of the 47r Group, some of whom I overlapped only briefly: Skip Vander Molen, Omar Bjarki, Robert Pak, Nathan Stone, Stefan Grevy, Mike Miller, and the wonderful REU students we’ve had over the years. Some of my long discussions with them, in particular Omar and Skip, have been among the most memorable times in my graduate career. And as is customary in acknowledgements, I would like to mention that I, too, worked many overnight Shifts during problematic experiments. What makes the NSCL such a unique place to matriculate is its unpresumptuous way of being the best without intimidating its students and visitors. My friends at other universities only wish they were so lucky. The support staff at the NSCL was just as dynamic and unwavering as it was purported to be in the acknowledgements of past graduate students, and many friendships were formed in the process. I would like to thank Walt Benenson and Ed Kashy for not only sharing their ideas and enthusiasm for teaching but for cultivating my own interest. They were both valuable advisors to a young apprentice full of energy, and I hope I can emulate some of the things I learned from them. . Ted Forringer literally has been by my side as a friend, roommate, and fellow graduate student since Day 1 at MSU. It is astonishing to sit back and think about all of the things we have been through together and all of the times one of us was there when the other one was in need. He and I have grown so much, together and apart and together, that I cannot imagine how I would have gotten to the point of vi typing my act graduation. bi graduate studc . very appreciate _ banter. I only E I would like and my extensi‘ me feel like the climbing. I tha piano lessons. I music career. I creativity has In my parents for t' Finally. I my! understated her eyerything slre II that. more than doing Something of my hectic sch: test of our lives. I . .or IE’SIlng with l typing my acknowledgements without him. I only hope that it doesn’t end with my graduation, because I would never find someone to replace him. Also, my fellow graduate students at MSU, in particular my office mates Erik, Jenny, and Dan, are very appreciated. Through no fault of their own they were subjected to my ongoing banter. I only hope I was at least half as good of a listener as they have been. I would like to thank my family: my parents, my sister Shelline, my grandparents, and my extensive set of relatives, all of whom lofted me above their heads and made me feel like the center of attention, providing some of the motivation to continue climbing. I thank my parents especially, for supporting and paying for 12 years of piano lessons. I used to joke that I was studying physics only to finance my budding music career. Instead, what I now get from music is the feeling that my artistic creativity has not been compromised by my choice of profession. I am indebted to my parents for that, for helping me to become the well-rounded person I am. Finally, I owe so much to Katrina Shields that by listing her here I have already understated her role. Perhaps a special section is in order! Instead of trying to list everything she has done to help me complete my graduate career, I will simply say that, more than any other non-physics person I’ve known, she makes me feel like I’m doing something extraordinary and valuable. Her unbelievable patience and tolerance of my hectic schedule and my inexplicable mood swings will be recompensed for the rest of our lives. And to my cat, Ally McBeal, for watching me write this beast and for resting with me on the in-between days. vii a CON'IL llST 0F TA] LIST OF F IC 1 Introductic ll l<00l031 l? 'lhe 3h 1.2.1 1.2.2 L3 Proper 1.! Collect 1-5 Transit 1.5.1 1.5.2 16 Surnm; 2 EXDerimen 21 Befinl; 2-2 The 4; 2.2.1 2'3 El‘l’t‘trc. 2.3.1 24 Rf‘tlufi 2.4.1 2.4.2 2.5 3mm Characteri; . Impafft "attic OIIECI 3.3.1 3.3.2 34 343:3 Umma CONTENTS LIST OF TABLES x LIST OF FIGURES xi 1 Introduction 1 1.1 Nuclear Physics: Introduction and Status ................ 2 1.2 The Nuclear Equation of State ...................... 4 1.2.1 Nuclear Compressibility ..................... 8 1.2.2 Neutron Stars ........................... 10 1.3 Properties of the Nuclear Medium .................... 12 1.4 Collective Flow in Nuclear Collisions .................. 13 1.5 Transitions in Collective Flow at Intermediate Energies ........ 19 1.5.1 Disappearance of Directed Flow ................. 22 1.5.2 Disappearance of Elliptic Flow .................. 23 1.6 Summary and Thesis Structure ..................... 24 2 Experimental Setup and Data Reduction 26 2- 1 Beam and Target Parameters ...................... 27 2- 2 The 47c Array ............................... 27 2.2.1 Detector Subsystems ....................... 29 2- 3 Electronics and Raw Data ........................ 32 2.3.1 Writing Raw Data ........................ 36 2- 4 Reducing Raw Data to Meaningful Quantities ............. 36 2.4.1 Particle Identification ....................... 37 2.4.2 Physics Quantities ........................ 38 2-5 Summary ................................. 40 3 Characterizing the Collisions 41 3-1 Impact Parameter ............................. 42 3-2 Reaction Plane Determination ...................... 45 3-3 Correcting for Experimental Biases ................... 49 3.3.1 Angle Smearing .......................... 50 3.3.2 Repairing the Reaction Plane Distribution ........... 51 3.3.3 Reaction Plane Resolution .................... 56 3.4 Summary ................................. 59 viii 4 Disappea. 4.1 lligs‘tt 4.? hluti‘. 4.2.1 4.2.2 4.2.3 4.3 Expo! 4.3.1 4.3.2 4.3.3 4.3.4 4.4 TheB 4.5 Ellec‘tl 4.6 .5\u+.-\ 4.6.1 4.6.2 4.6.3 4.6.4 4.7 Surnrn Disappear 5'1 Hlsir‘ir 5-2 .\loti\-; 5.2.1 5.2.2 5.3 EXper 5.3.1 5.3.2 5.3.3 5.4 Com}, 5'5 Sumn 6 CODCIUSiO A 47‘ Array 4 Disappearance of Directed Flow in Au+Au Collisions 4.1 History of Directed Transverse Flow ................... 4.2 Motivation for Au+Au Experiment at N SCL .............. 4.2.1 Extension of the Mass Dependence of Ebal ........... 4.2.2 Previous Studies of Directed Flow for Au+Au ......... 4.2.3 Can the Balance Energy Be Measured for Au+Au? ...... 4.3 Experimental Results ........................... 4.3.1 Caveats to Measuring the Balance Energy ........... 4.3.2 Transverse Momentum Analysis ................. 4.3.3 Directed Flow Over a Wide Range of Energies ......... 4.3.4 Summary ............................. 4.4 The BUU Model ............................. 4.5 Effect of Coulomb Interaction on the Balance Energy ......... 4.6 Au+Au as a Probe of the Nuclear EOS ................. 4.6.1 Independence of Eba] from Impact Parameter ......... 4.6.2 Independence of Ebal From am, ................. 4.6.3 Isolation of Nuclear Compressibility ............... 4.6.4 Extrapolation of am, ....................... 4.7 Summary ................................. 5 Disappearance of Elliptic Flow at Intermediate Energies 5.1 History of Elliptic Flow .......................... 5.2 Motivation for Present Study ...................... 5.2.1 Recent Theoretical Findings ................... 5.2.2 Previous Studies of Elliptic Flow for Au+Au .......... 5 .3 Experimental Results ........................... 5.3.1 Fourier Expansion ........................ 5.3.2 The Transition Energy for Au+Au ............... 5.3.3 Mass Dependence of the Transition Energy ........... 5- 4 Comparison to Theoretical Work .................... 5- 5 Summary ................................. 6 Conclusion A 471' Array Parameters LIST OF REFERENCES ix 60 61 66 67 68 71 74 76 77 87 89 89 97 99 100 100 103 105 105 107 108 111 112 113 114 115 117 123 123 129 132 135 140 LIST OF TABLES 2.1 3.1 4.1 4.2 4.3 5.1 A.1 A.2 A.3 A.4 A.5 A.6 A-7 Effective 197Au beam energies for all beam and target combinations used in this thesis. ............................ 28 Values for the correction to the reaction plane dispersion ........ 59 A list of the other Au+Au directed flow experiments. ......... 89 Parameter sets used for the density-dependent mean field U. ..... 92 A comparison of recent values for the compressibility K for a variety of measurement techniques. ....................... 105 Au+Au transition energies extracted from logarithmic fits to the data in Figure 5.8. ............................... 122 Solid angle subtended by the Ball Phoswiches. ............. 135 Solid angle subtended by the HRA Phoswiches. ............ 136 Phoswich Scintillator Specifications .................... 136 Low Energy Thresholds for the HRA ................... 137 Low Energy Thresholds for the Ball Telescopes. ............ 137 Mean angles for the ball phoswiches. .................. 138 Mean angles for the HRA phoswiches ................... 139 LIST 11 13 14 I6 1.8 An 01 “orhl I)llkt>t‘ hqunl glUUIl by a t stars a (a) Fr densu llaxir comp [13} 'Ihe ) llt‘lpa long“ the tl “The 1 VPrse denor dlCati fragrr in‘ple; Sc-hCI TWO OVera If’ring Syhen Elliott" instar the CL Schen ling-ti LIST OF FIGURES 1.1 1.2 1.3 1.4 1-7 1.8 An illustration of the way in which nuclear physics enters into our world at different length scales. ..................... Phase diagram of nuclear matter. The three distinct phases are the liquid phase (green), the hadron gas phase (light blue), and the quark- gluon plasma phase (white). The evolution of the universe is indicated by a downward trajectory along the temperature axis, and neutron stars are thought to be at very low temperatures. ........... (a) Fraction of a neutron star’s total mass which falls below a given density for three different values of the nuclear compressibility [16]. (b) Maximum mass of neutron stars plotted as a function of the neutron compressibility for several different assumptions about the nuclear EOS [15]. .................................... The Nuclear Fireball Model proposed by Westfall et al. [23]. The par- ticipant region formed by the overlap of the colliding nuclei moves at a longitudinal velocity between the two spectator region velocities, and the three “sources” can be treated as three separate pieces ....... The four major types of azimuthal anisotropies, viewed in the trans- verse plane and looking in the direction of the beam. The target is denoted by T and the projectile by P, and blue (green) arrows in- dicate the preferred emission directions of projectile-like (target-like) fragments. (a) Negative directed flow; (b) positive directed flow; (c) in-plane elliptic flow; and (d) out-of—plane elliptic flow. ........ Schematic representation of positive directed flow in the reaction plane. Two nuclei are incident in the center of mass frame collide, and the overall repulsive interaction causes the flow of nucleons to positive scat- tering angles. ............................... Schematic illustration of the collision of two Au nuclei and the resulting elliptic flow of the participant zone. Time shots are shown for an instant before the collision (a), early in the collision (b), and late in the collision (0). taken from Ref. [95] ................... Schematic behavior of directed flow (left) and elliptic flow (right) as a function of the beam energy in the lab frame. ............. xi 15 19 19 1.10 At tlti 3.3 3.4 (alff syrni and and.: (1)) 1 I flu p: the n Collar. energi enerni l (all3a by ear Ilui4 fronr' Array [Bl Sr (b) C Sinipl a fiJrn Dittgr gates 13) Se AE—I Sinip] plots Gt’unj Plane I301nt; Rt‘tlut beam Htost [IIle ; Idle 1 Sperm} iSIri Plan... Of 11m (‘1an 931:1.“ ORpfk the re. 1.9 1.10 2.1 2.2 2.3 2.4 2.5 2.6 2.7 3-1 3-3 3.4 3.5 (a) Elliptic flow excitation function for Au+Au near transition 3. Filled symbols represent experimental data. The dashed curve (open circles) and the solid curve (open squares) represent calculations for a soft and a stiff EOS, respectively. Taken from the E895 Collaboration [27]. (b) The rapidity dependence of the directed (v1) and elliptic (v2) flow for protons. The points below midrapidity have been reflected from the measurements in the forward hemisphere. Taken from the NA49 Collaboration [71]. ............................ At the balance energy, attractive dynamics, dominant at low incident energies, cancel the repulsive interactions which dominate at higher energies. .................................. (a) Basic geometry of the MSU 47r Array. (b) Polar angles subtended by each of the detector subsytems used in this thesis. ......... The 41r Array, located in the N2 vault of NSCL. (a) View of the array from the entrance to the vault. (b) View looking at the High Rate Array face of the detector. ........................ (a) Schematic diagram showing the components of a 47r Array module. (b) Geometry of the High Rate Array. ................. Simple schematic of the method for converting the detector signals into a form which can be stored on magnetic tape. ............. Diagram of the phoswich signal and gates. The width of the two FER-A gates selects the appropriate section of the signal. .......... (a) Sample AE—E spectra from HRA phoswich detector. (b) Sample AE—E spectra from BCC-fast plastic pair. ............... Simplified diagram that shows the entire process from raw data to final plots used in this thesis. ......................... Geometrical description of the impact parameter and the reaction plane. The impact parameter vector 5 lies in the reaction plane and points to the projectile side. ....................... Reduced total transverse kinetic energy (El/Eproj) for three different beam energies of Au+Au. The hatched area corresponds to the 20% most central events, as explained in the text ............... The azimuthal correlation method of reaction plane determination. The line that minimizes the sum of the distances d.- squared corre- sponds to the reaction plane ........................ Distribution of the azimuthal angles between the individual reaction planes for an event and the event average. Removing the flow “particle of interest” leads to N reaction planes for each event. ......... Relationship between the reaction planes determined including and excluding the particle of interest. The tall row at $313301 me, = ¢Rp,p01 Exc. shows the minimal effect that excluding the POI has on the reaction plane determination. .................... xii 23 29 30 33 37 39 43 45 46 48 49 l 36 EIhi‘l the d for 1.: 3.7 pI I“ smear 3.8 I'llt‘ti! 3170“ detec’r 3.9 Ellet't Fourio used 1 3.10 Absolt two su 4.1 Georru (Ereinc' 4.2 First e 4-3 Preyio flow. 4.4 Energy array; "3 Errergj Elitfrgj V1851“ E’Ilt‘rg‘x- 4'8 Transy for q”, 50 .\I(-. 4.9 Si'lrem n9ar.e 4.10 Dire”. 3.6 3.7 3.8 3.9 3.10 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4-9 4-10 4-11 4-12 4.13 4.14 4.15 4.16 Effect of smearing the polar angles of particles over the active area of the detector as opposed to assigning the detector’s geometric center for (a) 0 and (b) 4). ............................ pl. vs. (y/yproj)cm for (a) before angle smearing and (b) after angle smearing for Au+Au at 33.1 MeV/nucleon ................ Uncorrected reaction plane distribution for Au+Au at 33.1 AMeV. The arrows along the top of the plot correspond to the 15 most forward detectors in the Main Ball ......................... Effect of repairing the reaction plane distribution by making use of a Fourier expansion for Au+Au, 35 MeV/nucleon. n = 30 terms were used in the expansion. .......................... Absolute value of the azimuthal angle between the reaction planes for two subevents in Au+Au collisions. ................... Geometrical parameters of the model used by Scheid, Muller, and Greiner to predict transverse expansion. ................ First experimental evidence for the disappearance of transverse flow. . Previous mass dependence of the disappearance of directed transverse flow. .................................... Energy dependence of directed flow from data taken with a plastic wall array at Berkeley .............................. Energy dependence of directed flow from E08 and Plastic Ball data. Energy dependence of directed flow from FOPI data. ......... Vlasov-Uehling—Uhlenbeck calculations by Zhou et al. of the balance energy with and without the Coulomb interaction included ....... Transverse momentum 12,, as a function of the reduced rapidity y / ypmj for quantum molecular dynamics calculations of the Au+Au system at 50 MeV/nucleon by Soff et al. . ..................... Schematic representation of directed flow at incident energies below, near, and above the balance energy ............. , ....... Directed flow near the balance energy, as determined by theoretical and experimental studies. ........................ Schematic of the transverse momentum method for determining di- rected flow in symmetric nuclear collisions ................ The acceptance of the 47r Array for Z =2 fragments in Au+Au collisions at 38.3 MeV/nucleon for: data, unfiltered isotropic simulation, and filtered isotropic simulation. ....................... Average transverse momentum vs. the reduced rapidity in near-central 52 53 54 55 57 62 64 68 69 70 71 72 73 75 77 79 81 Au+Au collisions ((b/bmax) = 0.28). Z =2 is the flow particle of interest. 82 Average transverse momentum vs. the reduced rapidity in near-central Au+Au collisions ((b/bmax) = 0.28). Z =1 is the flow particle of interest. 84 Average transverse momentum vs. the reduced rapidity in near-central Au+Au collisions ((b/bmax) = 0.28). Z =3 is the flow particle of interest. 85 Extracted flow values in near-central ((b/bmax) = 0.28) Au+Au colli- sions for Z = 1, 2, 3 POI .......................... xiii 86 4.17 Extra 115101. 110% hot 418 Exp- as P1 are ('1 419Dum Of 1111' 4.20 811' for ’0. irnpat 4.21 MI (I for 1" duced 4.22 Bl'l'i MMdu 4.33 Balant dmks With at 4QiBalaric S}'Sterrr t0 guicl 4.2.5 BEL. I] reducti ”19018 I 4‘26 BU. b rarneip Ar+s(._ AU+A t: with 9r; First 9x 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 5-1 5.2 5.3 Extracted flow values in near-central ((b/bmax) = 0.28) Au+Au col- lisions for Z = 1,2,3 POI. Flow values below the minimum are re- flected about the x-axis, and dotted (dashed) lines represent linear least-squares fits with (without) the 44.5 MeV/ nucleon value reflected. Experimental flow values for near-central Au+Au collisions with Z = 2 as POI. Solid squares are directly measured values, and open squares are corrected for dispersion in the reaction plane determination. Directed flow normalized by the fragment masses plotted as a function of incident beam energy for several Au+Au experiments. ....... BUU calculations in coordinate space projected onto the reaction plane for 40Ar+"5Sc at an incident energy of 40 MeV/ nucleon and a reduced impact parameter of b/bmax = 0.28. ................... BUU calculations in coordinate space projected onto the reaction plane for 197Au+197Au at an incident energy of 150 MeV/nucleon and a re- duced impact parameter of b/bmam = 0.28. ............... BUU calculations of the average transverse momentum vs. the reduced rapidity in near-central Au+Au collisions (b/bmax = 0.28) ........ Balance energy as a function of combined system mass. Open squares / circles are experimental data, and triangles represent BUU calculations with and without the Coulomb interaction included in the calculation. Balance energy as a function of the reduced impact parameter for four systems. Data are taken with the 47r Array, the linear fits are intended to guide the eye. ............................. BUU model calculations of the mass dependence of Ebal for different reductions of the in-medium cross section an“. Experimental measure- ments of Ebal are shown as solid squares. ................ BUU balance energies plotted as a function of the arm reduction pa- rameter —a for three different values of the compressibility K for Ar+Sc. The experimental measurement is represented by a flat line with error bars. .............................. BUU balance energies plotted as a function of the arm reduction pa- rameter -a for three different values of the compressibility K for Au+Au. The experimental measurement is represented by a flat line with error bars. .............................. First experimental evidence for the transition from in-plane to out-of- plane emission in intermediate-energy collisions. Data are taken with the Nautilus detector at GANIL [92]. .................. Impact parameter dependence of the anisotropy ratio, in which positive RA values correspond to predominantly out-of—plane emission. Data are taken with the Miniball/ALADIN detector [93]. ............ Isospin-dependent BUU calculations of the excitation function of (a) the proton directed flow and (b) the proton elliptic flow in 48Ca-l-48Ca collisions at an impact parameter of 2 fm.Taken from Ref. [79]. . . . . xiv 87 88 90 94 95 96 98 101 102 103 104 109 110 112 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 Elliptic flow in Au+Au collisions from 90 MeV/nucleon to 1.49 GeV/ nucleon as measured by the FOPI Collaboration at GSI-Darmstadt for (a) a high-pt cut and (b) all pt included. ................ Schematic representation of distributions for (a) predominantly in- plane and (b) predominantly out-of-plane emission. .......... Azimuthal distribution of particles in near-central Au+Au collisions emitted near midrapidity ([(y/yproj)cml _<_ 0.5). Blue lines indicate the experimental spectra, and green curves represent Fourier fits of the form of Eq. 5.1 ............................... Azimuthal distribution of particles in semi-central Au+Au collisions emitted near midrapidity (|(y/ypmj)cm| _<_ 0.5). Blue lines indicate the experimental spectra, and green curves represent Fourier fits of the form of Eq. 5.1 ............................... Elliptic flow plotted as a function of incident beam energy for Au+Au collisions. Data are for four different impact parameter bins, and solid and dashed lines correspond to logarithmic fits to the data ....... Azimuthal distribution of particles in semi-central 40Ara-"58c collisions near midrapidity (|(y/yproj)cm[ g 0.5). Dashed curves represent Fourier fits of the form of Eq. 5.1. ........................ Azimuthal distribution of particles in semi-central 58Ni+58Ni collisions near midrapidity (|(y/ypmj)cm| S 0.5). Dashed curves represent Fourier fits of the form of Eq. 5.1. ........................ Azimuthal distribution of particles in semi-central 86Kr+93N b collisions near midrapidity (|(y/ypmj)cm| S 0.5). Dashed curves represent Fourier fits of the form of Eq. 5.1. ........................ Elliptic flow for four systems as a function of incident beam energy in near-central collisions (b/bmax < 0.28) ................... Elliptic flow for four systems as a function of incident beam energy in semi-central collisions (0.28 < b/bmam < 0.39). ............. Comparison of isospin-dependent Boltzmann-Uehling—Uhlenbeck (IBUU) calculations of 48Ca+48Ca from Ref. [79] with 40Ar+"5Sc experimental data taken with the 47r Array. ...................... XV 114 118 119 121 124 125 126 127 130 Chapt Introc This 1119815, 0 918195.” can 1 0! Equation c modi'namieg Other dl'narr 1119313 by Cm Charged Ila: calculations Darticuyarly Chapter 1 Introduction This thesis, entitled “Transitions of Collective Flow Observables at Intermediate En- ergies,” can be classified as a study of the nuclear equation of state. The connotation of equation of state when used in nuclear physics is the same as when used in ther- modynamics, I'.e., the relationships between pressure, temperature, density, and any other dynamic variables. The nuclear equation of state (EOS) is studied in this thesis by comparing experimentally measured variables obtained from detecting the charged fragments of nuclear collisions with the predictions of theoretical models and calculations. Specifically, transitions in these measured variables, or observables, are particularly useful because their comparison to theory is less-hindered by unavoidable experimental biases and irrelevant qualities of the theory itself. The two observables used in this thesis, directed flow and elliptic flow, are both types of “collective flow,” ordered motion characterized by the strong correlation between particle positions and momenta. Directed and elliptic flow both relate to particles emitted from the centers of nuclear collisions, where compressed nuclear matter leads to higher temperatures and densities. Therefore, these observables are well-suited for studying the EOS. The data set used primarily in this thesis is 197Au+197Au at incident energies between 25 and 59 MeV/nucleon. Au+Au is the system of choice because of large system size (423941 studying the i In this Ch; status of nuclv the nuclear Et presented. alo: define and (list I remaining chat 1.1 Nuc The science of r which makes up world we see ar of nuclei and tin structure of nor that fuel the sta The field of nucl did not discover “'35 discovered a At its core. search for Under Figure 1.1. prod the Board On ph “Wins Kr “‘a“S . . . that R.“ . ..tablished lentr l _ I 5 . :l» who 11] . altos ream size (A2394) approximates infinite nuclear matter. Other data sets are included for studying the influence that system mass has on these observables. In this chapter we will introduce the nuclear EOS in the context of the present status of nuclear physics. In particular the nuclear compressibility, a parameter of the nuclear EOS, is discussed. Then, some important traits of nuclear matter will be presented, along with the influence these traits have on EOS studies. Finally, we will define and discuss the transitions in directed and elliptic flow that are studied in the remaining chapters. 1.1 Nuclear Physics: Introduction and Status The science of nuclear physics concerns itself with the properties of “nuclear matter,” which makes up the massive centers of the atoms that account for 99.95 percent of the world we see around us. The main properties under investigation are the structure of nuclei and the interaction between nucleons (protons and neutrons), as well as the structure of nucleons themselves. Nuclear physics also explores the nuclear reactions that fuel the stars, including our Sun, which provides the energy for all life on Earth. The field of nuclear physics has existed for less than one hundred years, as Rutherford did not discover the atomic nucleus until 1911, the same year that superconductivity was discovered and atomic crystal structures were first observed. At its core, research in nuclear physics is a quest for knowledge as well as a search for understanding, and its reach extends from the very small to the very large. Figure 1.1, produced in 1999 by the Committee on Nuclear Physics, established by the Board on Physics and Astronomy of the National Research Council [1], shows the various ways that nuclear physics enters into scientific research over the entire range of established lengths. Nuclear physics not only advances the frontiers of knowledge but also makes remarkable contributions to the needs of society, such as nuclear energy MONO bone 599 um 1°”), Hod coll p? »3 10 cm lube Figure 1.1: An illustration of the way in which nuclear physics enters into our world at different length scales. Taken from Ref. [1]. ' and nuclear medicine. The current realm of nuclear physics is quite extensive, and some recent develop- ments provide strong momentum as the field moves into the next century. In 1999, researchers at Berkeley’s 88” Cyclotron discovered element 293118 [2] by increasing the mass of the projectile (that is fused with 208Pb) from A = 70 to A = 86, far beyond what had been deemed viable and opening the window to even heavier nu- clei. The result has yet to be reproduced at GSI heavy-ion research laboratory in Germany. Also in 1999, the Relativistic Heavy Ion Collider at Brookhaven performed its commissioning run and soon will be delivering beams to the four experimental devices hoping to detect and study the quark-gluon plasma [3]. Of local interest, the Committee on Nuclear Physics has endorsed an ISOL (Isotope Separation On- Line) facility, which pending financial support will be built in this decade, possibly at Michigan State University. And numerous other recent advances in nuclear structure, nuclear astrophysics, and nuclear reactions paint a positive picture for the advances to be made in the coming years. The 1999 Committee on Nuclear Physics identified four major priorities for the next decade, presented here as a matter of general interest: 1. Study of the quark-gluon structure of matter as well as the structure of hadronic matter at increasing energy densities; 2. Construction of a high-intensity accelerator facility for producing radioactive beams to study exotic nuclear matter; 3. Investment in instrumentation for research, including future upgrades of detec- tor systems at both large and small laboratories; and 4. Evaluate research priorities through the continued formation of National Science Advisory Committees (NSAC) at regular six-year intervals. 1.2 The Nuclear Equation of State When two 197Au nuclei collide with enough energy to overcome the Coulomb repulsion between them, a system of 394 nucleons is formed for a very short time (t ~ 10’22 s), occupying a volume of 10‘36 cm3. Despite its miniscule dimensions, the tiny laboratory of the Au+Au collision is nearly the largest that can be created on Earth. However, nature also provides nuclei of virtually infinite size in the form of neutron stars, which are typically 1016 cm3 and contain ~1055 nucleons. While observational astronomy can aid in understanding the behavior of a neutron star as a whole, such as its rotatitt' from finite ni example. kno nucleon intern on measurern. Nuclear r a realistic two- absence of rep" liquid having I 2.0 X 10” g/(‘] Protons and tie However. a hen matter. even tl can be related the nuclear equ :ln e(Illatim t-leyl. temperate” €121 mple, are d, Where lt - ’ IS a . . CO. as its rotational speed or its electromagnetic spectrum, much of what is learned from finite nuclei is crucial in explaining the properties of such infinite matter. For example, knowledge of the nature of the nucleon-nucleon force, the role of many- nucleon interactions, collective excitations, and the nuclear equation of state all rely on measurements made at terrestrial accelerator laboratories. Nuclear matter is a uniform medium in which pointlike nucleons interact via a realistic two-body interaction; the electromagnetic effects are not included. In the absence of repulsive Coulomb forces, the ground state of nuclear matter is a uniform liquid having equal numbers of protons and neutrons [4] with a density of about 2.5 x 1014 g/cm3 (0.16 nucleons per fm3). In reality, a liquid of equally abundant protons and neutrons cannot exist because of its prohibitively large Coulomb energy. However, a heavy nucleus such as 197Au can be seen as a stable, small drop of nuclear matter, even though its density is not uniform. Furthermore, some of its properties can be related to those of infinite nuclear matter, the most encompassing of which is the nuclear equation of state. An equation of state is a relationship between the pressure (or energy per parti- cle), temperature, and density of the matter under consideration. Atomic gases, for example, are described very well by the ideal gas law, which can be put in the form PU") = —p(T)T(T) , (M) where k is a constant, m is the mean molecular weight, P(r) is the pressure, T (r) is temperature, and p(r) is the density of a sphere of radius 1'. While the ideal gas law is convenient and simple, candidate expressions for the nuclear equation of state (EOS) are more lengthy and often times require large approximations or extremely small amounts of nuclear matter in order to be used. Two typical examples of nuclear EOS. includt. which is dett‘! and which lflClUdt' expressions fozl complex. The. this thesis. th. theories. :1 schemat it areas of a two-(l of the medium. with tern perat u The graph shoy exists below ab expected. tlltfot' densities lower 1 the black horde to be fully char.- EOS, included only for demonstration, are P: :01 [ TJ3/2(n,)/J1/2(77,) + V0 + 21521/1 + V2] which is determined from the two-body interaction using ground state properties [5], and _gs—l 2 3me 0° ~ n n=l which includes the Skyrme interaction and Hartree-Frock theory [6]. While both expressions for the pressure contain T and p, the functional dependence of P is quite complex. Though a theoretical analysis of the various EOS is beyond the scope of this thesis, the reader is referred to Refs. [7, 8] for discussions of the status of EOS theories. A schematic way of representing an equation of state is a phase diagram, in which areas of a two—dimensional plot of EOS variables correspond to different physical states of the medium. Figure 1.2 is a highly schematic graph of the phases of nuclear matter, with temperature plotted vs. baryon density (normalized by ground-state density, p0). The graph shows three distinct phases of nuclear matter. The liquid phase, which exists below about T = 20 MeV, is the phase for normal nuclei. Nuclear matter is expected, theoretically, to undergo a transition from a liquid to a gaseous phase at densities lower than pm. The first-order liquid-gas phase transition, represented by the black border around the liquid phase, has been studied extensively but has yet to be fully characterized. At sufficiently high temperature and density, a transition from normal hadronic matter to a quark-gluon plasma phase is expected due to an increase in the degrees of freedom among the constituents. The nature of the hadron- quark/gluon phase transition (i. 6., whether it is first-order or second-order) is not known and will be one of the interesting subjects studied at RHIC [9]. Also drawn in Figure 1.2 is the evolution of the universe as a downward trajectory alongside the vertical axis of the phase diagram. From the time of the Big Bang, 6 Early universe Quark-Gluon plasma ‘v L V u an ' ‘s A‘- I Neutron star Temperature (MeV) P/Po Figure 1.2: Phase diagram of nuclear matter. The three distinct phases are the liquid phase (green), the hadron gas phase (light blue), and the quark-gluon plasma phase (white). The evolution of the universe is indicated by a downward trajectory along the temperature axis, and neutron stars are thought to be at very low temperatures. the early universe cooled as it expanded. For the first microseconds, the temperature was at least hundreds of MeV and matter existed as a quark-gluon plasma. The matter of the early universe was dominated by photons, and as the universe cooled, the quark-gluon plasma merged into hadrons (such as nucleons). If this transition was first-order, than quarks came together as hadrons in the same way that water forms rain drops. The form of the nuclear EOS at these phase boundaries is especially interesting to experimentalists and theorists alike. One way to probe the EOS is by modeling the interactions between nucleons in a nuclear collision with a theory that treats the inter-nucleon potential as two separate pieces: a mean field, which represents the long-range, collective influence of all the nucleons on a single nucleon; and nucleon—nucleon scattering, which incorporates the hard collisions among the nucleons. The Boltzmann-Uchling-Uhlenbeck (BUU) model [10], used substantially in this thesis, implements an inter-nucleon potential with mean-field and collision terms; see Sec. 4.4 for a thorough description of the BUU model. 1.2.1 Nuclear Compressibility In order for a theory of the nuclear EOS to be valid, it must reproduce the empirical properties of nuclear matter. These properties include: 1. saturation density, which is the maximum density of nucleons in finite nuclei. The commonly accepted value is p0 = 0.16 :l: 0.02 nucleons/fm3. 2. binding energy per nucleon, which can be determined by using a nuclear mass formula that includes volume and surface terms. From studies made on finite nuclei, E3 = 16 :l: 1 MeV. 3. mean free path, which is related to the complex part of an optical model potential and signifies the distance a nucleon can travel before undergoing a collision. The knowledge of the mean free path A for infinite nuclear matter is poor because, in a nucleus, /\ is mostly determined by surface effects. An empirical property that is often used to characterize the equation of state of nuclear matter in its ground state is the nuclear compressibility K, which is a measure of the stiffness of nuclear matter against changes in density. Nuclear com- pressibility is analogous to the inverse of the bulk modulus in solids that characterizes variations of the volume of a material as a function of the applied pressure [11]. In terms of equation of state variables, K =9[1 (52%)] (1.2) dp dp P=Po where K is evaluated at the saturation density, p0. It is also conventional to express K in terms of the Fermi momentum, which is the maximum value of a nucleon’s momentum in a free Fermi gas. The Fermi momentum1 kp can be expressed in terms Of p02 37r2 ”3 and the corresponding expression for the compressibility is [4]: _ 2 dEB K is the slope in the variation of binding energy per nucleon as a function of kp. The value of K is of great practical interest because it significantly influences the extrapolation of the nuclear EOS from p = p0 to larger p [12]. It is worthwhile to stress that K corresponds to the compressibility only when p z p0. Several techniques for determining K have been employed. The first and most common method is to excite the isoscalar giant monopole resonance in nuclei [13], since the resonant frequency is directly related to the compressibility. However, there are inherent difficulties in using a nucleus’ breathing mode, such as the influence of surface effects on the resonant frequency. Another method for finding K is to use mass formulas that involve thermodynamic variables [14]. If certain assumptions are made about the density and pressure, then measurements of the nuclear size and mass can lead to good estimates of K. This is also somewhat problematic, because the value of K is very dependent on the structure of the mass formula. In this thesis, K is the EOS parameter of interest. The Boltzmann-Uehling- Uhlenbeck (BUU) model, presented in Sec. 4.4, determines values for K based on the model’s implementation of the nuclear mean field; see Equations 4.4 and 4.5. By using the BUU model to generate quantities for comparison to experimental data, values for K can be inferred. 1Here, the wave number kp is called the “Fermi momentum” in keeping with convention, despite its units of inverse length. The value for kp at p0 corresponds to pp = 2.91 x 10’15 MeV/c (from p = ’11:). 1.2.2 Neutron Stars As mentioned at the beginning of this Section, much of what is learned from EOS studies of finite nuclei is crucial in explaining the properties of astrophysical objects. These properties include the maximum mass and radius of a neutron star and the densities needed for supernova explosions to occur. The EOS is also important in the process of big-bang nucleosynthesis. As an instructive example, the relationship between the EOS and structural properties of neutron stars are discussed in this Section. A neutron star is the cosmological equivalent of a giant nucleus. Neutron star masses are roughly equal to a solar mass (MG), packed into a sphere 10-20 km in diameter. A neutron star is formed when atoms are literally crushed out of existence by intense gravitational forces which cause protons and electrons to combine into degenerate, unbound neutrons. The mass and radius of the star is mostly limited by the counterplay between gravity and the nuclear compressibility (K), and values of the maximum mass and radius obtained from theory studies have been shown to depend directly on the value of K [15]. Figure 1.3(a) shows how the density profile of a neutron star of mass Mmax depends on K, as calculated from simple assumptions about the binding energy of neutron- rich matter [16]. The curves represent different values of the nuclear compressibility, and the area below the curve is the fraction of the star’s total mass which falls below a given density. As K decreases, larger amounts of the mass are seen to be at higher density. This is an expected trend, since a very high compressibility would lead to a rigid and nearly constant density profile. Calculations of Mmax vary greatly depending on the parameters of the nuclear EOS. Figure 1.3(b) shows Mmax as a function of the neutron compressibility2 for 2The neutron compressibility is related monotonically to K, the compressibility of symmetric nuclear matter, by means of the neutron-to-proton ratio. As expected, for ratios near 1, K N z K. 10 0.8 — Ko 120 MeV '7‘. g 0.6 *- - 2 x >. P i E 93 2 =5 — - — 0.2 - (a) l l L l l I 1 l l l 1* o 2 4 s a 10 12 n(R)/O.16 Figure 1.3: (a) Fraction of a neutron star’s total mass which falls below a given density for three different values of the nuclear compressibility [16]. (b) Maximum mass of neutron stars plotted as a function of the neutron compressibility for several different assumptions about the nuclear EOS [15]. several different EOS [15]. For compressibilities around 200 MeV, me is seen to vary by more than a factor of 2, depending on the EOS. Of course, some EOS can be ruled out by observational astronomy, since known neutron stars have been observed to be as large as 1.5 M0. While a detailed explanation of the role that the nuclear EOS plays in neutron stars is beyond the scope of this thesis, it is clear that constraints can be placed on their characteristics by studying the nuclear compressibility. It is also worth noting that observational astronomy might be useful for constraining the nuclear EOS itself, although the properties of neutron stars, such as density profile, neutron-to—proton ratio, and even the recent suggestion of a quark-gluon plasma core (so-called strange stars, see Ref. [17]) make extracting useful information directly from neutron stars difficult. 11 1.3 Properties of the Nuclear Medium The process of extracting information about the nuclear equation of state by compar- ing theoretical calculations of nuclear collisions to experimental data is obscured by secondary features of nuclear matter. In particular, the behavior of particles in the nuclear medium (i.e., the “material” composed of nuclear matter) differs from their behavior in free space, leading to complications in the theory-data comparison. The differences between the nuclear medium and free space include: 1. the effective pion mass in nuclear matter, which increases slightly as the nuclear density p/po increases; 2. the peak energy and width of nucleon resonances in nuclei, which are affected slightly by the Fermi motion of the in-medium nucleons; 3. the cross sections of pion-nucleon, kaon-nucleon, etc. interactions in-medium show enhancements when compared to free space. A quality of nuclear matter which plays a role at intermediate energies is the reduction in the in-medium nucleon-nucleon cross section, on“. The main reason for a reduction in am, compared to free space is the Pauli Exclusion Principle, which disallows certain low-momentum processes from occurring [19]. The result is fewer nucleon-nucleon collisions in areas of high nucleon density (such as the interiors of heavy nuclei). The effect of a reduced cross section is substantial when trying to extract EOS information from nuclear reactions, since nucleon-nucleon collisions are a dominant part of the reaction. In intermediate-energy heavy ion collisions, most of the nucleon collisions scatter nucleons at the surface of the initial Fermi spheres. Therefore, the nucleons are often treated as hard spheres in theoretical models. The functional form 12 - i of the density-dependent reduction in am, is commonly written 0m = 0m (1+ afl) (15) P0 [18], where Ufree is the nucleon cross section in the vacuum, and a is a parameter between 0 and —1. Eq. 1.5 is not intended to be capable of representing the in- medium cross section at all densities; for example, it is clear from inspection that a density of p = 4pc limits a to values above -0.25. However, the representation has been used with success by numerical models such as BUU (presented in Sec. 4.4) in accounting for the reduced cross-section at intermediate energies and near p0. 1.4 Collective Flow in Nuclear Collisions A primary motivation for experimental studies of nuclear collisions is the hope that measured quantities can be compared to and modeled by theory unambiguously and that this comparison leads to better understanding of the empirical properties of nuclei or nucleons themselves as well as astrophysical (stars) and cosmological (Big Bang) entities. As discussed in Sec. 1.2, the empirical property of interest in this thesis is the compressibility of nuclear matter at normal nuclear density p0. The value for the compressibility has direct consequences on the mass limit of neutron stars as well as the stability and lifetimes of supernovae. When two nuclei collide and are compressed to densities higher than p0, a flow pattern will develop as the system subsequently expands. During the decompres- sion stage, the directions and speeds of the constituent particles are influenced by pressure gradients, i. e., particles tend to flow to regions of lower pressure. This pressure-dependent correlation between particle positions and momenta is known as collective flow. First proposed in the 1950’s [20] and experimentally discovered in the mid-1980’s [21], collective flow has been studied at beam energies ranging from 13 tens of MeV/ nucleon to hundreds of GeV/ nucleon for various emitted particle types, including nucleons, light charged particles, pions, kaons, and lambdas. In classical physics flow is described in the language of hydrodynamics, where the simple conservation laws (momentum, mass, energy) are linked to fundamental properties of the fluid, such as the equation of state and viscosity. In nuclear physics, the approach is similar: the final states of particles are linked to dynamical prop- erties of the collision, such as the in-medium nucleon cross section and the nuclear compressibility. In addition to contributions from random thermal motion and the initial kinematics of the collision, the contribution of collective flow to the final states is modeled by so-called transport models which propagate particles through space and time as they undergo interactions. The integrity of the model depends on its treatment of collective flow. Primitively speaking, collective flow is a measure of the transverse motion im- parted to particles and fragments during the collision of two nuclei. The transverse nature of collective flow is its most alluring quality to experimentalists, because it enables the measurement of flow for particles in different domains of the collision. This is best exemplified in the geometry of the Nuclear Fireball Model, proposed by Westfall et al. in the mid-1970’s [23] and illustrated in Figure 1.4. In the Fireball Model, nuclear collisions with non-zero impact parameter consist of three domains: the participant piece from the overlap region of the nuclei, and two spectator pieces from the projectile and target portions which do not overlap. Because the longitu- dinal velocities of the three domains are different (see Figure 1.4), particles from the domain of interest can be selected by their longitudinal velocity components without interfering in the measurement of their collective transverse motion. In this thesis, collective flow of the participant region, where the compression is expected to be the largest, is the quantity being studied. 14 Figure 1.4: The Nuclear Fireball Model proposed by Westfall et al. [23]. The par- ticipant region formed by the overlap of the colliding nuclei moves at a longitudinal velocity between the two spectator region velocities, and the three “sources” can be treated as three separate pieces. Generally, there are three types of collective flow which are discussed in nuclear collisions, termed radial flow, directed flow, and elliptic flow. Radial flow, or ra- dial expansion, which is not studied in this thesis, arises in central collisions and is characterized by increased yields in the kinetic energy spectra of particles emitted near 0cm, = 90° relative to the beam axis. The existence of radial flow indicates that kinetic energy spectra differ from the expected spectra of a compressed thermal fluid expanding outwards. In intermediate-energy collisions, radial flow has been shown to account for as much as half of the emitted particles’ energies [24], which is incom- patible with the assumption of a purely thermal distribution. The existence of radial flow has implications in the temperature gradient of hot nuclear matter as well as the effects of relativistic hydrodynamics; for a recent summary of work to date, please see Reisdorf and Ritter, Ref. [25]. 15 /\ Figure 1.5: The four major types of azimuthal anisotropies, viewed in the transverse plane and looking in the direction of the beam. The target is denoted by T and the projectile by P, and blue (green) arrows indicate the preferred emission directions of projectile—like (target-like) fragments. (a) Negative directed flow; (b) positive directed flow; (c) in-plane elliptic flow; and (d) out-of-plane elliptic flow. Directed Flow The remaining two types of collective flow, both of which are studied in this thesis, are illustrated schematically in Figure 1.5. Directed flow, also called “in-plane” or “sidewards” flow in the literature, refers to the preferential emission of particles within, and to a particular side of, the reaction plane. (The reaction plane is defined as the plane which contains the beam axis and a line joining the centers of the two nuclei.) Stated another way, directed flow is the deflection of nucleons and fragments to finite scattering angles as a result of nontrivial interactions during the collision. The transverse components of the particles’ momenta projected into the reaction plane are used to quantify directed flow. Figure 1.6 demonstrates the preferential emission of particles in the reaction plane, characteristic of directed flow. In Figure 1.6(a), two heavy nuclei in the center of mass frame are incident upon each other with non-zero impact parameter (i.e., a noncentral 16 projectile . . K. figure 1.6. ._ Two llll('l(‘l ax interartinu m mllisiun). and pirija'tile [ta] in figure 1.6:, which are pe of identical l1 in the center distribution. As memi i5- Pxpected . “gig“ are x i5 3 Lorem. of diff-Kiel) 3“ harm-1,.i l . . Where F l\ measure (l projectile O. [ g Q/ <——— o .. -" target Az/W Figure 1.6: Schematic representation of positive directed flow in the reaction plane. Two nuclei are incident in the center of mass frame collide, and the overall repulsive interaction causes the flow of nucleons to positive scattering angles. collision), and the reaction plane is taken as the plane of the paper. After the collision, projectile (target) nucleons are directed mostly in the +r (—r) direction, illustrated in Figure 1.6(b). In the transverse plane this corresponds to azimuthal (I) distributions which are peaked at 0° (180°) for projectile-like (target-like) remnants. In collisions of identical nuclei such as Au+Au, the (1) distribution for particles at forward angles in the center of mass frame must be totally anti-symmetric to the backward—angle (1) distribution. As mentioned above, directed flow of the participant region, where the compression is expected to be the largest, is the quantity of interest. Particles from the participant region are selected by making analysis cuts on the center of mass rapidity, ycm, which is a Lorentz scalar and a measure of a particle’s longitudinal motion. The strength of directed flow usually is measured by plotting the average transverse momentum of all particles projected on the reaction plane ([pi| cos (1)) as a function of rapidity and by taking the slope of this function at gm, 2 0: (1([pil cos ) F: dy (1.6) grin-:0 where F is the strength of the directed flow. A more thorough explanation of how to measure directed flow is presented in Sec. 4.3.2. \l’ wa dis ref (Cl; ilf Elliptic Flow Whereas directed flow is anti-symmetric with respect to the distributions for for- ward rapidity (ycm > 0) and backward rapidity (ycm < 0), elliptic flow has the same distribution in both rapidity regions, at least for symmetric systems. Elliptic flow refers to the anisotropy of the (1) distribution at midrapidity and its value indicates whether or not particle emission is in-plane or out-of-plane. Azimuthal distributions which are peaked at 0° and 180° exhibit predominantly in-plane emission, while (I) distributions peaked at i90° signify out-of—plane emission. The term elliptic flow has replaced old naming conventions such as “squeeze-out,” ”rotational motion,” or “anisotropic flow” because the shapes of d) distributions at midrapidity resemble el- lipses with a major axis along the x-axis (in-plane emission) or y-axis (out-of—plane emission). Out-of—plane elliptic flow, which can occur in non-central collisions, is an inter- esting phenomenon which has been observed at incident energies ranging from 100 MeV/nucleon to ~4 GeV/nucleon. Out-of-plane emission at midrapidity depends primarily on two factors: the pressure built up in the compression stage compared to the energy density, and the passage time for removal of the pro jectile- and target-like spectators of the Fireball Model. If the participant region emits particles at an early stage of the collision, the spectator pieces may still be close enough to cause pressure gradients in the out-of-plane direction. Figure 1.7 illustrates the formation of the compression region in a noncentral nuclear collision and the subsequent emission of particles from the midrapidity participant region. Since the midrapidity distribution due to elliptic flow is symmetric about both the reaction plane and the plane at (I) = :l:90°, the four-fold symmetry resembles a cos(2) function. In fact, directed and elliptic flow can be treated as the first and 18 “an ' ’ ' V (c) Figure 1.7: Schematic illustration of the collision of two Au nuclei and the resulting elliptic flow of the participant zone. Time shots are shown for an instant before the collision (a), early in the collision (b), and late in the collision (c). taken from Ref. [95]. second harmonics of a Fourier expansion of (I): F() 2 Co + cl cos() + c2 cos(2) , (1.7) where the coefficients c1 and c2 correspond to the strength of the directed and elliptic flow contributions, respectively. See Section 5.3.1 for a thorough discussion of the use of a Fourier expansion when studying elliptic flow. 1.5 Transitions in Collective Flow at Intermediate Energies Directed and elliptic flow are both robust observables for studying the hot and dense region of nuclear matter formed in the center of heavy ion collisions. Their strength rests in their simplicity; neither observable requires complex calculations or gross approximations. The most detailed part of the analysis is in the reaction plane determination, but, as discussed in Sec. 3.2, the method for determining the reaction plane is well-established. One of the underlying objectives of science is to find experimental observables which can be compared unambiguously to theory. However, some complications arise 19 A é . 3 in-plm ,,,,,, ‘Ek‘ ; g 2 out-of-plane l l l l .; A ff 1 “'0 ‘00 E(AGeV) - f’" ' _‘0 ‘00 E + 27—0 3 ‘ O Attractive Repulsive Diflogpfifnce (low energy) (high energy) (balance energy) Figure 1.10: At the balance energy, attractive dynamics, dominant at low incident energies, cancel the repulsive interactions which dominate at higher energies. Figure 1.6. At incident energies around 10—20 MeV / nucleon, the nuclear mean field dominates the collision dynamics, resulting in mostly negative deflection. As the beam energy is increased, the mean field plays a reduced role, and hard repulsive scattering increases in importance. The disappearance of directed flow (corresponding to transition 1 in Figure 1.8) occurs when attractive and repulsive effects cancel each other. At that incident energy, termed the balance energy, the azimuthal distributions of pro jectile—like (target-like) fragments with respect to the reaction plane do not exhibit peaks at 0° (180°). This is illustrated in Figure 1.10. The balance energy for Au+Au is the subject of Chapter 4, and its utility in constraining the nuclear equation of state is presented therein. 1.5.2 Disappearance of Elliptic Flow Transition 2 in Figure 1.8 corresponds to the change in the direction of preferred fragment emission from the compressed participant region. At energies below the transition energy3, emission is primarily in the reaction plane (i.e. (I) distribution is peaked at 0°, 180°). Since such an in-plane enhancement is also produced in the de—excitation emission of a nucleus with a large angular momentum, this effect was 3Unlike “balance energy,” which is a generally accepted term for the disappearance of directed flow, “transition energy” is somewhat new terminology. It was first used in 1997 by the F OPI group [66], but “onset of squeeze-out” and other conventions are still used. 23 called a “rotation-like effect” or “rotational flow” in early papers [29]. In higher- energy collisions, a maximum in the direction perpendicular to the reaction plane on both sides (formerly called squeeze-out) appears because the compressed matter in the interaction region can escape preferentially in directions unhindered by the presence of the projectile and target spectators. The transition energy for Au+Au is one of the topics of Chapter 5. Also, the mass dependence of the transition energy is measured, and the transition energy for a light system is compared to published theoretical calculations. 1.6 Summary and Thesis Structure The main objective of this thesis is to compare experimental measurements of tran- sitions in collective flow observables to theoretical predictions in order to study the nuclear compressibility, since its value has direct consequences on the nuclear equa- tion of state. The primary system being studied is 197Au+197Au, for various reasons outlined in Chapters 4 and 5. Lighter systems are analyzed for the purposes of un- derstanding the effect of system mass and comparing to published calculations. The balance energy for Au+Au is interesting because it extends the mass depen- dence of the balance energy by a factor of two, and because its existence has been questioned by theoretical studies. Au+Au also proves to be a useful system for sep— arating the dual influence of the nuclear compressibility and the in-medium nucleon cross section on the balance energy. The transition energy for Au+Au presented in Chapter 5 has a very different value from previous experimental studies, and the tran- sition energy for lighter systems is presented in hopes that it can provide a second way for collective flow to isolate the nuclear compressibility. The following is a brief outine of the remainder of this thesis: Chapter 2: Experimental details of the MSU 47r Array, with a special emphasis on 24 the reduction of raw data to meaningful physics quantities. Chapter 3: Characterizing the events with impact parameter and reaction plane. Several corrections to the data are also discussed. Chapter 4: Disappearance of directed flow for Au+Au with comparison to BUU calculations. The BUU calculations are used to quantify the role of the Coulomb interaction for heavy systems as well as isolate the nuclear compressibility for the first time using the balance energy. Chapter 5: Elliptic flow transition from in-plane to out-of—plane emission for Au+Au as well as lighter systems. Comparison to isospin-dependent BUU calculations are made. 25 Chapter 2 Experimental Setup and Data Reduction In order to make measurements of collective flow in nuclear collisions, a detector sys- tem must be chosen that can detect and characterize a large fraction of the emitted particles. This is especially true for the directed and elliptic flow study in this thesis because (a) the accuracy of the reaction plane determination is related to the num- ber of detected particles; (b) both directed and elliptic flow depend on the impact parameter, which is related to event multiplicity and the summed transverse energy of detected particles; and (c) both observables involve particles emitted over a large angular range. Detectors with near-complete coverage such as the MSU 41r Array [30] are most suitable for studying flow at intermediate energies. At higher beam energies for fixed target experiments, detector systems may have very good coverage, even though they physically do not surround the target like the 47r Array, because the emitted particles are boosted forward more by the higher beam velocities. In this chapter, the parameters of the Au+Au experiment carried out for this thesis are presented. In particular, an emphasis is placed on the full procedure for converting 47r Array detector signals into meaningful physics quantities such as position and energy. 26 2.1 Beam and Target Parameters The Au+Au collisions studied in this thesis were produced by colliding 197Au beams with thin, stationary 197Au foils located in the center of the MSU 47r Array. The 197Au beams were accelerated by the K1200 cyclotron at the National Superconducting Cyclotron Laboratory (NSCL) by first stripping off between 35 and 44 electrons to create the high charge states needed. Over a five day experimental run, approximately 80 hours of active beam with intensities of approximately 10-100 electrical pA (~ 107 Au nuclei per second) were delivered to the 47r experimental vault at incident energies of 21, 25, 29, 35, 40, 45, 50, 55, and 59 MeV/nucleon. Some of the energies were obtained by degrading a primary 197Au beam in the A1200 beam analysis system without a significant loss of intensity. The 197Au beams were focused on round targets of 197Au foils 1.5 cm in diam- eter at the center of the 41r Array. Targets can be switched conveniently without venting the detector’s large vacuum chamber, and 197Au foils of thickness 2, 6, and 19 mg/cm2 were interchanged. (A 19 mg/cm2 Au target is 10 pm thick.) Because the energy loss in the target is substantial for the energies used in this thesis, some of the incident beam’s energy is lost as it passes through the target. Therefore, the effective beam energy is reduced by a small amount. This affects the kinematics of the collision, in particular the transformation to center of mass coordinates. Table 2.1 shows the effective 197Au beam energies for the beam and target combinations used in the experiment. 2.2 The 47r Array The MSU 4n Array [30], located in the N2 vault of the NSCL, is a powerful detec- tor configuration which nearly completely surrounds the collision. The 47r Array is 27 Primary Degraded Target Effective Beam Energy Beam Energy thickness beam energy (MeV / nucleon) (MeV / nucleon) (mg/cm2) (MeV / nucleon) 29 25 2 24.5 29 N / A 6 28.2 45 35 19 33.1 45 40 19 38.3 45 N / A 6 44.5 50 N / A 19 48.4 55 N / A 19 53.5 59 N / A 19 57.6 Table 2.1: Effective 197Au beam energies for all beam and target combinations used in this thesis. a charged-particle detector, i.e., gamma rays and neutrons are not identified. Its strength rests in three key traits: o The underlying geometry is a 32-faced truncated icosahedron, as shown in Fig- ure 2.1(a). This geometric configuration allows for detector modules to be packed close together, which provides nearly full coverage in the solid angle and reduces the amount of dead space between detectors. 0 Most of the detector array is logarithmic, meaning particles in different energy ranges (and therefore diflerent penetration distances in the detector) are iden- tified by using information from different layers in the detector. This enables a wider range of particle types and energies to be characterized while still main- taining the 41r solid angle. 0 The extensive electronics for the 47r Array are located very near the vacuum chamber, which minimizes signal losses due to long cables. The data acquisition system is built to accomodate high rates, and detectors are monitored during the experiment via a single ethernet cable between a vault computer and the 28 '38 TP 18° ~162' Figure 2.1: (a) Basic geometry of the MSU 41r Array. (b) Polar angles subtended by each of the detector subsytems used in this thesis. users’ area. These qualities make the 47r Array a suitable detector system for experiments that require high statistics and a large acceptance for charged particles. 2.2.1 Detector Subsystems For the data presented in this thesis, the 47r Array was operated in a configuration that includes three distinct detector arrays: the plastic scintillators of the Main Ball, known as phoswiches; Bragg curve gas detectors in front of the Main Ball scintilla- tors; and the set of plastic scintillators of the High Rate Array.l The three detector subsystems are explained below. The entire detector system is housed in a large vac- uum vessel two meters in diameter for which pressures of 2 x 10‘5 torr were reached during the experiment. Figure 2.2 shows the 41r Array around the time of the Au+Au experiment from two different vantage points. In Figure 2.2(a) the beam enters from 1Three additional 41r Array subsystems are available but were not used: Maryland Forward Array, Zero Degree Detector, and the Iowa Forward Array. The 41r Array is also configured to run with PPAC detectors on the fronts of the Main Ball modules, although they presently are not installed. 29 site" 'i ”,1 ‘a If”, :— Figure 2.2: The 47r Array, located in the N2 vault of NSCL. (a) View of the array from the entrance to the vault. (b) View looking at the High Rate Array face of the detector. the right side and exits on the left side through opposing pentagonal faces. Figure 2.2(b) shows the view from the exit side of the Main Ball. The exit face serves as the mounting plate for the High Rate Array. Main Ball Phoswiches The 41r Array is composed of 170 detectors in the Main Ball, distributed among twenty hexagonal and ten pentagonal modules. (The remaining two pentagonal faces of the truncated icosahedron are the entrance and exit faces.) Each hexagonal (pentagonal) module contains a subarray of six (five) two-layer, close-packed plastic scintillators, and the polar angles subtended by the array of ball detectors is 18° ,3 010., ,E, 162° (see Fig. 2.1). The individual detectors in the main ball are triangular pyramids which subdivide either hexagons (60°, 60°, 60°) or pentagons (72°, 54°, 54°). Each phoswich consists of a 3 mm thick layer of fast plastic scintillator followed by a 25 cm thick block of slow plastic scintillator, which is optically coupled to a photo—multiplier tube (PMT). Physical specifications of the phoswiches are given in Appendix A. When a charged particle is incident upon the scintillator elements of a phoswich, light is produced [32], which is collected by the PMT and turned into a single current 30 9‘ \\~ \ ‘99» Q9 :/ BCC slow plastic L (a) fast [13m .- Figure 2.3: Schematic diagram showing the components of a 4n Array module. (b) Geometry of the High Rate Array. '\ r. \9 1'.— i l“ M v' I ‘\ ’9 pulse. The contributions of the phoswich’s individual layers to the current pulse can be separated because the fast and slow plastic have different response times (hence the names “fast” and “slow”). The separation is done in the electronics, discussed in Sec. 2.3. The thin layer of fast scintillator acts as a AE detector, while the thick, slow scintillator records the residual energy E when a charged particle is fully stopped. Bragg Curve Counters Mounted in front of each of the 30 Main Ball phoswich modules is a gas ionization chamber known as a Bragg Curve Counter (BCC) [31]. The BCCs primarily are used to detect 0 particles (Z=2) as well as intermediate mass fragments (i.e., particles with charge 3 5 Z ,3 20). These heavier fragments often do not “punch through” the thin fast plastic and could not be identified without the BCC. The five most forward hexagonal modules have BCCs with anodes of the same segmentation as the phoswiches; otherwise, a single BCC chamber is used for an entire module’s phoswiches. Thus, there are 55 separate BCC detectors in the Main Ball. Figure 2.3(a) shows the cross section of one of the 30 ball modules. When particles such as IMFs stop in the thin fast plastic, the BCC serves as the AE detector and 31 the fast plastic serves as the E detector. The threefold combination of detectors is logarithmic, since the purpose of each layer changes as a function of an incident particle’s penetration depth. Detectors of this nature are often called telescopes. High Rate Array Mounted to the exit plate of the 47r Array vacuum vessel is a 45-segment fast/slow plastic scintillator array known as the High Rate Array (HRA) [33]. The HRA, built by Robert Pak, provides a sturdy, close-packed subsystem of sufficiently high granularity to handle the large fraction of particles emitted at small polar angles (due to the center of mass boost). The 45 phoswich detectors span the polar angles 3° ,3 0,0,, ,3 18° and detect particles from Z = 1 to Z = 14. Figure 2.3(b) shows the basic geometry of the HRA; each of its five faces is tilted slightly so that incident particles impinge perpendicular to the detector’s face. The HRA functions the same way as the fast / slow phoswiches in the Main Ball. An incident particle produces light in both layers of the plastic, and the light is collected by a single PMT. The separation of the PMT signal into AE and E components is possible because of the different response times of the two layers of plastic. In the following section, the procedure for converting the detector signals from the 41r Array to the raw data, which is stored on magnetic tape, is detailed. 2.3 Electronics and Raw Data Figure 2.4 illustrates the general route taken by a candidate phoswich or BCC signal through the electronics of the 41r Array, with a possible final destination on a raw data tape. The term “candidate” applies to the signals because several conditions need to be satisfied for a signal to be recorded. This section outlines the basics of the 4n Array’s electronics setup. The High Rate Array’s electronics are nearly identical to 32 Data Ac uisition Method Used In This Thesis 4n Vacuum Chamber AE-E Bragg-A E higher energy particles that OR low energy or high Z particles stop in the slow plastic that stop in the thin fast plastic Discriminator signals above a threshold \oltage are measured Quad Shaper shapes and amplifies signals from Braggs for input to Silena Splitter splits phoswich signal three ways Summer/Trigger signals are summed. which is used to determine iftrigger requirement is satisfied l SILENA ADCs peak-sensing ADC converts the peak height to a channel number FERA ADCS Signals from fast or slow plastic are integrated and converted to a channel number . 3. DLT Tape Drive 9‘3 (36 ‘f‘p ESL... tam C 0 m p U te r Parameters VME level Figure 2.4: Simple schematic of the method for converting the detector signals into a form which can be stored on magnetic tape. 33 the phoswiches of the Main Ball and do not need separate discussion. The discussion is more qualitative than technical in order to complement the rigorous explanations of previous theses with nearly identical experimental setups [34, 35]. Phoswich signals The PMTs of the 215 Main Ball and HRA phoswich detectors, located in the vacuum vessel, are supplied their voltage via a single cable which also transmits the detector signal. These high-voltage cables are connected to splitter box modules where the high voltage is separated from the phoswich signal and the signal is split into three signals: AE, E, and time. The time signal initially goes to a 16-channel discriminator which checks to see if the signal’s (negative) amplitude is larger than the pedestal level for that channel. If it is, then a voltage signal proportional to the number of hits in the discriminator is sent by the discriminator to a summer which adds the voltages of the Main Ball (and/or HRA) channels. If the summed voltage signal for all channels exceeds the requirements of the trigger (i.e., if enough detectors fire discriminators), then a gate is generated which allows the signals from all detectors, including the BCCs, to be recorded to tape. The current pulses representing the AE signal and E signal are identical when leaving the splitter box. However, the signals are gated differently by their respective FERAs, which are analog-to-digital converters (ADCs) with 2048-channel resolution. The general shape of the current pulse and the AE and E gates to separate the fast and slow components of this signal are schematically shown in Figure 2.5. Data from the phoswich FERAs are zero—suppressed, i.e., only channels with nonzero values are recorded to tape. For all channels, the pedestal level, which represents the minimum threshold amplitude a particular channel’s signal needs in order to be nonzero, is determined by operating the electronics with the detectors turned off to determine 34 l(-—250 nsec-——)l Figure 2.5: Diagram of the phoswich signal and gates. The width of the two FERA gates selects the appropriate section of the signal. the noise level of that channel. The pedestals are passed as parameters from the UNIX computer to the F ERAS where they are stored in memory. The use of zero suppression and pedestals reduces the amount of dead time as well as the amount of tape used. Including the AE and E for both the Main Ball and HRA, there are 645 channels of electronics in the FERAs. BCC signals Because the response time for a Bragg Curve Counter signal is on the order of mi- croseconds (as opposed to 1003 of nanoseconds for the phoswich signals) and the arrival time varies by microseconds, BCCs are not used in the trigger. The BCCs are supplied their voltage via a separate cable. A BCC signal first passes through a quad shaper, which amplifies the signal and produces a very clean voltage signal. The amplified signal then goes to a Silena, a peak-sensing analog-to—digital converter with 4096-channe1 resolution. Like FERAS, Silenas are zero-suppressed so that only non-zero channels are written to magnetic tape. In total, there are 55 channels of Silenas used for BCC electronics. 35 2.3.1 Writing Raw Data The 47r Array trigger can be configured to include the phoswiches of the Main Ball and the HRA, and the trigger level can be adjusted to select certain types of events. A trigger level of 5 preferentially would select more central events, since peripheral events produce fewer particles (see Sec. 3.1). If a trigger is satisfied (i.e., enough phoswich pulses produce voltage signals in the discriminators) then electronic gates are generated and sent to the ADCs, and the digitized signal information is recorded and sent via a VME crate to a UNIX computer. In the 47r Array’s data acquisition configuration, raw data is written to magnetic tape by the UNIX computer which also serves as the gateway for ethernet access from outside the experimental vault. On-line monitoring of detector counts and en- ergy spectra are done via the ethernet connection. The computer also monitors the electronics and can set threshold levels to the individual FERAs and discriminators. Raw data are written on an event-by-event basis to tape. The on-line monitoring programs and analysis software perform the task of selecting and matching signals from an event that correspond to the same particle. 2.4 Reducing Raw Data to Meaningful Quantities The basic task of the electronics is to convert detector signals into numbers that can be stored on tape in a structured manner. Analogously, the task of translating these numbers into physics variables (such as energy and charge) is performed by a series of software routines designed to identify and characterize charged particles from the raw data numbers which includes signals from cosmic rays as well as neutral particles (such as neutrons and high-energy gamma rays). The result of these software routines is a conversion and reduction of raw data to “physics” files, which are much smaller 36 256 I l l l I l l E 192 — ° 3 E g 5 —1 v a i _o 128 ‘r o i U 0 ‘“ ‘ é a 64 —] g on _] 0 1 0 64 128 192 256 0 64 128 192 256 Slow Plastic (channels) Fast Plastic (channels) Figure 2.6: (a) Sample AE-E spectra from HRA phoswich detector. (b) Sample AE-E spectra from BCC—fast plastic pair. and permit quicker analysis of the data. 2.4.1 Particle Identification When the integrated signals from the thin fast plastic are plotted versus the cor- responding signals from the thick slow plastic, particles with different charges and masses fall into different bands in the resulting AE—E spectrum. The same is also true for particles that stop in the fast plastic of the Main Ball, although in that case the BCC signal serves as the AE. Figure 2.6 shows examples of raw two—dimensional spectra produced by (a) a HRA phoswich and (b) a particle stopping in the fast plastic. Some of the qualities of fast/slow spectra differ from those of BCC/fast spectra. Fast/slow spectra (Figure 2.6(a)) are bound by thick diagonal lines, which result from the fact that the same detector signal is sampled for both the AE and E. The nearly vertical line, called the punch-in line, contains signals from particles which stopped in the fast plastic; the diagonal line that forms the bottom border of the spectra arises 37 from cosmic rays or neutrons that scintillate in the thick slow plastic only. In contrast, the BCC/ fast spectra (Figure 2.6(b)) do not have these lines, since there is no overlap between the two different signals. A decision is made for particles with both phoswich and BCC signals as to which spectra type should be used to identify the particle. If the particle lies in the punch-in line of the fast /slow spectra (as determined by a 2-D gateline), then it is identified using the BCC / fast spectra. Figure 2.7 shows the process for reducing raw data used in this thesis. The first step, corresponding to the blue boxes, is to sort the raw data by detector into fast / slow spectra and BCC/fast spectra. These spectra then are gain-matched to existing 2-D templates by adjusting the offset from the origin and by stretching/shrinking each spectra to account for voltage and gate differences between the detectors and the reference template. The templates for each spectra type (HRA fast/slow, Main Ball fast/slow, Main Ball BCC/fast) were created when the particular subsystem was added to the 47r Array, and their purpose is to map experimental raw data numbers into particle type and incident kinetic energy. The two gain-matching programs (MT K and BRAGGMATCHQ) generate parameter files, known as lines files, that contain the offsets and multiplicative factors for each axis for all 2—D spectra. 2.4.2 Physics Quantities Once the lines files for all of the detectors are generated, the raw data are converted to physics data with PHYTAPE, a VMS-based program that uses an assortment of tables and parameter files to assign physics quantities to each real particle. This assortment includes: 0 lines files generated by the gain-matching programs; 2The BCC/fast spectra were matched using a MAC-based application written by Gary Westfall for the first time. This replaced a VMS—based program whose graphics package was phased out by DEC. 38 Data Reduction and Anal sis Method Used In This Thesis Raw Data Raw data is stored in 16k buffers on an event-by-event basis. Not all signals recorded on a Raw Data tape are identified as particles. (.evt files) CD CD ADF BRAGGSORT sorts phoswiches sorts braggs PH YTAPE into 2-d spectra files into 2-d spectra files Using .lns, bins, and various (.ch files) (on files) parameter files, candidate particles are identified or thrown ® ® away. Events with identified particles are then written to .b2 MTK BRAGGM ATCH or SE2 files, depending on interactive program in Mac-based interactive experimental trigger. Each which user matches program; user matches identified particle has a Z’ A, 9’ spectra in .c2d files to spectra in .txt files to 4” E, and detector #- existing templates and existing templates and generates .Ins files generates .hlns files @ ® V ANA AV A ANA controls the reading of the physics files and calls numerous customizable subroutines. Histograms and/or ntuples are generated and stored as list files. (5) V initial pass of.b2 or s2 files is used to generate impact-parameter cuts, stored in .bs files PAW PAW is a CERN -developed graphical analysis © package that reads .hst files and manipulates —§ the data. PAW can perform a number of statistical tasks, and it generates final graphs. Figure 2.7: Simplified diagram that shows the entire process from raw data to final plots used in this thesis. 39 o detector parameter files which map detector number to their lab polar angles (6’, 45); o look-up files which contain 2-D arrays for mapping the matching templates to particle type and energy. For each event with at least one particle, identified particles are written to a physics file by their parameters Z, A, 0, d, E, and detector number. Some particles are flagged because of their location in the AE-E spectra. For example, candidate particles that fall in the punch-in line of the fast/slow AE-E spectra but do not have BCC signals are assigned a mass of A = 0 so that they can be included/excluded in the analysis. Also, particles that overflow a particular axis (which happens when the maximum channel number of an analog-to—digital converter is exceeded by a detector signal) are flagged with negative mass so that their inclusion in the analysis can be decided later. 2.5 Summary Once the experiment has been performed and the raw data have been converted to useful physics variables, the analysis can begin. Figure 2.7 indicates the basics of the analysis in red boxes. One primary aspect of the analysis in this thesis is the use of millions of collision events to increase statistics and to decrease fluctuations in analysis variables. The next chapter details some of the analysis techniques used in this thesis to describe and classify the collisions. 40 Chapter 3 Characterizing the Collisions Before anything can be said about the physics of nuclear reactions, it is first necessary to determine the parameters of each collision. For example, to first order, the number of particles produced in a collision depends on four parameters: three chosen by the experimentalist (beam energy, projectile particle type, and target particle type) and one selected by the collision itself (the impact parameter, see Sec. 3.1). If a theorist wishes to predict the number of particles that will be produced in a collision, he or she needs to specify these four parameters. Determining the nature of each collision requires manipulating the experiment’s measured quantities. In the case of the 41r Array, the measured quantities for each particle are charge, mass, final position, and kinetic energy. Using these variables, we can calculate a particle’s momentum, rapidity (an invariant measure of longitudinal velocity), transverse kinetic energy, and other useful quantities. We can also sum certain quantities over all measured particles in an event to produce even more useful quantities, such as total transverse kinetic energy, excitation energy, etc. For the present analysis, several tasks need to be completed before the results can be presented. Some of these tasks characterize the collisions, others were performed to understand or reconcile confusing results, and still others were merely diagnostic. In 41 this chapter, the impact parameter and reaction plane determination are presented. Also, several corrections to the data are made in an efl'ort to remove some of the experimental bias from the 47r Array. 3. 1 Impact Parameter As expected from the Nuclear Fireball Model [23] (see Fig. 1.4), what transpires dur- ing a nuclear collision depends strongly on the impact parameter of the collision. Two nuclei that merely graze each other will produce a tiny participant zone of com- pressed matter, while a head-on collision of zero impact parameter will lead to higher compression and larger pressure gradients because nearly all the initial kinetic energy is deposited into the participant source. As a result, both collective flow observables being studied in this thesis have been shown to depend on impact parameter. In the case of the disappearance of directed flow, Pak et al. showed that the balance energy increases linearly as a function of the impact parameter for a light system such as Ar+Sc [58]. For the transition to out-of—plane elliptic flow, data from Popescu et al. showed the energy of transition to increase with impact parameter for 64Zn-l-58Ni [92]. Figure 3.1 illustrates the definition of impact parameter 5 as the vector pointing from the center of the target to the center of the projectile’s path (beam points along the z-axis). A maximum impact parameter of bmax = Rpmj + Rtarg is defined and is especially useful for studying the system mass dependence of a quantity, since the basic geometry of the collision can be preserved. The 41r Array is well-suited to act as an impact parameter filter due to its large acceptance. Impact parameter is not a directly accessible experimental quantity, but there are several centrality variables which have been shown to be strongly correlated with the impact parameter. These include total charged-particle multiplicity [36], midrapidity 42 Looking down on Looking in the the reaction plane reaction plane [‘I‘t \] L‘cl l iL‘ . y ’1 reaction plane Figure 3.1: Geometrical description of the impact parameter and the reaction plane. The impact parameter vector b lies in the reaction plane and points to the projectile side. charge [37], and the total transverse kinetic energy E, [38]. In this thesis, E, is the chosen centrality variable because impact parameter binning can be more precisely controlled [39] and because the acceptance of the 47r Array over the range of Au+Au energies is weak at backward midrapidity; see Figure 4.12. The total transverse kinetic energy of an event with N identified particles is defined N E, : 2 E,- sin2 0,- , (3.1) i=1 where E,- is the kinetic energy and (9, is the polar angle of the ith particle in the lab frame. To obtain a quantitative estimate of the impact parameter from E, (or any other centrality variable), a geometrical prescription owed to Cavata et al. [36] assumes that E, is monotonically related to the impact parameter b, i.e., as E, increases from zero to its maximum value, b continually increases or decreases, but not both. The following relation expresses this: 27rbdb ”b2 : —f(Et)dEt . (3.2) m 3X 43 In this expression f (Et)dEt is the probability of detecting a collision with a value between E, and E, +dEt. The function f (Et) is normalized to unity. A consideration of b = 0 and bum,x leads to the assumption that b —> bmax corresponds to E, —-> 0, which causes the minus sign in Eq. 3.2. The correct divisions in the E, spectra for placing events into impact parameter bins can be obtained by integrating Eq. 3.2 from b to bmax: bmax max A $51—13 = — [W ) f(E;)dE;. (3.3) max E¢(b) The integration limit E¢(bmax) can be replaced with 0 as stated above. After in- tegrating the left-hand side, the reduced impact parameter b/bmam can be expressed as 0 b/bmax = \/1 + f(El)dEf ' (34) Eta?) This expression directly relates the E, spectra (normalized to unity) to b/bnm,x through the integral.1 For example, a reduced impact parameter of b/bmax = 0.5 corresponds to the (0.5)2 x 100 = 25% events with the largest E. To separate the data into impact parameter bins, equal partitions of the E, spectra with respect to the number of events are constructed and the limits of the bins are determined from Eq. 3.4. Figure 3.2 shows sample reduced E, (Et/E'pmj) spectra for Au+Au at three different energies. The hatched area represents the 20% of all events with the largest Et. Using Eq. 3.4, this corresponds to b/bmx S 0.44, which is roughly 6 fm using the formula R = 1.2/1V3 fm for the radius of each Au nucleus. In Chapters 4 and 5 different impact parameter binning is used, depending on the observable and whether it is being plotted with data from previous studies. 1The value of the integral in Eq. 3.4 is always between 0 and -1 because the integration is carried out from left to right in the E, spectra. Therefore, b/bmax is between 0 and l, as required. 44 24.5 MeV/n ' 38.3 MeV/n 53.5 MeV/n 4 z . .~ 10 .. . db ’9‘ I... = 6.0.. ”fl: ”0‘ : flfifi 000‘ gkfi ' %%*% 10’ :ozozozo. is». e ms ...t >...t : -..... ¥%%%% 3$$03 “ %%~%%‘ ¥%%%&A afififie 3 Qflflfiflfi 2 ...... ..... ...... ...... >....< ...... 10 ....... .....~ g- 900006u ......~ >..... _ ...... »...... ...... _ ....... same». mm = em»: ’fiflfihflfifi flfififlflflt ’ ttfififififlu >....... ......« ........ 10 afifififlfififii ¥¥%%%%& g" accesses ........ >......« : .......4 >........ ....... - ........ ......... >......4 - .......< vvuvuwusw hfififififlfl’ b %%%%%VV%“ »........4 .......g ........' l fifififififlflflfl ”WVWVVV. successes 9.05.0...5...¢.8’£ l ’A’A‘i’l’AoaQs.‘ l . '5‘... ’a’.’s‘a’¢’§ 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 Et/Eprot Figure 3.2: Reduced total transverse kinetic energy (Et/Epmj) for three different beam energies of Au+Au. The hatched area corresponds to the 20% most central events, as explained in the text. 3.2 Reaction Plane Determination The observation and quantization of directed and elliptic flow require the accurate determination of a reaction plane, since both observables rely on the azimuthal angle between the flow particles and the reaction plane. In the thesis, the convention (I) = (152' —¢RP- is chosen to mask the (bar: angle, since it is the orientation of particles with respect to the reaction plane that is of importance. Historically, two standard techniques of reaction plane determination have been used most commonly: the sphericity tensor method [40] and the transverse momentum method [52]. The sphericity tensor method uses the existence of flow to calculate a kinetic energy tensor which represents an ellipsoid, and the reaction plane is formed by the plane which contains the beam and the principal axis of the ellipsoid. Danielewicz and Odyniec [52] showed that the sphericity tensor method is poor when the flow signal is weak, because any spherical momentum distribution of a finite number of particles will yield a reaction plane and 45 Beam is out of page ’ ‘l’nr .. > px pf J. Reaction Plane i P; Figure 3.3: The azimuthal correlation method of reaction plane determination. The line that minimizes the sum of the distances d,- squared corresponds to the reaction plane. non-zero flow, even if generated randomly. In contrast, the transverse momentum method uses only the transverse components of the particles’ momenta to determine the reaction plane: N i=1 where the weight w,- is chosen to be positive (negative) for particles emitted in the forward (backward) center-of-mass hemisphere. The reaction plane is defined by the beam axis and P, which points toward the positive side of the plane. Both of these techniques were shown to be inferior to the azimuthal correlation method, introduced by Wilson et al. [41], for cases when flow is weak. The method is based on the observation that particle emission is strongly enhanced in the reaction plane. Thus, this technique involves finding the plane that aligns best with the particles themselves. Figure 3.3 illustrates the azimuthal correlation method for a small event of five particles. The sum of the deviations d,- of the particles’ transverse 46 momenta in the event from a line which passes through the origin is given by: N N D’ = Zldfl = Z [(2225): + (p3)? — (”3‘ + mp” ] . (3.6) l-i-m2 where m is the slope of the line in Figure 3.3. The value of m which minimizes D2 is determined by setting the derivative of D2 with respect to m equal to zero and finding the two roots, which are m = zines? - (pm 2c \/<2:::.I21)2+ «zimw (3 7) 22.1mm ' ' Substituting the two m roots back into Eq. 3.6 determines which root minimizes D2 and maximizes the in-plane enhancement. (253p is found by taking the arc tangent of m and picking the direction for the positive side of the reaction plane using Eq. 3.5. A necessary step taken to avoid autocorrelation between (flap and (15,- when measur- ing the flow of the ith particle is to remove that “particle of interest” (POI) from the determination of 4531:. This must be done for each flow particle in an event, resulting in N different reaction planes for an event of N particles. To examine the effect of excluding POIs from the (tap determination, Figure 3.4 shows the spread of an event’s N reaction planes with respect to the average (1531: for all events. The narrow distribution peaked at 0° demonstrates the validity of using a separate reaction plane for each POI, which also greatly enhances the statistics of the method. Another way to see the effect of excluding the P01 is to view the relationship between the reaction planes determined with and without the P01. Figure 3.5 shows this relationship by means of a three-dimensional surface plot. A very strong correla- tion exists, since a tall row of peaks exists at 4531),”); INC = ¢RPJJOI EXQ. The small rows of bumps to the left and right of the strong correlation correspond to the slight probability that a reaction plane’s direction flips to the other side of the beam, (i.e., rotates by 180°). The peaks at (90°, 270°) and (270°, 90°) illustrate an interesting and small side effect of the azimuthal correlation technique. Reaction planes aligned 47 Au+Au, 38.3 MeV/n soooo _— T 7.... ;. some E 500007 moo f- ’ 90 A ‘ 180 (hm, - ¢Rp(event ave.) (deg) Figure 3.4: Distribution of the azimuthal angles between the individual reaction planes for an event and the event average. Removing the flow “particle of interest” leads to N reaction planes for each event. near the y-axis are slightly more likely to appear flipped to the other side when the P01 is excluded due to the more likely chance that a small rotation of the (011}? angle will move the direction into a neighboring quadrant. It should be emphasized that the above method can only estimate the true reaction plane. The true reaction plane is concealed by an imperfect detector system, a finite number of particles, and thermal fluctuations. Because of the difference between the true and estimated reaction planes, the observed flow projected into the estimated reaction plane will always be smaller than the actual flow. A measure of the accuracy of the reaction plane determination is the reaction plane resolution, discussed in Section 3.3.3. 48 150 ¢ 106 ’ ’ 20° 250 RP, POI INC. 6 o ¢RP, POI EXC. Figure 3.5: Relationship between the reaction planes determined including and ex- cluding the particle of interest. The tall row at ¢Rp,p01 INC, 2 ¢Rp,p01 EXC. shows the minimal effect that excluding the POI has on the reaction plane determination. 3.3 Correcting for Experimental Biases One of the underlying objectives of nearly all science is to find experimental observ- ables which can be compared unambiguously to theory. Unfortunately, in nuclear physics the comparison is obscured by features of the detector system. The primary quality of nuclear experiments that acts to bias the data is the finite position and momentum resolution of all detectors. This is caused by either the physical dimen- sions and properties of the detector elements (such as the 47r Array) or the error associated with reconstructing a particle’s path through a detector (track resolution). Experimenters go to great lengths and expense to improve position and momentum resolution by building larger detector arrays with more detector elements. For exam- ple, the ALICE detector currently being built at CERN will have more than a million separate detector elements [42] to improve the detection resolution for the anticipated 50,000 particles per collision. In this section, corrections made to the data presented 49 in this thesis are discussed. 3.3.1 Angle Smearing In total, the 47r Array contains 215 geometrical detector elements: 45 phoswiches in the High Rate Array and 170 BCC-phoswich telescopes in the Main Ball. The 215 detectors almost completely surround the target at the center of the array so that each detector covers an appreciable amount of solid angle. The position of an incident particle within the detector cannot be determined for detectors of these types. Therefore, previous studies of collective flow with the 47r Array assigned the geometrical center of the detector as the particle’s (6, (b). This leads to discrete bands in both anguular distributions. In addition, the 47r Array has five-fold symmetry in the azimuth due to its truncated icosahedronal shape, leading to slight anisotropies in the 43 distributions. Assigning a particle’s position to be the geometrical center of the detector affects the isotropic symmetry of the reaction plane distribution. It also is slightly erroneous because the “most likely” location for a particle to strike depends slightly on the center of mass boosts for different beam energies. In order to reduce the effects of detector granularity in the 47r Array, the as- signed 0 and 43 angles for each particle are smeared over the range of angles covered by the detector. The angular probability distributions for each detector are gener- ated by making use of the 47r software filter which describes the geometry of the 47r Array. To generate the (0, (b) histograms that describe each detector’s acceptance, isotropic distributions of protons were produced in the center of mass by a simple event generator and boosted to several different beam energies. The distribution was passed through the 47r software filter, and reference histograms were created. CERN’s Physics Analysis Workstation (PAW) software was used to assign a variable based on 50 the distribution of a 1-D histogram. Then, each particle’s (0, go) in the experimental data was smeared by making use of the histogram look-up table for the particular detector. Figure 3.6 shows the effect of angle smearing on the 0 and d) distributions for Au+Au at 40 MeV/nucleon. The discrete blue bands represent the centers of the detectors, and since the 47r Array has five-fold symmetry there are five times fewer discrete bands as there are detectors. The green spectra represent the smeared angular distributions. The (1) distribution is nearly flat as expected, with the exception of five periodic dips. These irregularities correspond to the corners of the pentagon created by the five most forward modules, evident from inspection of the mean to angles in Table A.6 in Appendix A. The 0 distribution is smoother and is peaked at forward angles due to the velocity of the beam projectile. In order to check the effect of angle smearing on the analysis, it is practical to investigate the change in the pat vs. (y / ypmj)cm, since 19,, is the quantity of interest in the directed flow analysis of Chapter 4. Figure 3.7(a) shows the pi. us. (y/ypmj)cm distribution when discrete angles are assigned to the particles. When the angles are smeared, as in Figure 3.7(b), the distribution’s color contour lines are smoother but the general shape of the plot is unchanged. Therefore, smearing the particles positions according to the 47r software filter is an effective method for dealing with the granularity of the 47r Array. 3.3.2 Repairing the Reaction Plane Distribution For a perfect detector system, one expects an isotropic reaction plane distribution because beam projectiles are incident upon target projectiles with no preferred side. However, the locations and shapes of detector elements as well as dead channels and imperfect calibration lead to anisotropies in 433p. Figure 3.8 shows the 453p 51 Blue = before angle smearing Green = After angle smearing A 3 v 50000 30000 20000 IDTIITIFFTIIDTIfiIIIIIrTj 10000 IIIT 1--.] 80 100 120 140 160’H180 Assigned 0 (degrees) c 8 g . 8 14000 (b) 12000 10000 8000 frTIIIIITIITIITII 1) 4000 at" .rwv NVW/‘WrV‘f‘W 2”: 1L 1. 11 ll Ill 0 so 100 150 200 250 300 350 Assigned 4) (degrees) Figure 3.6: Effect of smearing the polar angles of particles over the active area of the detector as opposed to assigning the detector’s geometric center (a) 6 and (b) (b. 52 Px (MeV/c) 0.5‘ i 1.5 i 11.5‘91P103‘0’ 0.5%? 11.5 L 2 (y/Yprojl Figure 3.7: p, us. (y/yproj)cm for (a) before angle smearing and (b) after angle smearing for Au+Au at 33.1 MeV/nucleon. distribution for Au+Au data at 33.1 MeV/nucleon. The arrows pointing downward along the top of the figure indicate the geometric centers of the fifteen most forward detectors in the Main Ball. Since these detectors are subject to higher counts as well as higher-momentum particles, they have a substantial influence on the reaction plane distribution. Also shown in Figure 3.8 is the azimuthal location of module 9, which had a dead Bragg Curve Counter for the duration of the experiment. The effect of module 9 is also seen at 180° from its azimuthal location, since module 9 detects particles from the backward hemisphere which contribute to the reaction plane determination with a 180° phase (see Sec. 3.2). Biases due to the finite acceptance of the detector can be removed by making the distribution of reaction planes isotropic in the laboratory [43]. There are several different methods to remove the effects of anisotropy which have been used, and each of them has some disadvantages. (see Ref. [43] for a discussion of the different methods.) The method used in this thesis, adopted from the E877 Collaboration at the AGS [44], is to fit the unweighted 05m: distribution, summed over all events, to a Fourier expansion. An event-by-event shifting of (hp needed to make the final 53 U T to § I MEG-mod! Figure 3.8: Uncorrected reaction plane distribution for Au+Au at 33.1 MeV/ nucleon. The arrows along the top of the plot correspond to the 15 most forward detectors in the Main Ball. distribution isotrOpic is performed by using the coefficients of the Fourier expansion. A new angle is defined as (bin) = $11? + A ¢RP , (33) where A0531: is written in the form AQSRP = E [An cos(n (Imp) + B, sin(n 05313)] . (3.9) fl. Requiring the vanishing of the nth Fourier moment of the new distribution, the coef- ficients An and B,, can be evaluated by the original distribution: 3,, = %(cos(n ¢Rp)) , (3.10) An = -%(sin(n 453p» , (3.11) where the angular brackets refer to an average over events. These coefficients are written to a look-up file for each set of data, since the reaction plane distribution 54 8000 6000 — 4000 r‘ 2000 7 — Uncorrected distribution * — Corrected distribution 0 1 11.1L111.lrxrnerxLLaranrilrlrrxrl 1 0 50 100 150 200 250 300 350 4’" Figure 3.9: Effect of repairing the reaction plane distribution by making use of a Fourier expansion for Au+Au, 35 MeV/nucleon. n = 30 terms were used in the expansion. changes slightly for each beam energy. The final distribution is given by ¢’Rp = mm + 2 % [—(sin(n 45313)) cos(n 053p) + (cos(n 05%)) sin(n (Imp) ]. (3.12) The number of terms in the Fourier expansion needs to be large enough to account for periodic features of the (tap distribution, and the adjusted o’Rp distribution needs to be checked to make sure it indeed becomes isotropic. In the present analysis, n = 30 was used to account for the 15 periodic peaks in the aim: distribution and the ten- fold symmetry of the High Rate Array. Figure 3.9 shows the aim» distribution before and after the Fourier correction for Au+Au at 35 MeV/ nucleon. The distribution is sufficiently flattened for n = 30. Redistributing the reaction planes removes possible trigger biases, such as imperfect calibration, dead channels, or any other asymmetry, from the data. 3.3.3 Reaction Plane Resolution The reaction plane determined in Sec. 3.2 is only an estimate of the true reaction plane. The disparity between the measured reaction plane and the true reaction plane is due to imperfect detectors as well as a finite set of particles per collision. Since both observables studied in this thesis, directed flow and elliptic flow, depend on the azimuthal angle between the reaction plane and the particle whose flow is being measured, the data must be corrected for the effects of reaction plane dispersion. The form of the correction is Observed Flow (cow... - a» ’ (3'13) Flow 2 where (am and 05, are the experimentally measured and true reaction plane angles,2 respectively, and the brackets signify the mean over all events. The mean cosine values are less than one and thus this correction always increases the measured flow. Of course, in its current form the correction term (cos(q§m - qb,)) contains the unknown 45,. Therefore, the correction must be obtained by an indirect method. One approach is to recognize that when flow is present, a correlation exists between the azimuthal angles of the particles from a particular collision. In other words, an event’s particles themselves can be used to determine the accuracy of the reaction plane determined from all particles. This approach relies on the fact that the reaction plane resolution is directly related to the flow; the stronger the flow, the better the resolution. When an event of multiplicity N is divided randomly into two equal “subevents” of multiplicity N / 2, the reaction planes for the subevents are correlated due to the flow present in the collision. Figure 3.10 shows the absolute value of the azimuthal angle between the two reaction planes for Au+Au. The peak at 0° is evidence that 2The subscript ‘RP’ is dropped from the symbols for the reaction plane in this section due to overcrowding. All 42 angles in this section correspond to reaction plane angles. 56 ., I5“ [1“ A A‘ ._ ‘A “A -_ “ 7 “A A ‘ - ‘ A - A A “ _ A“ _ A“‘ ““‘A _ ‘A“‘ __ “Ml 24.5 MeV/n : 28.2 MeV/n : 33.1 MeV/n I I I I I Li L - I I i L I I I I - I I L I I J L I 0 30 60 90 0 30 60 90 0 30 60 90 E‘AA“ E“A“‘ “A.“ A ‘ — A AA . “ _ A“‘ A“‘A - A - “‘AAA‘] : A “‘ - “w : 38.3 MeV/n : 44.5 MeV/n ‘ ; 48.4 MeV/n ’- I I I I I I I I _ L I I I L I I I I +- I I I I I I I I 0 30 60 90 0 30 60 90 0 30 60 90 AA““ AA““‘ A“‘ A“““ ‘A“‘A AA 53.5 MeV/n 57.6 MCV/ll I L LI I I I I I I I I J I I I 0 30 60 90 0 30 60 90 ¢’-¢” Figure 3.10: Absolute value of the azimuthal angle between the reaction planes for two subevents in Au+Au collisions. 57 there exists a correlation. The correlation weakens slightly as the beam energy is increased, which is most likely due to the strengthening of the out-of-plane elliptic flow signal for Au+Au, as discussed in Sec. 5.3.2. The angle between the two constructed reaction planes is related to the true reaction plane by the expression (coswi‘. — at.» = (cos(¢:. — <0» >< (cos(¢£’. — ¢r)) , (3.14) where (0:, and 42%, are the measured reaction planes for the subevents.3 Then, recog- nizing that the correction terms on the right-hand side of Eq. 3.14 should be nearly equal due to the random division of the events, (cos(¢‘,’n — 05.)) z (cos(¢fn — 05.)) , (3.15) Eq. 3.14 can be turned around to read 450 reactions exists. However, the Au+Au system is the system of pOpular choice for another reason: it is isotopically pure in nature, making it less expensive than other heavy elements (such as lead) that need to be isotopically purified. The purity of the projectile and target is particularly important for studying directed flow because the flow analysis is done in the center of mass, and variations in the system mass introduce error bars in the transformation. As an added benefit, gold is the most malleable metal, allowing for thin targets to be made with relative ease. 4.2.1 Extension of the Mass Dependence of Ebal As mentioned at the outset of this chapter, the initial purpose for performing an Au+Au experiment at the NSCL was to study the balance energy for a heavy system and extend the mass dependence of the balance energy beyond 86Kr+93Nb. The prospect of studying Au+Au collisions over the energy range necessary to measure the balance energy was made possible by ion source developments and corresponding A1200 calculations which showed that the high charge states needed to produce high energy Au beams could produce sufficient currents. Figure 4.3 shows the mass dependence of the balance energy, obtained with the 47r Array and published in 1993 [57]. Solid squares are experimental data, and open points are Boltzmann-Uehling-Uhlenbeck model calculations from Ref. [18] (see Sec- tion 4.4 for a discussion of the BUU model). The balance energy is seen to decrease as the system mass is increased, in agreement with the calculations. The lines in the figure represent power law fits of the form Ebal 0: A”, and the solid line corresponds to 7' = 1/3. A simple scaling argument suggests the power law relationship arises 67 150 C+C ”a O 0 "‘3 5 100 l ‘ > . 0 E . SB 0 b 4 G LL] 8 . g "."BUU(al=0) _ ‘ \ c3 50 » ‘NBUU(°’1='°°” -\ 'A'BUU(a‘=-O.2) + ‘1 20 50 100 200 Mass of Combined System Figure 4.3: Previously published mass dependence of the disappearance of directed transverse flow [57]. Data are taken using the 41r Array at the NSCL, and BUU model calculations are from Ref. [18]. from two terms: the nuclear mean field, which scales roughly with the surface area of the nuclei, A2/3; and the repulsive nucleon-nucleon scattering, which scales with the number of nucleons present in the collision, A. However, visual inspection of Fig. 4.3 shows that the value of r in the power law fit may increase as the system mass in- creases. Extending the mass dependence of Ebal to Au+Au is useful for understanding the trend. 4.2.2 Previous Studies of Directed Flow for Au+Au Another motivation for studying directed flow in Au+Au collisions at the NSCL is to add to an already extensive set of data which spans several orders of magnitude in beam energy. Flow measurements at intermediate energies can be compared with flow data taken at SIS, AGS, SP8, and (soon) RHIC, and it is hoped that a complete flow excitation function, or flow vs. beam energy plot, would provide insight. In order to plot finite flow values from different experiments on the same graph, dispersion in 68 0.|0 - ~ . ----.. - - - ---- l 4 > AU - AU [ 008: Z > 0 a :- (Y/YP)c.m.2 o_3 , ' A TRUE J A ESTIMATED A 0.06 " ] a * i ’ l x> > f 4 V 0.04- r . 0.02 r . L i ‘ l 0.00 4 --- IOl [o3 BEAM ENERGY. E (A MeV) Figure 4.4: Energy dependence of directed flow for data taken with a plastic wall array at Berkeley [64]. An extrapolated a value of Ebal = 47 i 11 MeV/ nucleon was obtained using a logarithmic fit. the experimental reaction plane needs to be corrected for in a consistent manner (see Section 3.3.3 and Appendix A). Since the balance energy for Au+Au is a significant feature of the flow excitation function, three collaborations previously estimated the value of Ebal by extrapolating from flow measurements taken at higher beam energies. Using a plastic wall array at the Bevalac, Zhang et al. [64] measured the average in-plane transverse velocities and corrected for the reaction plane dispersion by adopting the method of Ref. [52]. They obtained a value of Ebal = 47 :l: 11 MeV/ nucleon by fitting data between 75 and 400 MeV/ nucleon to a logarithmic function (see Figure 4.4). However, the large error was associated only with the uncertainty in the interaction energy of the collision between the projectile and the target. Inspection of Fig. 4.4 shows that the magnitude of dispersion correction affects the extrapolated balance energy greatly. The EOS Collaboration measured directed flow in Au+Au collisions at the Bevalac 69 0.3 _ ’3 \ E 0.2 _- ... E A .— 3 . o I- E I FL 0.1 :- 3 —-—eos Data : 0 Plastic Ball 0. IjllllllllllLlIJllllllIlll 0 0.2 0.4 0.6 0.8 l 1.2 1 .4 Beam Energy per Nucleon (GeV) Figure 4.5: Energy dependence of flow for EOS data [65] and Plastic Ball data [63]. The EOS data extrapolated a value of Ebal = 47 i5 MeV/ nucleon using a logarithmic fit. with beam energies ranging from 0.25A to 1.15A GeV [65]. By comparing their results with a quantum molecular dynamics model, they found that neither the “soft” nor the “hard” equation of state described the data over the entire range of beam energies. Figure 4.5 shows their data plotted together with data from the Plastic Ball [63]. All of the data is for a semi-central impact parameter bin. Like Zhang et al., they extrapolated Ebal for Au+Au by fitting their dispersion-corrected flow parameters to a logarithmic function and measuring the abscissa. They obtained a value of 47 i 5 MeV/nucleon. Most recently, the FOPI Collaboration at GSI in Darmstadt measured the ex- citation function of directed flow at seven incident energies between Ebeam = 100 MeV/nucleon and Ebeam = 800 MeV/nucleon. Figure 4.6 shows the flow (in this case, normalized by fragment mass) in semi-central collisions for four different frag- ment types, Z = 2 — 5. The solid lines correspond to fits to the data points up to 400 MeV/nucleon with Fermi functions. All of the extrapolated balance energies are 70 (Z7 (0) 0.6 :— 0.5 3 i? 0.4 E L. 0.3 L 0.2 _- fi-lt D>

> \.\ " 100*- i100.\5\\\ K a : +a=0 Z \\ L +0=0 .\ E r 0.0:“ E 1 «GU-=1, 1 “P3?” 1 oo=136ar \ 3°” *2. ... _ +exp. \- exp. 40 1 1 1 1 1111 1 1 4.0 1 1 1 20 40 100 200 20 40 100 200 COMBINED SYSTEMMASS COMBINED SYSIBMMASS Figure 4.7: Vlasov-Uehling—Uhlenbeck calculations by Zhou et al. [67] of the mass dependence of the balance energy with and without the Coulomb interaction included. Experimental data are from the 47r Array. The influence of Coulomb forces on Ebal was first studied by Zhou et al. [67], using model calculations based on the Vlasov-Uehling-Uhlenbeck (VUU) approach in which the mean field is determined by the coupling constants between nucleons and mesons. The energy of vanishing flow was a result of the complex interplay between nuclear and Coulomb forces, even for medium-sized systems. Figure 4.7a shows the experimental mass dependence of Ebal, taken from 47r Array data, and the result of VUU calculations without the Coulomb interaction. A soft EOS is used. One can see that Ebal decreases slowly as a function of system mass. With Coulomb included, as in Fig. 4.7b, the mass dependence of the VUU-calculated balance energy decreases more rapidly, as expected. The calculations were carried out for system sizes up to A = 200. An interesting prediction arose from quantum molecular dynamics (QMD) cal- culations performed by Soff et al. at the University of Frankfurt [68]. They re- garded the existence of negative flow, and therefore of a balance energy, for Au+Au as an open question due to the strong Coulomb repulsion. Figure 4.8 shows the transverse momentum pm as a function of the reduced rapidity y/ypmj, the slope of 72 15 Au(50AMeV)Au p, (MeV/c) O without Rutherford-trajectories I with Rutherford-trajectories -l5 ‘ -l.5 -l.0 -0.5 0.0 0.5 1.0 1.5 yIYpmj Figure 4.8: Transverse momentum p: as a function of the reduced rapidity y/ypmj for quantum molecular dynamics calculations of Au+Au at 50 MeV/nucleon by Soff et al. [68]. Squares represent calculations in the lab frame, circles are calculations in a rotating Rutherford frame. which corresponds to the directed transverse flow (as discussed in the next section). When Coulomb interactions are included in the QMD calculation (labeled “with Rutherford-trajectories” in the figure), the transverse momentum is nearly flat at 50 MeV/nucleon. But when the projectile’s motion due to the long-range Coulomb interaction is accounted for, a negative lepe is recovered, representing overall attrac- tive collision dynamics. In effect, the reference frame becomes a rotating reference frame so that Coulomb influence is removed. The conclusion that Soff et al. reached was that Eb,“ could not be observed experimentally because Coulomb efl'ects would overwhelm the low-energy attractive regime of the flow excitation function. In their 1997 review of collective flow [25], Reisdorf and Ritter pointed to the importance of determining the balance energy for Au+Au. For very heavy systems such as Au+Au, efforts to determine the bal- ance energy so far are inconclusive, since the balance energy has not been 73 reached experimentally... Coulomb effects are not negligible and could change the simple size scaling considerations (of the mass dependence of Ebal) made before. In this chapter, experimental evidence for the balance energy is presented. In addi- tion, Boltzmann-Uehling—Uhlenbeck (BUU) simulations are carried out to investigate the role that Coulomb interactions play in the collision dynamics. Also, using BUU calculations, Eb,“ for Au+Au is seen to be insensitive to both the impact param- eter and the in-medium cross section, allowing the EOS parameter K, the nuclear compressibility, to be extracted for the first time. 4.3 Experimental Results As discussed in Sec. 1.5.1 of the introductory chapter, the disappearance of directed flow occurs at an incident beam energy, termed the balance energy, where the at- tractive and repulsive dynamics of the collision balance each other. At low energies (~ 10 MeV/ nucleon), the attractive part of the nuclear mean field dominates, leading to the deflection of matter to negative scattering angles. At higher energies (~ 150 MeV/nucleon), repulsive nucleon-nucleon scattering dominates and deflects matter to positive scattering angles. At the balance energy the attractive and repulsive in- teractions, which are both energy-dependent, cancel, leading to zero transverse flow in the reaction plane. Physically the balance energy Ebal occurs when nuclear matter in the collision’s overlap (or participant) region does not expand preferentially in a particular quadrant of the reaction plane. However, the balance energy should not be misunderstood to represent necessarily isotrOpic emission of participant matter in the transverse direction. Figure 4.9 shows a schematic representation of directed flow in the center- of-mass frame at incident energies below, at, and above Ebal. At the balance energy, 74 Looking onto reaction plane in cm. Looking down beampipe (z-axis» y y cm Figure 4.9: Schematic representation of directed flow at incident energies below, near, and above the balance energy. The left column is looking down on the reaction plane (z—z plane), and the right column is looking down the beampipe (z-axis). In the right column, particles are grouped by their rapidities to show the preferential emission of forward (backward) particles to positive (negative) angles. 75 emission at midrapidity is effectively symmetric about the y-axis, as expected from the balancing of the attractive and repulsive interactions. This corresponds to a fragment ¢-distribution about the beam axis which is symmetric about 7r/2. As discussed in Sec. 5.3.1, it could be that the (xi-distribution of participant fragments is never azimuthally symmetric for non-central collisions. The utility of the balance energy is the ease with which experimental data can be compared to theoretical models. Since Ebal has been shown to be independent of particle type [57, 69, 58], the need for accurate fragment formation in the model is removed. And since the balance energy represents “zero” flow, experimental biases such as detector acceptance and reaction plane dispersion affect Ebal weakly [26]. That is not to say that these biases can be ignored, however, as discussed in the following paragraphs. 4.3.1 Caveats to Measuring the Balance Energy Measuring directed flow requires determining the reaction plane as well as assigning the reaction plane direction. Direction can be taken to mean the side of the target that the projectile strikes or the direction that matter flows in the collision. Either way, determining the direction requires using the particles in the collision themselves, of course. The result is that one cannot differentiate between negative (attractive) and positive (repulsive) flow. Figure 4.10 shows the difference between experimental and theoretical study of directed flow. Because the theorist knows the reaction plane a priori, he or she can assign negative flow correctly. However, the experimentalist only sees positive flow, based on the reaction plane direction’s determination (see Sec. 3.2). There- fore, the balance energy corresponds to the minimum of the flow excitation function, as opposed to the crossover energy. Alternatively, the experimentalist can reflect 76 Theog Experiment 3 e G + Q I ’ 5 Beam E Beam Energy Energy Figure 4.10: Directed flow near the balance energy, as determined by (a) theoretical and (b) experimental studies. Experimental studies of flow cannot discern between negative and positive flow because the reaction plane is determined using the same particles. points below the minimum about the art-axis to represent the attractive regime, but a systematic error is introduced since there are one or two points whose reflection is questionable. The other caveat to measuring the balance energy is the inherent difficulty of trying to measure “zero,” since the flow should be zero at Ebal. Because there are a finite number of particles in the collision, and because of imperfect detectors and imperfect angular resolution, detected flow will be nonzero even at the balance energy [52]. To circumvent this effect and to avoid autocorrelation, each detected particle is removed from the reaction plane determination (see Sec. 3.2), resulting in N separate reaction planes for an event with multiplicity N. 4.3.2 Transverse Momentum Analysis The technique for extracting the directed flow at midrapidity is due to the work of Danielewicz and Odyniec [52] who first proposed transverse momentum as a way of isolating collective motion in high-energy nuclear reactions. Previous methods, such as two-particle correlations [70] and sphericity analysis [21], were hindered by 77 statistical fluctuations and finite-particle effects. In the case of sphericity analysis, where any anisotropy in the reaction is estimated by measuring how spherical the particles’ momenta are on an event-by-event basis, it is easy to see that any spherical momentum distribution of a finite number of particles will yield a reaction plane and non-zero flow, even if generated randomly [52]. In contrast, the transverse momentum method successfully isolates collective mo- tion by: 1) treating each particle individually and determining a reaction plane from the remaining particles; 2) separating participant and spectator matter in the colli- sion by plotting vs. the rapidity; and 3) summing over many events to distinguish between statistical fluctuations and flow effects. The transverse momentum method has been the standard technique for virtually every directed flow analysis over the entire range of available beam energies. The method is quite straightforward. For an event with N detected particles, each particle is treated as a “particle of interest” (POI) and removed from the de- termination of a reaction plane (see Sec. 3.2). In the reaction plane determination, a direction is assigned to be positive based on the summed transverse momenta of the particles. The transverse momentum ng of the POI is then projected onto the reaction plane: where pf is the transverse momentum of the POI and Q,- is the unit vector of the reaction plane for that POI. Since the reaction plane includes the z-axis, Q, lies in the transverse plane (icy-plane in Fig. 4.9). Expressed another way, P? = ID;L | COS(¢.- - $11.13), (4-2) where d),- and (15,31) are the azimuthal angles of the POI and the reaction plane, respec- tively. 78 r--—-o (MeV/c) MId-rapldlty (ch=o) Figure 4.11: Schematic of the transverse momentum method for determining directed flow in symmetric nuclear collisions. The slope of a least-squares fit over a range of midrapidity (p3) corresponds to the flow. A strong flow signal is characterized by particles in the forward hemisphere of the reaction (ycm > 0) With mostly positive pf ’s, and particles in the backward hemisphere (yam < 0) with mostly negative pf’s. The rapidity for each POI is calculated and transformed to the collision’s center-of-mass, and P013 are then placed into rapidity bins. For an experimental run with Nrun particles of a certain type (such as p or a) and rapidity bin, (pm) = ' (4.3) where (103) is the average transverse momenta. Figure 4.11 shows a rough schematic of the transverse momentum method. (1),) is plotted vs. (y/ypmj)cm, the center-of- mass rapidity normalized by the projectile rapidity. Under the useful normalization, ypmj = 1 and ymg = -1. The directed transverse flow is defined as the slope of a least squares fit to (p3) over the midrapidity region. Before presenting the Au+Au transverse momentum analysis, detector acceptance effects need to be considered. To investigate acceptance effects on the analysis pre- sented in this chapter, a two-dimensional histogram of pf vs. (y/ypmj)cm is generated for the data. Also, a simple event generator is used to simulate isotrOpic events which are passed through a software replica of the detector array, as discussed in Sec. 3.3. 79 Comparing the data to “filtered” isotropic events will uncover any unexpected accep- tance issues. Figure 4.12 shows the two-dimensional acceptance plots for (a) 38.3 MeV/ nucleon Au+Au data, (b) unfiltered simulated isotropic events boosted to 40 MeV/nucleon, and (c) filtered isotropic events. The acceptance at backward rapidities is hindered by detector energy thresholds. This is due to the low lab-frame energies that the particles at backward rapidity strike the detectors. Preliminary transverse momentum analysis showed a strong effect due to these acceptance effects. In particular, the direction of the deduced reaction plane was inaccurate, as judged by the negative slope of (1);) obtained for all but the highest beam energy. Therefore, to minimize acceptance bias, only particles in the forward hemisphere of the collision (ycm > 0) are used in the transverse momentum analysis presented here. Assuming forward/backward symmetry in the reaction, the (p;) values are reflected about ycm = 0 for graphical purposes, and the least squares fit is carried out only for the actual (and not the reflected) (px). The practice of reflecting (pr) values to regions of poor acceptance is very common at higher energies [66, 71]. Figure 4.13 shows the average transverse momentum (p1) plotted as a function of the reduced rapidity for Z = 2 POI in Au+Au collisions at all of the available beam energies. Data are for the most central impact parameter bin ((b/bmax) = 0.28, see Sec. 3.1). In these plots, open squares represent experimental values, and closed squares are the experimental values reflected about yc.m. = 0 by assuming forward/ backward symmetry. The error bars are statistical, and the solid lines cor- respond to linear least square fits for the midrapidity region 0.0 S (y/ypmj)c.m_ S 0.5. Fragments emitted in this region are emitted from the excited participant volume cre- ated by the projectile-target overlap. The slope of this fit is defined as the directed transverse flow, which is a measure of the amount of collective momentum transfer in 80 2000 1000 -1ooo — . _2000 111111111111111111 l 1l1111l11111 1111111 -1.5 -1 -0.5 0 0.5 1 1.5 2 r1: 2000 1 000 Px (MeV/c) O -1000 1000 -1 000 11111 1 l1111l1111 -200011111111111 1 -2 -1.5 -1 -o.5 o 0.5 1 1.5 2 (V’Ypropm. Figure 4.12: The acceptance of the 47r Array for Z :2 fragments in Au+Au collisions at 38.3 MeV/nucleon for: (a) data, (b) unfiltered isotropic simulation, and (c) filtered isotropic simulation. The acceptance is measured in terms of the transverse momen— tum 112 in the reaction plane vs. reduced c.m. rapidity. The detection efficiency is weak at backward mid-rapidity, and the effect is reproduced well by the software filter. 81 Px (MeV/c) 5 G 24.5 MeV/n [[Iva‘ IIJIIIIILIIIII 28.2 MeV/n 3.3 l MeV/n I I I L I I I I l I I I I I I D Erfl NIH] | . IIII : :38. 3 MeV/n _ I I I I I I I I I I I I I I I O - 348.4 MeV/n , ; $44.5 MeV/n l I I I I l I I I I I I I I I I — I IIIIIMLLIIMLLIIII 1 353.5 MeV/n fi _ 0 I I I I I I I I I I I I I I I 57.6 MeV/n I ..1' . [ a data "- - reflected I IIILIIIIIIIIIIII - - r 1 ‘ -l 0 l -l 0 1 (y/yproj)c.m. Figure 4.13: Average transverse momentum vs. the reduced rapidity in near-central Au+Au collisions ((b/bmx) = 0.28). Z = 2 is the flow particle of interest. Open squares are experimental measurements, and solid squares are reflections about y = 0 assuming forward/ backward symmetry in the collision. 82 the collision. Clearly the slope is largest for the lowest (Ebeam = 24.5 MeV/ nucleon) and highest (Ebeam = 57.6 MeV/nucleon) beam energies. The transverse momentum is expected to pass through pat = O at yam. = 0. The negative offsets in Fig. 4.13 has been seen previously in many systems [58, 26] and is attributed to two experimental biases. By removing the flow particle from the reac- tion plane determination (to avoid autocorrelation), an inherent lack of momentum conservation is present in the assigned reaction plane direction. However, this effect is small, because the 47r Array does not detect all particles. Furthermore, the expected reduction of this effect with increasing system mass is not seen experimentally. A larger effect results from the likelihood of double hits in detectors which lie in the direction of the reaction plane. If a flow particle of interest (P01) is directed in the reaction plane, it is more likely to contribute to a double hit and be undetected than a POI directed to negative angles. This effect was seen in EOS data [72] and corrected by using negative-rapidity particle spectra and by considering the two-track resolution. However, these corrections are more difficult in a phoswich array such as the 47r Array, as the location of a particle cannot be pinpointed precisely enough; see Sec. 3.3.1. Plotted in Figs. 4.14 and 4.15 are (pm) vs. (y/ypmj)c,m, for Z = 1 and Z = 3 POI fragments, respectively. For the case of Z = 1, (p35) is nearly flat at midrapidity for all beam energies. This is attributed to weaker flow for lighter POI, as expected from previous experiments. The extracted flow values (slopes from the transverse momentum plots) are plotted vs. incident beam energy in Figure 4.16 for Z = 1, 2, and 3. For Z = 1, the extracted flow is weak for all of the measured energies. This is partially due to the well-known fragment mass dependence of flow and the higher energy thresholds for Z = 1, since the Bragg curve counters in the main ball of the 47r Array do not detect these; see 83 Px (MeV/c) : '3 O 24.5 MeV/n iTTIT 28.2 MeV/n ( IIIJIIIII‘IIIIIII Va‘W‘afiWfi . ' '9 It 4 I I I I I ;I I I I I I I I I I :IgLJ I I I I :L I I I I I I I J I 33.1 MeV/n :- 38.3 MeV/n IIIIIIIIIJkIIJIII IIIITUII ooooooooooooooooooooooo ‘oouuu-an-oou-ano-unonunion-ooou-nuoso ......IIOIIDOCIIIOIIDOIOI§IIOIIIOIIIIII..I...IIIOOIOIOODIICOIIOO‘ 44.5 MeV/n 48.4 MeV/n '~W%Wfi' IIIIIIIIIIIIIIIIIJIIII_ILJ'IIIIIIIII I I I l I I I I I I I I I I I I -l 0 1 -1 0 l (y/yproj)c.m. 53.5 MeV/n 57.6 MeV/n IIIIIIIILILIIIIII Figure 4.14: Average transverse momentum vs. the reduced rapidity in near-central Au+Au collisions ((b/bmax) = 0.28). Z = 1 is the flow particle of interest. Open squares are experimental measurements, and solid squares are reflections about y = 0 assuming forward/ backward symmetry in the collision. 84 G ”$1: I I I l I I I I I l I L l+ 333.1 MeV/n IIIIIIILIIIIII 3.28 2 MeV/n oooooooooooooooooooooooooooooooooooooooooooooooooooooooo 344. 5 MeV/ {ll 1 3‘ ° ' I1.i.i..141..".3].|.1..!....|..|l ........................... %%-.%.MsX/.3." 13.5761“ . . -'l t MI; MMM .. .f .. {ll Eodata o .01 f - reflected ' I I I I I 1 Li I l l I I I I I I I I I I I I l I I I LLI l -1 0 l -1 0 l (y/yproj)c.m. Figure 4.15: Average transverse momentum vs. the reduced rapidity in near-central Au+Au collisions ((b/bmax) = 0.28). Z = 3 is the flow particle of interest. Open squares are experimental measurements, and solid squares are reflections about y = 0 assuming forward/ backward symmetry in the collision. 85 30 E_ a) 2:1 3_ b)Z=2 3_ c)Z=3 3 E E E 5+... % 20E- :- E '+ ..'+ E 5*} J'E I, q..- B 10 r r ,Ai :- ' """ a : - "'-§....+«I’ : o” i ' 2.1.1.1.IC.|.1.|.”.1.141L 304050603040506030405060 Beam Energy (MeV/nucleon) Figure 4.16: Extracted flow values in near-central ((b/bmax) = 0.28) Au+Au collisions for Z = 1,2,3 POI. Sec. 2.2.1. However, for Z = 2 and Z = 3 the plot clearly shows that the flow goes through a minimum. Because measurements are unable to distinguish between the negative attractive scattering which dominates below Ebal and the positive repulsive scattering which dominates above Ebala such a minimum is indicative of the balance energy for the system. The dotted curves are parabolic fits, included to guide the eye. To assimilate the experimental data with theoretical predictions and to account for the attractive regime below the balance energy, the flow data below the minimum are reflected about the :1: axis and plotted in Figure 4.17. The dashed (solid) linear least squares fits are with the 44.5 MeV/nucleon data point reflected (not reflected). For Z = 2, this corresponds to a balance energy of 42.5 :t 3.5 (43.1 i 3.6) MeV/ nucleon. For Z = 3, the numbers are 41.6 :1: 4.3 (41.9 i 6.1) MeV/nucleon. The errors above are statistical. However, the use of two different fits introduces a small systematic error of 1 MeV/ nucleon, resulting in a balance energy of 42 :l: 4 MeV/nucleon. The balance energy for Z = 3 is the same as for Z = 2 to within error bars, in agreement with previous experimental studies that showed no dependence on particle type [57]. 86 Flow (MeV/c) G 40 _ _ b) Z=2 30 20 r , 10 it. 5- .n.../ ; 5_ -30 1 l 1 l 1 l 1 1 l 1 l 1 l_1 " l 1 l 1 l 1 30 40 50 60 30 40 50 60 30 40 50 60 Beam Energy (MeV/nucleon) Figure 4.17: Extracted flow values in near-central ((b/bmx) = 0.28) Au+Au collisions for Z = 1, 2, 3 POI. Flow values below the minimum are reflected about the x-axis, and dashed (solid) lines represent linear least-squares fits with (without) the 44.5 MeV/ nucleon value reflected. 4.3.3 Directed Flow Over a Wide Range of Energies The 47r Array flow data presented to this point have been corrected for dispersion in the reaction plane determination, which arises from the fact that there are a finite number of particles in the collision. As discussed in Sec. 3.3.3, the dispersion correc- tion leads to an increase in the measured amount of directed flow. Figure 4.18 shows the eflect of the dispersion correction for Z = 2 P01. Solid squares indicate the ex- perimental measurements from Fig. 4.16, and open squares are dispersion-corrected. Once dispersion correction is made, the directed flow data presented in this chapter can be compared to Au+Au flow data taken over a wide range of energies. Table 4.1 lists other Au+Au experiments that have produced directed flow measurements over the past fifteen years. When plotting the various data on the same graph, it should be noted that small differences in analysis techniques are difficult to account for. Some of the differences present in a few of the experiments are: 87 N 9 D dispersion-corrected I not dispersion-corrected 1‘ U] ’+ s I I O. ’ I. ’ O I 3 I .-° I ’ 0.. H i """ + Flow(MeV/c) p.11 O TIIIIIVIITYTTIIIUIT 5 LllJllJlllllllllllllllllllllllLlllllll 25 30 35 40 45 50 55 60 Beam Energy (MeV/nucleon) Figure 4.18: Experimental flow values for near-central Au+Au collisions with Z = 2 as POI. Solid squares are directly measured values, and open squares are corrected for dispersion in the reaction plane determination. o impact-parameter binning o dispersion correction method 0 global experimental cuts on pi due to acceptance 0 inclusion/exclusion of POI in reaction plane determination o normalization of flow variable Figure 4.19, shows the directed flow normalized by the fragment masses over the beam energy range 25A MeV to 11.8.4 GeV. The 47r Array data agree well with the general trend from higher energies. The flow excitation function appears to peak at Ebeam ~ 2A GeV, and recent results from NA49 at CERN indicate near-zero directed flow in 208Pb+2°8Pb collisions at Ebeam = 158A GeV [71]. 88 Experiment ( Location l Date I Energy Range 1 Reference Plastic Wall / Ball Berkeley 1983 150 AMeV - 1.05 AGeV [50] EOS TPC Berkeley 1995 250 AMeV - 1.15 AGeV [65] FOPI Darmstadt 1997 100 AMeV - 800 AMeV [66] E877 Brookhaven 1997 11.8 AGeV [73] E895 Brookhaven 1997 2 AGeV - 8 AGeV [27] Table 4.1: A list of the other Au+Au directed flow experiments included in the excitation function of Fig. 4.19. 4.3.4 Summary In the remaining sections of this chapter, the BUU model is used to investigate two aspects of Ebal over the extended range of system mass: the increasing strength of the Coulomb interaction and its influence on the mass dependence, and the loss of sensitivity of the balance energy to both the impact parameter and the in-medium cross section. The latter is a significant finding, since the equation of state is isolated for the first time using the balance energy. 4.4 The BUU Model The ability to extract information about the equation of state from studying nuclear collisions rests upon the agreement between observables found in experimental data and produced in numerical models. A model derives its strength from its ability to reproduce experimental quantities when initial conditions are varied. For example, it is well known that the total number of fragments produced in a collision increases with decreasing impact parameter, increasing beam energy, and increasing system size. A model that claims to simulate accurately fragmentation in nuclear collisions must reproduce these dependencies. As discussed in Sec. 1.5, a numerical implementation of the Boltzmann-Uehling— 89 300: . ...... . . ......, ~ f-fi..., ‘ z o 4n,NSCL E 250 g u FOPI data, 815 4A 3 E A EOS data, Bevalac + : 200 ..- o E895,AGS '3 A 5 k E877, AGS * 5 a 150 -_— $ l 1 % t E ' = E 100 :— ¢ -2 g 5 ¢ 3 : ¢°° : 0 _ . : ¢° : _50 2.. ¢ .2 E¢ E _100 " . .11...1 . . ......1 . . 11141.1 ‘ 102 103 10“ Beam Energy (MeV/nucleon) Figure 4.19: Directed flow normalized by the fragment masses plotted as a function of incident beam energy for several Au+Au experiments. 90 Uhlenbeck (BUU) model [54, 74] has been successful in reproducing various collective flow phenomena such as the balance energy. The BUU model propagates test particles (which represent nucleons in a collision) through phase space as they interact with each other via the inter-nucleon potential: a hard-core repulsion at short distances (1‘ S 1fm), a strong attraction at intermediate distances (1 S, r S, 2fm), and a weak attraction at long distances (7‘ > 2fm). The Pauli Exclusion Principle is approximated by monitoring the densities for both coordinate-space and phase-space. BUU is useful when the behavior of nuclear matter collectively is of interest; however, BUU does not coalesce nucleons into fragments and therefore is not suited to reproducing multiplicity distributions or fragment energy spectra. In the current implementation, the inter-nucleon potential is split among two mechanisms: a nuclear mean field for low-momentum processes, and hard scattering for modeling the strong, repulsive, high-momentum processes. The form of the nuclear mean field potential U (p) is U(p) = a (155) + b (i), (4.4) where U represents the average of the potentials of the surrounding matter on a single nucleon. In Equation (4.4), pg is normal nuclear matter density, a is attractive, b is repulsive, and a > 1. The parameters a, b, and a are constrained by knowing the density p0 and saturation binding energy of ground-state nuclear matter and by initializing all nucleons with the Fermi momentum pp: 2 K =9(%%+a+ab), (4.5) where K is the compressibility of nuclear matter [54]. (See Sec. 1.2.1 for a discussion of the compressibility as the relationship between energy and density.) Typical values in the current study for the compressibility K are shown in Table 4.2 [19]. A soft EOS corresponds to K = 200 MeV, and a hard EOS corresponds to K = 380 MeV. 91 Model K (MeV) a (MeV) b (MeV) a Soft 200 -356 303 7/6 Medium 235 -213 164 4/3 Hard 380 -124 70.5 2 Table 4.2: Parameter sets used for the mean field U. For a given value of K, [a] is always greater than lbl. Therefore, when p = p0, U (p) is attractive. As p increases, the repulsive term increases in relative strength, and above a density p = H)“ p0 (4.6) U (p) is repulsive. This density-dependent effect contributes to the crossover from negative to positive flow with increasing beam energy. The hard scattering of nucleons at short distances is parameterized by am, 2 Ufree (I + a—p—) , (4.7) P0 where am, is the in-medium nucleon-nucleon cross section, Ufree is the nucleon cross section in vacuum, and a is a parameter between 0 and -1 [18]. The efl'ect of reducing the cross section is to make nucleus-nucleus collisions less repulsive on average, since there are fewer collisions. The compressibility K and in-medium cross section am, are the main quantities of interest in the BUU model. It is hoped that K and am, can be constrained by comparison to experimental data. Modeling the Balance Energy with BUU The BUU model is well-suited for investigating the energy dependence of directed flow, since the inter-nucleon potential is represented by density-dependent variables (K and am). At low energies, the attractive portion of the nuclear mean field U (p) 92 dominates the collision dynamics, leading to negative directed flow in the reaction plane. BUU successfully reproduces this effect. Figure 4.20 shows BUU calculations in coordinate space projected onto the reaction plane for 40Ar+45Sc at an incident energy of 40 MeV/ nucleon and a reduced impact parameter of b/bmam = 0.28. Each frame represents a time step of 15 fm/c (~ 5 x10”23 8) in the evolution of the collision. After maximum density is reached in the overlap region of the nuclei, participant and spectator matter deflects mostly to negative scattering angles, indicative of negative flow. As the obtained density p increases with increasing beam energy, the repulsive portion of the mean field U (p) and the hard scattering due to 0m, gradually dominate the reaction dynamics. Figure 4.21 shows BUU calculations for 197Au+197Au at an incident energy of 150 MeV/ nucleon. Each frame represents a time step of 20 fm/c (~ 6 x 10‘23 s). After maximum compression, nuclear matter mostly deflects to positive scattering angles, and the signal for positive directed flow is clearly evident. The procedure for extracting the balance energy for a given system and set of input parameters (K, dun, b/bmax) is the same as for experimental data, except that for BUU the reaction plane is known a priori and exclusion of a flow particle of interest is unnecessary. Nucleons are placed into rapidity bins, and the average transverse momenta projected into the reaction plane, pm, is plotted as a function of the reduced c.m. rapidity, (y/ypmj)cm. Figure 4.22 shows the results for Au+Au collisions for several energies encompassing the balance energy. The flow at each energy is extracted by fitting the midrapidity range |(y/ypmj)cm] < 0.5 to a least squares fit, and the balance energy is determined by plotting the extracted flow values as a function of incident energy. 93 vi,- 15. IIIIVIIIY1‘I—I‘l' IIIVIYIIIIII'II "l""l"'ll" lJllllllllllll IIIIIIIIIIIILL 111-11.11111111 Figure 4.20: BUU calculations in coordinate space projected onto the reaction plane for 40Ar+45Sc at an incident energy of 40 MeV/ nucleon and a reduced impact param- eter of b/bmax = 0.28. Each frame represents a time step of 15 fm/c (~ 5 x 10‘23 s) in the evolution of the collision. 94 'Ivvl [Irrl1TT11 Ill! IIIII T ll—llLllLllllll l 'l"_llA-.‘"l Figure 4.21: BUU calculations in coordinate space projected onto the reaction plane for 197Au+‘97Au at an incident energy of 150 MeV/nucleon and a reduced impact parameter of b/bmax = 0.28. Each frame represents a time step of 20 fm/c (~ 5 x 10'23 s) in the evolution of the collision. 95 Px (MeV/c) .0q S 11111 I illLlll4lllllllllJlll 20 r 7o MeV/n [[0 :J so MeV/n [[]* ++l+ll +++l]l ............................. Figure 4.22: BUU calculations of the average transverse momentum vs. the reduced rapidity in near-central Au+Au collisions (b/bmax = 0.28). 96 4.5 Effect of Coulomb Interaction on the Balance Energy As discussed in Sec. 4.2.3, Zhou et al. investigated the effect of excluding the Coulomb interaction from VUU calculations of the balance energy for light systems. The general finding was that Ebal is higher when Coulomb effects are turned off, which is expected since the Coulomb interaction acts to increase the repulsive dynamics of the collision. However, the VUU calculations were carried out only for systems up to A = 200. Westfall et al. [57] postulated that the nuclear mean field can be associated with the surface of the two interacting nuclei and hence should scale as A2/3. Also, the re- pulsive nucleon-nucleon interaction should scale as A, the number of nucleons present. They observed that the competition between these two effects leads to the A‘l/3 de- pendence in the experimental data. The role of Coulomb repulsion was not included in their discussion because of the small system masses. Figure 4.23 shows the bal- ance energy plotted as a function of the system mass for an accumulated set of 47r Array datas, represented by open squares. Data are for near-central events with (b/bmax) = 0.28. A power law fit of the form Ebal o< A‘T agrees well with the data. However, TData = 0.44 :l: 0.03, which is not in agreement with the suggested A’l/3 dependence. To investigate the role of Coulomb forces in the system mass dependence of Ebal, BUU calculations were performed for systems of various masses, from N e+Al (A = 47) to Au+Au (A = 394). For the present calculations, a soft equation of state (K = 200 MeV) and a reduced in-medium cross section (a = —0.2, see Eq. 4.4) was used. The Coulomb interaction is included or excluded by way of an input flag to the k 8The 12C+uC point from the original mass dependence paper by Westfall et al. [57] is excluded from Fig. 4.23. 97 T l l I I I l I r A 1‘ ., e 3 100 s‘ _. ‘2) 90 I. .3 x‘ n - \ > 80 - ,‘ - G) u. E 70 - . C ‘s .5. a + Q l' ‘\ ...°°. " a 60 ‘s‘ ..’+ o x g 50 - q .5. V BUU, NO Coulomb ‘ g A BUU, Coulomb ON . 40 Li: Data, 41t-NSCL 4 0 Data, GANIL l L l l l I l l l 50 70 100 200 300 400 System Mass Figure 4.23: Balance energy as a function of combined system mass. Open squares/ circles are experimental data, and triangles represent BUU calculations with and without the Coulomb interaction included in the calculation. 98 BUU program. Figure 4.23 also shows the results of the BUU calculations (solid triangles). With the Coulomb interaction included in the calculation (dashed line), the simulation agrees well with the data (TBUU = 0.41 :l: 0.03). When the Coulomb interaction is removed (dotted line), 7' z: 0.31 :l: 0.03, very close to the anticipated value of 7' = 1/3 from scaling arguments. 4.6 Au+Au as a Probe of the Nuclear EOS The disappearance of directed flow was suggested initially as a powerful probe of the EOS [76]. However, numerous model calculations have demonstrated that the balance energy, while sensitive to the nuclear compressibility K, was also sensitive to the in-medium cross section am, [77, 26, 67]. Zheng et al. recently used an isospin- dependent BUU model for the 48Ca—l—‘mCa system to show that the same balance energy is obtained with a stiff EOS and vacuum cross section as with a soft EOS and reduced 0,", [79]. In fact, Ebal was shown to have a weak dependence on K for light systems [69]. Also, Eba] was shown to depend strongly on the impact parameter [58, 68, 69], further hindering study of the EOS. However, all of these studies were carried out for systems with A f, 200. In this section we show for the first time that the compressibility K can be isolated using the balance energy. We reveal that the impact parameter dependence of Ebal weakens as the system mass increases and nearly vanishes for a heavy system such as Au+Au. We employ BUU model calculations [78, 18] to show that the dependence of Eb,“ on 0m, weakens as well for heavy systems. These findings, together with a strong dependence on the compressibility, allow for the first time extraction of EOS properties with the balance energy, which is particularly beneficial because the balance energy is a relatively model-independent observable [26]. Finally, the extended system mass dependence of Ebal can then be used to examine the magnitude of an". 99 4.6.1 Independence of Ebal from Impact Parameter Previously, the balance energy was observed to increase linearly as a function of impact parameter b for light systems [58]. This dependence was attributed to the need for a larger incident energy to overcome effects of the mean field as the participant zone gets smaller (with decreasing b). Figure 4.24 shows the balance energy as a function of the reduced impact parameter b/bmax for the four systems Ar+Sc, Ni+Ni, Kr+Nb, and Au+Au. Data were recorded at the National Superconducting Cyclotron Laboratory with the 47r Array in a consistent configuration which included a 45-element High Rate Array in the forward direction. Details of the experimental setup can be found in Chapter 2 and in Refs. [58, 59, 80]. Not all impact parameter bins are shown due to detector acceptance effects at low incident energies and for less-central collisions. The lines represent linear fits to the data, included to guide the eye. As the system mass increases Eb,“ exhibits a weaker dependence on b/bmax, and for Au+Au the dependence nearly vanishes. This could be due to the increasing role of the Coulomb interaction as b/bmax increases, counteracting the attractive mean field. Therefore, for near-central collisions b can be regarded as a model-independent parameter when comparing Ebal to model calculations for Au+Au. 4.6.2 Independence of Ebal From am, We present results of a comprehensive and systematic study of the balance energy using the BUU model for a wide range of system sizes, 63 < A < 394. For each system size, several energies near the anticipated balance energy were chosen, and several combinations of (K, oz) were selected: K = 200, 235, and 380 MeV and 0: = 0, -0.1, -0.2, and -0.3. Each set of parameters was calculated using four different random number seeds to build statistics and to minimize any eflect from the choice of seed. For all systems, an impact parameter corresponding to b/bmx = 0.28 was 100 140 - H N G l 1—1 8 I I I Balance Energy (MeV/nucleon) 80 — r L -+ 60 ._ fl [ l A J 40 ]— 1 Yl ] l l 1 fl 0 2 0 3 0.4 0.5 0 6 0.7 0.8 b/bmax Figure 4.24: Balance energy as a function of the reduced impact parameter for four systems. Data are taken with the 41r Array, the linear fits are intended to guide the eye. 101 fir K = 200 Mev (som' b/bmx = 0.28 (most central) § —4 8 a 6 Q G O I O I.- o O Q ~‘ 0 ‘ ‘Q ~ ~ . ~ Balance Energy (MeV/nucleon) S I it (X=-.3 50‘ A (1:..2 ‘ D =-.l O freecrosssection 40 1 1 E 1 1 l 1 60 80 100 200 300 400 System Mass (A) Figure 4.25: BUU model calculations of the mass dependence of Eb,“ for values dif- ferent reductions of the in-medium cross section ann. Experimental measurements of Eb,“ are shown as solid squares. The calculated balance energy for Au+Au depends very weakly on the value of a. used in order to compare to the most central bin of our experimental data (for which the mean b/bmx z 0.28). Note that momentum-dependent mean fields were not included in the present numerical implementation, since Ebal is affected very little by momentum dependence at low beam energies and in near-central collisions [60]. Figure 4.25 shows balance energies extracted from BUU calculations as a function of the system mass for four different cross sections, assuming a soft equation of state. Lines represent power law fits to the simulated values, as suggested by the experimen- tal mass dependence of Ebal [57, 81]. The error bars are associated with the linear fit of the flow excitation function. The balance energy clearly shows a strong depen- dence on a for light systems, in agreement with previous theoretical work. However, as the system size increases, the a dependence of Eb,“ nearly vanishes. For A = 394 102 § l I I fl C o K3380 MeV (stiff) ~ ’5‘ 95 L I x=235 MeV ; 8 : A K2200 MeV (soft) j '5' Z I Dltl : 3 9° .— t \ _ _. Q 85 L La". I l E 80 L -' ”3.5:-mg; —‘ g F +‘,-u .’.-- 3 “I : ,.-- : 2 7o -— —: I’ll)...u =0.28(Inost central) 65 1 1 1 1 0 0.1 0.2 0.3 41 Figure 4.26: BUU balance energies plotted as a function of the arm reduction parame- ter —a for three different values of the compressibility K for Ar+Sc. The experimental measurement is represented by a flat line with error bars. (Au+Au), all of the BUU balance energies are well within error bars. The gradual loss of the sensitivity of Ebal to am, can be attributed to the change in the collision dynamics at lower beam energies and for larger system masses. As A increases, hard scattering processes play a lesser role in the dynamics of the collision due to Pauli blocking [68]. Therefore, for heavy systems the balance energy is due mostly to combined effects of the attractive mean field and the repulsive Coulomb interaction. Without the Coulomb interaction included in the BUU calculations, Eba, for Au+Au is ~10-15 MeV/nucleon larger (see Sec. 4.5). 4.6.3 Isolation of Nuclear Compressibility Since the balance energy for Au+Au is nearly independent of am, (over a range of a’s) and the impact parameter, BUU predictions can be compared directly to the exper- imental value of the balance energy to estimate the nuclear compressibility K. This 103 Q u: ,_ fl I I 1 d : o K=380 MeV (stiff) : ‘70 :- I K=235 MeV 1 t A K3200 MeV (soft) : 65 E 1 K=180 MeV g z I Data ‘ S l | _',_ _.'_ 1111L+lllll Balance Energy (MeV/nucleon) a ‘3 ‘2: 1 [pr 1 l n 3 I |__ b/bm = 0.28 (most central) j l 1 l l 0 0.1 0.2 0.3 -(1 U 0a Figure 4.27: BUU balance energies plotted as a function of the arm reduction pa- rameter —a for three different values of the compressibility K for Au+Au. The experimental measurement is represented by a flat line with error bars. is in contrast to lighter systems for which K could not be isolated. Figure 4.26 shows BUU balance energies for Ar+Sc (A = 85) as a function of the cross section reduction parameter —-a for three different values of the nuclear compressibility: K 2200, 235, and 380 MeV. Dashed lines are included only to guide the eye. The single experimen- tal value is plotted as a horizontal line with error bars. Depending on the a selected, all three K’s can agree within error bars of the experimental value for Ebal. In Figure 4.27, BUU balance energies for Au+Au are plotted vs. —a, and again the experimental value is represented by a horizontal line. Only K = 200 MeV, which corresponds to a soft equation of state, falls within error bars of the experi- mental measurement. The approximate value of K is in good agreement with other measurement techniques; see Table 4.3. Studies of the isoscalar monopole resonance indicate K =200:l:20 MeV [82], while recent Thomas-Fermi model calculations pointed to K =234 MeV [8]. Also, Pan and Danielewicz estimated that K lies between 165 104 Technique K (MeV) Date Author(s) Reference Isoscalar Monopole Resonance 200 i 20 1999 Chung et al. [82] Giant Monopole Resonance 231 :1: 5 1999 Youngblood et al. [83] Linear Momentum Transfer 210 1996 Haddad et al. [84] Thomas-Fermi Model 234 1998 Myers and Swiatecki [8] Transport Model 165-220 1993 Pan and Danielewicz [85] Table 4.3: A comparison of recent values for the compressibility K for a variety of measurement techniques. and 220 MeV by studying the dependence of sideward flow on multiplicity [85]. 4.6.4 Extrapolation of am, Once a value for K in the BUU parameterization is established, the system mass de- pendence can be used to investigate the magnitude of onn’s deviation from the vacuum cross section. Figure 4.25 shows the experimental data for the mass dependence of the balance energy (filled boxes). A reduction of a = —0.2 in the cross section agrees well with the data for light- and medium-sized systems, while a = -O.3 most closely reproduces the slope (power-law exponent) on the experimental mass dependence. 4.7 Summary In conclusion, we have presented the disappearance of directed transverse flow for Au+Au using the MSU 47r Array. Our results indicate that the balance energy is 42 :l: 4 MeV/nucleon. We have also shown that Ebal scales as A" for heavy systems as well as light systems. When the Coulomb interaction is removed from the BUU model calculations, 7' z 1 / 3, which supports previous scaling arguments about the competing roles of the nuclear mean field and nucleon-nucleon scattering. The measurement of the balance energy for Au+Au presented in this section extends the 105 system mass dependence of Ebal by a factor of two. We have shown that the impact parameter dependence of the balance energy nearly vanishes for heavy systems such as Au+Au, which we attribute to the increased strength of the Coulomb repulsion counteracting the attractive mean field as b/bmaLx increases. We have also performed a systematically complete set of BUU calculations to show that the sensitivity of Ebal to the in-medium cross section weakens as the system size increases and nearly disappears for Au+Au. This eflect is ascribed to the lesser role of hard scattering processes at lower beam energies. These two findings allow the EOS parameter K, the nuclear compressibility, to be estimated using the balance energy for the first time. BUU calculations for Au+Au with K =200 MeV, corresponding to a soft EOS, produce balance energies which lie within error bars of the recently measured value. The experimental mass dependence is then employed to estimate the cross section reduction parameter a to be -0.2 to -0.3. 106 Chapter 5 Disappearance of Elliptic Flow at Intermediate Energies In the past year, two major elliptic flow developments in very different energy ranges have generated a lot of discussion. In the 1-10 GeV/ nucleon range, measured changes in the shape of the elliptic flow excitation function have been used to suggest a softening in the nuclear equation of state. And at intermediate energies, transport model calculations of elliptic flow near the balance energy show a strong dependence on the nuclear equation of state. (See Sec. 1.5 for introductory discussion.) This chapter is structured similar to the previous chapter, except that the model calculations presented in Sec. 5.4 are published theoretical calculations. The full Fourier analysis of midrapidity emission patterns is carried out for four systems: 40Ar+45Sc, 58Ni+5’3Ni, 86Kr+93Nb, and 197Au+197Au. The first study of the mass dependence of the transition energy, introduced in Sec. 1.5.2, is presented, as well as elliptic flow measurements for Au+Au made over a wide range of energies. Finally, isospin-dependent model calculations are compared to data for our lightest system, Ar+Sc. 107 5.1 History of Elliptic Flow The first mention of midrapidity emission perpendicular to the reaction plane occurred in a 1982 theory paper by St6cker et al. [86]. Using a fluid dynamical model to study reactions at EM, 2 390 MeV/nucleon, they calculated that angular distributions of protons emitted from near-central collisions show out-of-plane jet structures at 0““, = 90°,4) = i90°. They attributed the out-of-plane emission to the fact that compressed matter perpendicular emitted in the reaction plane is hindered or blocked by “spectator” matter. In the same month that the Stéicker et al. paper appeared in Physical Review C, a paper by Gyulassy, Frankel, and Stbcker was published in Physics Letters B which employed a kinetic energy tensor to analyze flow patterns for the first time [49]. A 3 x 3 flow tensor is constructed from the emitted particles’ momentum components, weighted inversely by the mass of the fragments. The eigenvectors of the tensor represent the principal axes of the flow ellipsoid, and its shape and orientation relative to the reaction plane and the beam axis can be used to measure in-plane and out-of- plane emission. Experimentally observed out-of-plane emission, termed squeeze-out, was first ob— served in 1989 at nearly the same time by two competing collaborations. The Diogene Collaboration at the Saturne synchrotron in Saclay observed slight peaks in the az- imuthal distribution of particles at midrapidity in 800 MeV/ nucleon Ne-induced re- actions [87]. At the Bevalac in Berkeley, the Plastic Ball / Wall group observed out-of- plane emission in Au+Au collisions at 400 MeV/ nucleon [88]. They also characterized the emission patterns in terms of the kinetic energy flow ellipsoid, which could account for directed flow as well as out-of—plane flow. The strength of the Plastic Ball/ Wall Collaboration’s analysis technique was their use of the transverse momentum method to determine the reaction plane. This al- 108 . 0.5 0.3 _ + b.xp85‘m [_ b.‘pt7'9m 40 60 80 4o 60 80 Elnc(MeV/u) Figure 5.1: First experimental evidence for the transition from in-plane to out-of- plane emission in intermediate-energy collisions. Data are taken with the Nautilus detector at GANIL [92]. lowed the flow ellipsoid to be rotated into the reaction plane initially, followed by a second rotation so that the ellipsoid’s primary axis coincided with the beam axis. Without this two-fold rotation, the cp-distributions would show irregularities and would not be peaked at (I) = 90°,270°. The same collaboration also characterized squeeze-out as a function of projectile energy and mass, as well as the rapidity de- pendence by using a novel ratio of out-of-plane/in-plane emission [91]. The transition from in-plane to out-of—plane emission was first observed using 64Zn+58Ni collisions at the GANIL facility by the NAUTILUS collaboration in 1994 [92]. Figure 5.1 shows their data in the form of excitation functions of a2, a measure of the anisotropy. In-plane emission is indicated by positive a2, and out-of-plane emission is indicated by negative oz. The energy of transition increases as the impact parameter increases, similar to the balance energy for light systems (see Sec. 4.6.1). Also, the MINIBALL/ALADIN collaboration observed the onset of out-of—plane emission (the experiment was conducted at GSI) in Au+Au collisions of 100 MeV/nucleon [93], where out-of—plane emission is seen for central collisions while peripheral collisions 109 — 0 5E . o e ;: 009‘ 00 O: x 90" o.‘ ..... -Q-i - - - - - - — -&—I- * - - - - - - - 4 I: 1 4D 1 1» 4 null— 4 qt 4 l: 1 4) 4 ".7 1 1: 400 MeV 1 1) 4. l ‘ K I I a: : ' d d 1 1 1 3f 1- - some... 0.0 0.5 1.0 Figure 5.2: Impact parameter dependence of the anisotropy ratio, in which positive R1 values correspond to predominantly out-of-plane emission. Data are taken with the Miniball/ALADIN detector [93]. clearly show in-plane emission. Their comparison to Boltzmann-Uehling—Uhlenbeck calculations showed that as the beam energy is increased, the value for the reduction in the in-medium nucleon-nucleon cross section appears to change; see Fig. 5.2. In the past couple years, the field of elliptic flow1 has expanded to include mea- surements at Bevalac, SIS, AGS, and recently SP8. The out-of—plane energy regime, extending roughly from 80 MeV/ nucleon to 4 GeV/ nucleon, is especially interesting because information about the early high-density stage of the collision is preserved in the detected particles. In fact, the ratio of out-of—plane to in-plane emission appears to be very sensitive to some of the parameters of the equation of state in this energy 1The term elliptic (or elliptical) flow was introduced in 1997 by Heinz Sorge [89], a theorist at SUNY—Stony Brook who studied the elliptical emission patterns at midrapidity that were measured by the E877 Collaboration at Brookhaven [90]. 110 range, as demonstrated by Hartnack et al. [94]. But of greater interest is the recent finding by the E895 Collaboration published in 1999 [27]. Using elliptic flow measure- ments of protons for Au+Au from 2-8A GeV, a crossover from out-of-plane to in-plane flow is seen. Comparison of the crossover energy to a relativistic Boltzmann equation suggests a softening of the nuclear equation of state from stiff to soft around 4A GeV. Such a softening of the EOS could result from a number of effects, including the pos- sible onset of a nuclear matter phase change, discussed by P. Danielewicz et al. [95]. At intermediate energies, Zheng et al. at Texas A&M used isospin-dependent BUU calculations to show that the first crossover from in—plane to out-of—plane emission is dependent on the EOS [79]. This will be discussed in the next section. 5.2 Motivation for Present Study The azimuthal anisotropy of particle emission near 90° in the center-of—mass frame (which signifies elliptic flow) is an attractive quantity for two reasons. First, measur- ing the azimuthal distribution of protons and fragments at midrapidity is a relatively easy task for the experimentalist. At intermediate energies, the only delicate part of the analysis rests in the reaction plane determination.2 Second, as stated earlier, frag- ments emitted out of the reaction plane at midrapidity are uninhibited by projectile- and target-like fragments after the collision. With the addition of Au+Au, data are available for masses extending from A285 (Ar+Sc) to A=394. The detector setup is nearly identical for the four analyzed data sets. Therefore, a systematic study of the mass dependence of the transition energy can be performed without taking additional data. 2Another slightly involved calculation at higher energies is the determination of the flow angle; that is, the angle between the major axis of the flow ellipsoid and the beam axis. However, at intermediate energies, the major axis nearly coincides with the beam axis, and rotating into the frame of the flow ellipsoid is unnecessary. See Ref. [97] 111 20 --fi,-...,....,....,-...,....,....,.-- .""" ..,....,.. ..,-- ,-.. ,fir. . r : _.-'Oj . “Co+"Co,b= me 15 »_ + ‘ _1 1.5 1 o stiff cos - - i . ~ ~ 0 was: .0... I i I ’ A so w: ? 10 ’_ .0 ,1" J 1 L— “ C] no mean field -1 O : Nil +X : : $‘1 : g 5 :h #39. v I,” E f f? l i o .1 A ' +' ., ~ . : U : C) "1‘ , 5 O _ .: > —5:~ ] [A f j >~ , lv 1 ‘1’ : I ' [ ,1" 1 : l 1 2 ~ 3 1 l « -o.5 ~ [g -.1 ., __ l . . ¢-------+ i : u. : - “Ca+“Co.b=2fm . . ['5 t5 + . 15 : f e stiff EOS : _1 f + j - r 0 soft EOS r - " . [(3‘) A soft £105 with 0.880... . : (b) l + 3 111 11_1 111111 L111 1.] 1 1 1A 1111 111111111111111111111 2020 30 4o 50 60 7o 88 90 106 '520 :50 4o 56 60 7o 80 90 100 Em (MeV/nucleon) E..." (MeV/nucleon) Figure 5.3: Isospin-dependent BUU calculations of the excitation function of (a) the proton directed flow and (b) the proton elliptic flow in 48Ca+48Ca collisions at an impact parameter of 2 fm. Taken from Ref. [79]. 5.2.1 Recent Theoretical Findings Recently, theorists working at Texas A&M studied proton elliptic flow in collisions of 48Ca+48Ca at energies from 30 to 100 MeV/nucleon using an isospin-dependent Boltzmann-Uehling-Uehlenbeck (IBUU) model [79]. Their initial hope was that the elliptic flow transition would not show the dual dependence on the compressibility K and in-medium cross section om, exhibited by the disappearance of directed flow for light systems (see Sec. 4.6). Figure 5.3 shows the results of the IBUU calculations.3 In Figure 5.3(a) the directed flow excitation function is calculated for three different sets of parameters. The same balance energy (F = 0) is obtained for a stifl' EOS and a soft EOS with a reduced om. However, as shown in Figure 5.3(b), the proton elliptic flow excitation function for both the soft EOS and stiff EOS changes from positive (in-plane) at low energies to negative (out-of-plane) at high energies, and the energy at which this 3Although they chose neutron-rich nuclei, effects due to isospin asymmetry on the proton elliptic flow are not very strong, as discussed in Ref. [96]. 112 MM Tl] I I Ir - . pp) > 0.8 V protons , - . _ 1 I A A=1+2+3 : _ 0 M4 - ' ~ ~ - A M3 « : cud-Nana : .- fl : : 0. W " -o.05_— . - add-plan 4“ r ' _ 4.05; i % .1 g. IIII v a 01— fl '9 01— a ‘2‘ 39 2 E005— — E i a saw— 1 I111 -0. 15 _— —_ p 4 ; a) : -o. 1 — ‘9 ~ r 1 l L1 1 l l 1 l l 1 1 l l l I p l l 1 l 1 1 1111 l A l I l 4 -0.2 _, .1 10 1 1O 1 Beam energy (Ge V/nucleon) Beam energy (Ge V/nucleon) Figure 5.4: Elliptic flow in Au+Au collisions from 90 MeV/nucleon to 1.49 GeV/nucleon as measured by the FOPI Collaboration at GSI—Darmstadt for (a) a high-pt cut and (b) all pt included. Taken from Ref. [98] transition occurs differs for the two EOS. Zheng et al. concluded that the transition energy is more sensitive to the EOS than am, and suggested that the study of both directed and elliptic flow at intermediate energies would allow the extraction of both K and am. For present consideration, the 40Ar—l-45Sc system is very close to 48Ca+48Ca in size and proton number. Therefore, comparison of experimental data to the IBUU calculations can be useful for extracting EOS information. 5.2.2 Previous Studies of Elliptic Flow for Au+Au Similar to the directed flow analysis of Chapter 4, a strong motivation exists for studying elliptic flow for the Au+Au system in particular. To date, Au+Au elliptic flow has been studied in several experiments at SIS at GSI-Darmstadt, Bevalac at Berkeley, and AGS at Brookhaven. In particular, the FOPI Collaboration recently completed an extensive study of Au+Au elliptic flow between 90 MeV/ nucleon and 1.5 GeV/nucleon [66, 97, 98]. Figure 5.4 shows the elliptic flow for three different 113 in- out of in- out of in- in- out of in- out of in- plane plane plane plane plane plane plane plane plane plane (a) (b) 480 ~00 ' V T I f j 00 180 480 00 180 0 0 N (I) (degrees) (I) (degrees) Figure 5.5: Schematic representation of distributions for (a) predominantly in-plane and (b) predominantly out-of-plane emission. cuts on particle type and transverse momentum pt. The transition energy Em" occurs near 100 MeV/ nucleon for all three cuts, and the data with high-pt particles exhibits a lower Emu, than the inclusive data. Not only does this suggest that Em“ is significantly higher than Ebal, but it also suggests that the transition energy does not occur within the range of energies measured with the 47r Array. 5.3 Experimental Results As discussed in Sec. 1.5.2, the azimuthal anisotropy of particles emitted near midra- pidity exhibits a minimum at an incident energy, Em“. At energies below Etm, the azimuthal emission pattern at midrapidity is primarily in-plane; that is, the distribution ( = 43,- — dim) is peaked at 0° and 180°. Above the transition energy, the azimuthal emission pattern is primarily out-of—plane, signified by peaks at i90°. Figure 5.5 is a schematic representation of the distributions for both elliptic flow regions. 114 5.3.1 Fourier Expansion The basic technique for studying elliptic flow is owed to Gutbrod et al. [91] who recognized that the presence of directed flow hinders the extraction of the squeeze- out (out-of-plane flow) signal. Because elliptic flow is expected to be symmetric about the reaction plane as well as the perpendicular plane, fitting a (1) distribution with a cos(2) term could characterize the strength of the elliptic flow. To account for both the cos() nature of directed flow and the cos(2) nature of elliptic flow in the azimuthal distributions, a Fourier expansion of the form dN as = arm + ai’“’cos() + a‘é"cos(2<1>)}, (5.1) is used, where dN/d represents the azimuthal distribution of emitted fragments at midrapidity. In Eq. (5.1), a?" is a normalization factor, a?" is related to the in-plane directed flow component and agxp to the elliptic flow component. The superscript “exp” on the coefficients is to remind that the coefficients extracted from fitting the @-distribution are not corrected for reaction plane dispersion. This correction, discussed in Sec. 3.3.3, is made for a2 by dividing the experimental coefficient by the correction factor: a?" a2 : (cos(A¢rp))' (52) Therefore, dN d5 = a0{1+ a1 cos() + a2 cos(2)}, (5.3) after the dispersion corrections are made. Since the dispersion correction to a?" is made after fitting the Fourier expansion to the distribution, the other coefficients can be ignored in the rest of the analysis. Two different conventions are in practice for quantifying the elliptic flow in terms of the a2 coefficient. First, the ratio of out-of—plane to in-plane emission, called the 115 number squeeze-out ratio by Gutbrod et al. [91], is defined as the number of particles at midrapidity emitted perpendicular to the reaction plane divided by the number of particles emitted in the reaction plane and is given by R _ N(90°) + N(—90°) N — N(0°) +N(180°) ’ (5.4) where N ((1)) represents the summed number of particles in a 90° wedge centered at (I). Stated more explicitly, 135° dN —45° dN R _ f4 ° ddq> N _- 45° dN 225° dN ' -—45° (14> dd) + f135° d<1> dd) (5.5) According to this definition, RN < 1 and RN > 1 are related to a preferential emission of matter in the reaction plane and out of this plane, respectively, while RN 2 1 corresponds to a perfect azimuthally isotropic distribution. RN can be expressed in terms of the a2 coefficient from the Fourier expansion by taking into account that a1 = 0 at midrapidity, since directed flow should be zero at ycm = 0. Therefore, inserting Eq. (5.3) into Eq. (5.4): _ [1+ a2 cos(180°)] + [1+ a2 cos(-180°)] R _ , 5.6 N [1 + a2 cos(0°)] + [1 + a2 cos(360°)] ycm=0 ( ) which reduces to 1 — 02 RN _ 1 + a2 ' (5.7) More recent studies of azimuthal anisotropy and elliptic flow favor the use of the coefficient a2 itself, with some normalization. Taking the beam direction along the z axis and the reaction plane on the x-z plane as usual, the elliptic flow is determined from the average difference between the square of the a: and y components of particle transverse momentum, i. e., 2 2 p1: _ py> 2) == . 5.8 2 <fi+fi ( ) 116 222 is a measure of the aspect ratio of the flow ellipsoid, and 272 > 0 ('02 < 0) indicates in-plane (out-of—plane) alignment of the ellipsoid. 212 can be related to the Fourier coefficient a2 by recognizing that (pi—p5) = >> (5.9) pi + 19?, and since (cos(2)) corresponds to (lg / 2 in the Fourier expansion of Eq. (5.3), a m=§. (mm Eqs. (5.7) and (5.10) give the expressions for relating the Fourier coefficient oz to other methods for measuring anisotropy. In the present analysis, v2 is the chosen elliptic flow representation. 5.3.2 The Transition Energy for Au+Au Now that the technique for characterizing elliptic flow has been presented, the Fourier expansion of Eq. 5.1 can be applied to experimental distributions at midrapidity for any of the available systems and energies. As with the directed flow analysis of Sec. 4.3.2, each detected particle in a collision is treated separately, and a reaction plane is determined from the remaining particles in the collision. This removes any autocorrelation associated with the “particle of interest” (POI) and ensures that an out-of—plane signal is not manufactured by problems in the reaction plane determina- tion. Also, since the azimuthal distribution is the object of analysis, particles are not weighted in the (P-distributions. It has been suggested [100] that the distribu- tion should be weighted by the transverse momentum p} of the included particles. However, since this is not generally practiced, pf weighting is not carried out in the present analysis. 117 Au+Au, all particles, b/bmax < 0.28 l l IIIIIIIIIIIIIIIIIIIIIII e. ..... ... 1-- e I M mlo M mlo ..... m I-:::EiA::m t t mam a mdm -— __—-—_—___h_lF_|F_—h-P-I _ n _ — _ _ _ _ — — — _ .1 m1% m le m m1 -----------...(.-- ........... .V.--.«.h ............... .V.--.(.h 1 e i C 1 m10 M m10 M ml ..... m :ii:::?$::m :-:::35£::m H. .H 7 1" Ms” M mdm 5 mm ——-l_ — _ — _ — n _ _ — _ — —l _—— —————— _ — — — pl—l i u ................... ”In-“ ................... 1....-.“ m1% m ml” m ml .....1. ----------.(.h ............... V. ..... u --- ........... V. ..... u 1 e 1 e 1 m10 M mlo M m1 .................. m Iii:::§3::m :-:::§35::m 1“ .u 0 t 11% w ”is a mm l—l— ———-——-——————————l _——__——__—L pun —_-——_-—-»- $3.5 .923 937:. -90 0 90 (1) (degrees) -90090 Figure 5.6: Azimuthal distribution of particles in near—central Au+Au collisions emit- ted near midrapidity (I(y/'yproj)cm[ S 0.5). Blue lines indicate the experimental spec- tra, and green curves represent Fourier fits of the form of Eq. 5.1. Dashed red lines are shown to indicate in-plane and out-of—plane regions of particle emission. 118 Au+Au, all particles, 0.48 < b/blmax < 0.56 cl l 4| il n 1 m W W m uuuuuuuuuuuuuuu M. iii inuuuuuiuieunuinul. "i0 M Hi lllll n ruin: niliiiii-uAliu-ih l t u!” 8 uL 0 i . 4 U 1 uuuuuuuuuuuuuuu I. in: iiiiiiiiiiiiiiiiiuL __.____.____rr.r ________._ uuuuuuuuuuuu 4|...iil. r-ii iiiiiiiiuluiiiouliiu - w n 1 HI. / Hi ( i V l 1 llllllllllllllllllllll I. rill iiiiiliiuieiuiiuun "[0 M Hi .I i l 1 1H-.. ---- .......... 51“...-“ ul% 4 ul 0 i . 4 0 1 ——- -— —_————_———————————— I IL .. 1 t Uio n MI I 9 W i lllllllllllll h iiiiil ainiuiuuieiiiiilL ”[0 M "1.1 ... 1 IIIIIIIIIIIIIIIIIIIIIIIII 1 t i 1w t 1 l w 8 i. ll ll 0 i . 3 0 .__ t__r____~..___.. -90090 -90090 -90090 63:: 5.3V 637:. 6. 1 7 MI 5 0 i _ _ _ _ _ _ — _ —l m mi ............... .V.----h e l M mi ............... 5-----m 3 m1 5 U i i— i— _ — _ _ — i—iP l -90090 -900 90 (degrees) (I) Figure 5.7: Azimuthal distribution of particles in semi—central Au+Au collisions emit- ted near midrapidity ([(y/ypmj)cm| S 0.5). Blue lines indicate the experimental spec- tra, and green curves represent Fourier fits of the form of Eq. 5.1. Dashed red lines are shown to indicate in-plane and out-of—plane regions of particle emission. 119 For the present analysis, events were placed into impact parameter bins of equal width, following the geometrical prescription of Sec. 3.1. Figures 5.6 and 5.7 show the midrapidity Q distributions at all reduced incident beam energies for near—central (b/bmax < 0.28) and semi-central (0.48 < b/bmax < 0.56) Au+Au collisions, respec- tively. In these plots and all subsequent plots, the midrapidity region is defined as ‘0-5 S (y/yproj)cm S 0-5 - (5'11) For elliptic flow experiments in the regime of higher multiplicities and limited accep- tance, narrower midrapidity regions usually are selected [71, 27]. The green curves in Figures 5.6 and 5.7 represent fits to the Fourier expansion of Eq. 5.1. At low energies, the Q-spectra are peaked at 0° and i180°, indicating primarily in-plane elliptic flow. As the beam energy is increased, the Q-spectra become flatter. For the near-central impact parameter bin, the Q-spectra exhibit peaks at i90° for incident energies above 44.5 MeV/nucleon, indicative of primarily out-of-plane elliptic flow. For the semi-central bin, the elliptic flow is primarily in-plane over the entire range of beam energies studied. Before presenting the v2 excitation functions, it is fruitful to discuss some general traits of the Q distributions. These distributions show both a four-fold symmetry from the shape of the ellipsoids and a two—fold symmetry attributed to two effects. First, since the P01 is excluded from the reaction plane determination, it is less likely that the reaction plane will be in the direction of the POI. This leads to a slight suppression of Q near 0° when compared to Q near 180° as well as a small depression of Q at Q = 0°. Second, peaks and valleys in the out-of-plane region are not centered exactly at i90°. This two-fold asymmetry is due to the fact that the Q distributions are about the beam axis, not the flow axis. Near the transition energy, the flow axis nearly coincides with the beam axis, so that a coordinate transformation is not necessary. In addition, true flow angles are difficult to determine accurately in this 120 0.15 _ ; ‘1 ~ Au+Au 0.1 :— ............. 0.05 L ............. . >61 C ........ #9 0 r. ................................................................... $12-34 ............... : 0 0.48 < b/bm< 0.56 _0.05 -_ A 0.39 < b/bnm< 0.48 : CI 0.28 < b/bmx< 0.39 : o b/bmx < 0.28 .0.1 l I l l 30 40 50 60 70 Beam Energy Figure 5.8: Elliptic flow plotted as a function of incident beam energy for Au+Au collisions. Data are for four different impact parameter bins, and solid and dashed lines correspond to logarithmic fits to the data. energy range because of a finite number of detected particles. The second order coefficients (12 can be extracted from the Fourier fits for each beam energy and impact parameter bin. Figure 5.8 shows the v2 coefficients obtained from Eq. 5.10 for Au+Au collisions. Data for four different impact parameter bins are shown. The error bars are obtained by assigning statistical errors to each bin in the Q distribution and extracting the error from the convergence minimization routine HFITV, which is standard routine in the CERN library. The fits in Figure 5.8 are logarithmic, as suggested by the trend of the data. All impact parameter bins show an energy dependence of U2, and as the impact parameter is increased, the transition energy also increases. The monotonic relationship between b and Em" for Au+Au over the range of impact parameters here can be attributed to the energy deposited into the overlap (participant) zone of the collision. When going from peripheral to central collisions, 121 b Bin Emu, b/bmam < 0.28 41 :l: 3 MeV/nucleon 0.28 < b/bmax < 0.39 46 :l: 4 MeV/nucleon 0.39 < b/bmax < 0.48 56 :l: 3 MeV/nucleon 0.48 < b/bmx < 0.56 75 :l: 6 MeV/nucleon Table 5.1: Au+Au transition energies extracted from logarithmic fits to the data in Figure 5.8. the deposited energy becomes larger and the expansion takes place faster, leading to the development of an out-of—plane elliptic flow signal at a lower incident energy. In less central collisions, the excited participant zone needs a higher incident beam energy to generate expansion/emission patterns quickly enough to be hindered by projectile-like or target-like nuclear matter; otherwise, the participant zone will emit fragments unhindered because the spectator pieces will have left the collision zone beforehand. The impact parameter dependence of Em" is in stark contrast to the lack of an impact parameter dependence of Ebal for Au+Au (see Sec. 4.6.1). This strengthens the notion of two separate collective mechanisms at work in non-central nuclear collisions. Table 5.1 lists the values of Em“ obtained from the logarithmic fits in Figure 5.8. The error bars are associated with the variance in Em“ when data points are excluded systematically from the fit. The data is in strong disagreement with measurements of Etm, made by the F OPI group, as evidenced in Figure 5.4. For example, the “M4” impact parameter bin in Figure 5.4(b), which corresponds roughly to 0.39 < b/bmax < 0.56, has a value for Emu, that is greater than 100 MeV/nucleon, a 30-40 MeV/ nucleon disparity from the present measurement. 122 5.3.3 Mass Dependence of the Transition Energy Using the same analysis technique applied to the Au+Au system, the elliptic flow over a range of beam energies was studied in three additional systems: 40Ar+458c, 58Ni+58Ni and 86Kr+93Nb. The subsystem configuration selected in the data analysis for these three systems is the same for Au+Au, presented in Chapter 2.4 Figures 5.9, 5.10, and 5.11 show representative Q distributions of particles emitted near midra- pidity (|(y/yp,oj)cm| S 0.5) for the three systems mentioned above. Data are for semi-central collisions with 0.28 < b/bmam < 0.39, and dashed lines represent Fourier fits of the form of Eq. 5.1. At low incident energies for all three systems, emission is primarily in-plane, signified by peaks (valleys) at Q = 0°,180° (21:90°). As the beam energy is increased, a transition to primarily out-of plane emission, characterized by peaks (valleys) at :l:90° (Q = 0°,180°), is observed. The extracted or) values for each system are plotted vs. incident beam energy in Figures 5.12 and 5.13 for near-central and semi-central impact parameter bins, re- spectively. The curves are included only to guide the eye. The transition energy decreases as the system mass increases for both impact parameter bins. Complemen- tary theoretical studies of the mass dependence of Em" are needed to understand the effect. Interestingly, the mass dependence of Em," follows a trend similar to the mass dependence of Eb“. 5.4 Comparison to Theoretical Work Recent calculations based on an isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) model indicated that the crossover from in-plane to out-of-plane elliptic flow is sensitive to the nuclear equation of state. Calculations for 48Ca+48Ca at incident 4The 86Kr+93Nb physics files used in the present analysis contained identified particles from a fourth subsystem known as the Zero Degree Detector [34]. However, these particles were ignored in all aspects of the analysis, including impact parameter binning and reaction plane determination. 123 Ar+Sc, all particles, 0.28 < b/bmax < 0.39 75§Mevln ioutiiniouti ssiMev/n §out§ in Eouti 1:1L1:I [151] iii llel:lll:llI:l I:l [1:1 I1:L11:L -90090 -90090 -90090 45§Mevln Eoutiinéouté I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I1 I I I I I T I I I I <1, git IIIIIIIIIIIIIIII 115: MeiV/n 125: Mey/n EoutiinioutE foutiiniouti IEIIIEIIJEI I IS. lslllEI1LlS‘_Llsl 95 iMe‘if/n ioutiiniouti [ELIIEIIIEILLEI -90 0 90 -90 0 90 -90 0 90 IIIIIIWITITIFIIII—IT—I II IIIIIIIIIIITIIII IIII dN/dQ (arb. units) 14S Me2V/n 155: MeEV/n ioutéiniouté ioutéinéouti lglllslllzlllzl ULliEiliiiiiil lzlllslllslllzl -90 0 9o .90 o 90 -90 0 90 Q (degrees) 13$ Mey/n ioutiiniouti IIIIIIIIITIIIIII IIIIIIIIIIIIIIrII II Figure 5.9: Azimuthal distribution of particles in semi-central 40Ar+45Sc collisions near midrapidity (|(y/yp,oj)cm| S 0.5). Dashed curves represent Fourier fits of the form of Eq. 5.1. 124 Ni+Ni, all particles, 0.28 < b/bmax < 0.39 I...‘ 45§Mevln iouti in gout; 71:1IlilllellEl [Elllflllzlllil LULLLLIiEiiiin 35%Me‘iv’ln 40§Me\§'ln Eoutiiniouti Eoutiiniouti IIIIIIIIIIIIIIIIIIIII II -90090 -90090 -90090 A a - :3 1 '_ 3_ .5 - : I 3-1 - r- “ ' : : - : : I. : : V - 5 5 — 5 i '_ § § 6: : 50EMeY/n F 55SMeY/n - 65§MeYln E :ioutiiniouti :ioutiiniouti “goutiiniouti z Islllilllilllsl lElILElIlElIlEl bIElIIEIIlEIIlEI "5 .90 0 9o .90 o 90 -90 0 9o 75%Me‘if/n 90EMeI’ln 105: Me2V/n ioutiiniout§ ioutiiniouté Eout§in§out§ IEIIIEI LL31 I [E] lzl IIIIJ.1I Isl ISIIIEIIIEIIIEI -90 0 9o -90 0 9o -90 o 90 Q (degrees) IIIIIIIITIII IIIIIIVTTTIIII TIIIIIIIIIIII Figure 5.10: Azimuthal distribution of particles in semi—central 58Ni+58Ni collisions near midrapidity (|(y/yp,oj)cm| S 0.5). Dashed curves represent Fourier fits of the form of Eq. 5.1. 125 Kr+Nb, all particles, 0.28 < b/bmax < 0.39 f. 25§MeYIn 35§MeYIn 45§MeYIn IioutiinioutE Eoutiiniouté Eoutiinéouti _IEIIIEI I 1311131 131 I 15] LIiIIISL IEILIEI I IEJILEI -90090 -90090 -90090 65§Mevln Eouti in Eouti IEIIISIIIEI I IE] [5] I IEIIISI I IEI IEIIIEI I Isl I Isl -90090 -90090 -90090 ssEMev/n Eoutiiniouti 50§Mevln iontiiniouti r I I I I I I I IT1III1IIIIIIIIIIIIIII dN/dQ (arb. units) mWHIHHI'HFI ” 95§MevIn goutiiniouti 75 iMe‘if/n 85 éMeVi’ln I. ioutéiniouti tioutéiniouté lilllzl i 1:11:11 111113111311151 lilIlElIlzlll: .90 0 9o .90 o 90 .90 0 90 Q (degrees) IIIIIIIIIIII IITTIIIIIIII Figure 5.11: Azimuthal distribution of particles in semi-central 86Kr+93N b collisions near midrapidity (|(y/ypmJ-)cm| S 0.5). Dashed curves represent Fourier fits of the form of Eq. 5.1. 126 System mass dependence of v2, blbm < 0.28 0.2 - * 19'7Au + 19'7Au A “Kr + ”Nb 0 58Ni + 58Ni 0 ”Ar + ”Sc 4 .0.1 IllIIIJJJILJJJJJIIIIIIIIIlllIlllIIlLLlIllllIllll 30 40 50 60 70 80 90100110120 Beam Energy (MeV/nucleon) Figure 5.12: Elliptic flow for four systems as a function of incident beam energy in near-central collisions (b/bmax < 0.28). 127 System mass dependence of v2, 0.28 < b/bm < 0.39 0.2 - A “Kr + ”Nb - D 53M + 5”Ni 0 4"Ar + ‘5Sc -01 IIIILIIIIIIIIIIIIIIIIIIILIIIIlllIIIJlIIIIIIIIlII 30 40 so 60 70 80 90100110120 Beam Energy (MeV/nucleon) Figure 5.13: Elliptic flow for four systems as a function of incident beam energy in semi-central collisions (0.28 < b/bmx < 0.39). 128 energies between 30 and 100 MeV/ nucleon showed the transition energy to be lower for stiffer EOS (i.e., higher nuclear compressibility K) than for a soft EOS, even if the in-medium cross section is reduced [79]. This is in stark contrast to the balance energy, which shows almost no sensitivity to the EOS for medium-sized systems with reduced cross-section [18, 69] and the Opposite relationship between EOS stiffness and Ebal for vacuum cross section (see Sec. 4.6.3). Therefore, the transition energy may be more useful than the balance energy for resolving the nuclear EOS stiffness using medium-sized systems. Figure 5.14 shows the elliptic flow for 40Ar+"5Sc data taken with the 47r Array plotted with the IBUU calculations for 48Ca+48Ca, since these systems are very close in mass. The 40Ar+“58c data is for the most central impact parameter bin (shown in Figure 5.12) so that the impact parameter of the calculations closely matches the mean impact parameter of the data. For the data, (b/bmax) z 0.20, and for the calculations, b/bmam = 0.23. The curves for both the data and calculations are included to guide the eye. The transition energy for the 40Ar+45Sc data appears to agree most closely with the soft EOS and reduced on. This finding is in good agreement with the directed flow studies of Sec. 4.6. It should be noted that the magnitudes of the 02 values in Fig. 5.14 are questionable, since the IBUU calculations from Ref. [79] were performed in an independent-particle formulation with no fragment clustering. However, the energy of crossover is the observable of interest, and differences in the magnitudes are unimportant. Also, the isospin of the system may be important. More calculations are needed. 5.5 Summary In this chapter we have presented the disappearance of elliptic flow at intermediate energies for several systems. The transition energy for Au+Au shows a strong impact 129 [' ' ' l 1.5 — : Data :40Ar+45Sc 1 } — 0.5 } — 0 : _O.5 j '+ r ; . stiff EOS i + - —1 —osoftEOS ~ [ 1 soft EOS with 0.886,", " + + 7 _ fr 41: data - _1'5 1iiliii liiiiiiiiiiiiliii..i,i-1_mmLmLLi_Lr_1 20 30 4O 50 60 7O 80 90 100 Figure 5.14: Comparison of isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) calculations of 48Ca+45’Ca from Ref. [79] with 4°Ar+45Sc experimental data taken with the 47r Array. Comparison to IBUU calculations [Ill‘ 1 Till] Illii[v IBUU: 48Ca +48Ca Beam Energy (MeV/nucleon) 130 parameter dependence, which is in contrast to the balance energy results of Chapter 4. The values obtained for the Au+Au transition energy are also quite different from previously published values by the FOPI Collaboration at GSI. Also, the mass dependence of the transition energy was shown. Similar to the balance energy, the transition energy increases as the system size decreases. Previous theoretical calculations of 48Ca+48Ca by the Texas A&M group showed a strong dependence of Emu, on the nuclear EOS, and 40Ar+45Sc experimental data agrees most closely to the soft EOS with a reduced in-medium cross section. This is in good agreement with the findings of Chapter 4. Thus, an additional observable for probing the nuclear EOS using nuclear collisions is established, and future theoretical studies may constrain the nuclear EOS further by exploring the mass— and impact-parameter dependence of the transition energy. 131 Chapter 6 Conclusion In this thesis, we have studied two transitions in collective flow observables: the dis- appearance of directed flow at intermediate energies, or balance energy; and the first disappearance of elliptic flow Emu, also at intermediate energies, called the transition energy. The study of transitions, or zeroes, in flow excitation functions is especially useful because the energy of transition is not influenced strongly by experimental bi- ases such as detector acceptance and detector granularity. Also, these transitions can be modeled by theory without incorporating the complexities of fragment formation and experimental reaction plane dispersion. The balance energy for Au+Au is interesting because it extends the mass depen- dence of the balance energy by a factor of 2 in system mass. Previous experimental studies at higher energies were only able to extrapolate Ebal for Au+Au, and the extrapolated value depends on the reaction plane dispersion which alters the finite flow values at each measured energy. Also, QMD calculations suggested that the bal- ance energy for Au+Au might be obscured by the large Coulomb interaction for the heavy system. In this thesis we presented the balance energy for Au+Au and showed that the Coulomb interaction acts to decrease the balance energy. In fact, remov- ing the Coulomb term from Boltzmann-Uehling-Uhlenbeck (BUU) model calculations 132 restores the expected 7' z 1/3 exponent in the Ebal o: A" power-law relationship. The value of 7' z 1/3 is predicted by a simple scaling argument in which the balance energy arises from surface effects which scale as A2/3, and volume effects which scale with A. Also, previous theoretical studies of the balance energy showed a dual dependence on the nuclear compressibility K and the in-medium cross section am, for light- and medium-sized systems. The impact parameter dependence of Ebal also has been observed experimentally for light systems. However, BUU calculations in this thesis of the balance energy for Au+Au is nearly independent of on“, and the impact parameter dependence nearly vanishes for Au+Au as well. Therefore, for the first time the balance energy is used to constrain the nuclear equation of state, and comparison between the experimental Ebal for Au+Au and BUU calculations show agreement with a soft equation of state. The extended mass dependence then is used to show a reduction of 20—30% in the in-medium nucleon cross section. The elliptic flow transition energy for Au+Au was presented, and the values ob- tained with the 47r Array show substantial disagreement with previous measurements by the FOPI group at GSI. The transition energy for Au+Au shows a strong impact parameter dependence, in contrast to the balance energy. The mass dependence of Em“ was studied using four systems with a nearly identi- cal configuration of the 47r Array. As the system size increases, the transition energy decreases, similar to the balance energy. Comparison between experimental 4"Ar+45Sc data and 48Ca+“8Ca isospin-dependent BUU calculations show good agreement with a soft EOS, in accordance with the directed flow results. Therefore, two different phenomena have been shown for the first time to be use- ful for constraining the nuclear EOS. Both the balance energy for Au+Au and the transition energy for Ar+Sc agree with model calculations in the BUU framework 133 a?” that incorporate a soft equation of state and a reduced in-medium nucleon cross sec- tion. Further theoretical and experimental studies are warranted, as a primary goal of nuclear physics is to determine these two entities. It is ultimately hOped that nu- clear collisions may be useful for constraining the value of the nuclear compressibility and resolving some of the difference in values obtained from monopole and dipole resonance studies. 134 Appendix A 47r Array Parameters This Appendix contains many of the specifications for the 4% Array at the time of this this thesis experiment. Please refer to Chapter 2 for related discussion and diagrams of the 41r Array. Table A.1: Solid angle subtended by the Ball Phoswiches. Module Type Ideal (msr) True (msr) Hexagon (6x) 75.2 66.0 Pentagon (5x) 59.0 49.9 135 Table A.2: Solid angle subtended by the HRA Phoswiches. HRA Detector Number Solid Angle (msr each) 12, 15, 18, 21, 24 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 11,13,14,16,17,19,20,22,23,25 27,28,31,32,35,36,39,40,43,44 26,29,30,33,34,37,38,41,42,45 5.11 6.27 6.18 6.88 6.65 Table A.3: Phoswich Scintillator Specifications. Element Plastic Thickness (mm) Rise time (ns) Decay time (as) Ball Fast AE BC412 3.2 1.0 3.3 Ball Slow E BC444 250 20 180 HRA Fast AE NE110 1.7 1.1 3.3 HRA Slow E NE115 194 8.0 320 136 Table A.4: Low Energy Thresholds for the HRA. Particle Punch-in Particle Punch-in Particle Punch-in Type Energy (MeV) Type Energy (MeV) Type Energy (MeV) p 13 C 269 A1 877 17 N 341 Si 962 t 20 O 419 P 1079 He 50 F 515 'S 1170 Li 99 Ne 591 C1 1294 Be 152 Na 687 Ar 1455 B 212 Mg 767 Table A5: Low Energy Thresholds for the Ball Telescopes. Particle Punch-in Particle Punch-in Particle Punch-in Type Energy (MeV) Type Energy (MeV) Type Energy (MeV) He 12 N 74 Mg 163 Li 23 O 91 A1 184 Be 34 F 108 Si 202 B 46 Ne 123 P 224 C 59 Na 146 S 242 137 Table A.6: Mean angles for the ball phoswiches. Module C D 0 ¢ 0 ¢ 0 4) 0 4) 0 ¢ 0 d) 1 23.1 342.0 32.3 5.6 46.0 356.3 51.7 342.0 46.0 324.7 32.3 318.4 2 23.1 270.0 32.3 293.6 46.0 287.3 51.7 270.0 46.0 252.7 32.3 246.4 3 23.1 198.0 32.3 221.6 46.0 215.3 51.7 198.0 46.0 180.7 32.3 174.4 4 23.1 126.0 32.3 149.6 46.0 143.3 51.7 126.0 46.0 108.7 32.3 102.4 5 23.1 54.0 32.3 77.6 46.0 71.3 51.7 54.0 46.0 36.7 32.3 30.4 6 54.7 298.0 54.7 314.0 67.3 317.5 74.6 306.0 67.3 294.5 7 54.7 226.0 54.7 242.0 67.3 245.5 74.6 234.0 67.3 222.5 8 54.7 154.0 54.7 170.0 67.3 173.5 74.6 162.0 67.3 150.5 9 54.7 82.0 54.7 98.0 67.3 101.5 74.6 90.0 67.3 78.5 10 54.7 10.0 54.7 26.0 67.3 29.5 74.6 18.0 67.3 6.5 11 64.9 342.0 72.4 355.0 86.5 354.4 93.5 342.0 86.5 329.6 72.4 329.0 12 64.9 270.0 72.4 283.0 86.5 282.4 93.5 270.0 86.5 257.6 72.4 257.0 13 64.9 198.0 72.4 211.0 86.5 210.4 93.5 198.0 86.5 185.6 72.4 185.0 14 64.9 126.0 72.4 139.0 86.5 138.4 93.5 126.0 86.5 113.6 72.4 113.0 15 64.9 54.0 72.4 67.0 86.5 66.4 93.5 54.0 86.5 41.6 72.4 41.0 16 86.5 306.0 93.5 318.4 107.6 319.0 115.1 306.0 107.6 293.0 93.5 293.6 17 86.5 234.0 93.5 246.4 107.6 247.0 115.1 234.0 107.6 221.0 93.5 221.6 18 86.5 162.0 93.5 174.4 107.6 175.0 115.1 162.0 107.6 149.0 93.5 149.6 19 86.5 90.0 93.5 102.4 107.6 103.0 115.1 90.0 107.6 77.0 93.5 77.6 20 86.5 18.0 93.5 30.4 107.6 31.0 115.1 18.0 107.6 5.0 93.5 5.6 21 105.4 342.0 112.7 353.5 125.3 350.0 125.3 334.0 112.7 330.5 22 105.4 270.0 112.7 281.5 125.3 278.0 125.3 262.0 112.7 258.5 23 105.4 198.0 112.7 209.5 125.3 206.0 125.3 190.0 112.7 186.5 24 105.4 126.0 112.7 137.5 125.3 134.0 125.3 118.0 112.7 114.5 25 105.4 54.0 112.7 65.5 125.3 62.0 125.3 46.0 112.7 42.5 26 128.3 306.0 134.0 323.3 147.7 329.6 156.9 306.0 147 .7 282.4 134.0 288.7 27 128.3 234.0 134.0 251.3 147.7 257.6 156.9 234.0 147.7 210.4 134.0 216.7 28 128.3 162.0 134.0 179.3 147.7 185.6 156.9 162.0 147.7 138.4 134.0 144.7 29 128.3 90.0 134.0 107.3 147.7 113.6 156.9 90.0 147.7 66.4 134.0 72.7 30 128.3 18.0 134.0 35.3 147.7 41.6 156.9 18.0 147.7 354.4 134.0 0.7 138 Table A.7: Mean angles for the HRA phoswiches. 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