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A b s t r a c t
P r o d u c t i o n o f L i g h t N u c l e i in Relativistic
H e a v y I o n C o l l i s i o n s
Joseph V. Germani
Yale University
November 1993
A study of the production of light nuclei, including deuterons, tritons, 3He, and
alpha particles, in collisions of 28Si ions at 14.6 GeV per nucleon with targets of Pb, Cu,
and Al has been conducted. The nuclei measured are in the rapidity range of 1-2, and thus
are expected to be formed in the decay of the excited region of participant nucleons. The
mechanism by which nuclei are formed in such an extreme environment is not well
understood, but it is believed to be a process of coalescence. The relative yields of such
nuclei are related to the state of the system in which they were formed, and thus provide
insight into the properties of hot and dense nuclear matter.
The measurements were made using the E814 apparatus, which consists of a set of
detectors for characterizing the centrality of the collisions and a spectrometer for
measuring reaction products. The spectrometer accepts all particles produced within a
rectangular aperture centered on the beam axis, with a full width of 38 mr by 24 mr. A
scintillator hodoscope is used to measure time of flight and charge, and a set of three
tracking chambers is used for momentum measurements. A trigger based on the time of
flight of particles through the spectrometer allowed for detection o f light nuclei
independent of the impact parameter of the collision.
Measurements of the yield of deuterons, tritons, 3He, and alpha particles produced
at zero degrees are presented. Several models o f coalescence are applied to the data. The
classic momentum-space coalescence model, which works well at lower energies, is found
to be inadequate at AGS energies. Thermodynamic and improved coalescence models are
applied with more success and are used to calculate parameters related to the size o f the
system. Finally, a coalescence model based on an intranuclear cascade simulation is
examined, and is found to offer some promising insights.
P r o d u c t i o n o f L i g h t N u c l e i in Relativistic
H e a v y I o n C o l l i s i o n s
A Dissertation
Presented to the Faculty of the Graduate School
of
Yale University
in Candidacy for the Degree of
Doctor of Philosophy
by
Joseph Victor Germani
November 1993
A c k n o w l e d g m e n t s
I am happy to have the opportunity to express my gratitude to the many people who have
helped to make this work possible.
I would first like to thank my thesis advisor, Shiva Kumar, for taking me on as his
student after I had spent several years in a different field. His guidance and advice have
been helpful. Also, his hard work at building up a strong research group and excellent
computer facility made this work possible.
Jack Sandweiss has been an important guiding force in this experiment. His energy
and enthusiasm are unparalleled. I am indebted to Dick Majka for his indispensable help in
setting up the run and in the analysis.
This experiment would not have been possible without the many people who make
up the E814 collaboration. Specifically, I would like thank the spokesman, Peter Braun-
Munzinger, whose management of the collaboration was noteworthy. Tom Hemmick was
instrumental in my decision to join E814. I thank him for his hard work in setting up the
run and for the many fruitful discussions we have had. I would also like to acknowledge
Helio Takai and Sean McCorkle who were the backbone of the experiment, and to whom
I caused no small amount of torment during the run. I thank Jeff Mitchell for helping me
understand Quanah and Jamie Nagle for his work with RQMD which appears in this
dissertation.
I am grateful to the members of my thesis committee, Jack Sandweiss, Peter
Parker, Dimitri Kusnezov, and Malcolm Boshier for their careful reading of my
dissertation and helpful comments. I would also like to thank Tony Baltz, my outside
reader, and Carl Dover, who have provided several insightful conversations.
The staff o f WNSL was an important part of my graduate education. I would
specifically like to thank Allen Ouellette for his excellent work at keeping the computers
running. His patience and calmness in the face o f chaos have been admirable. I am
grateful to Peter Parker, the director of WNSL for his support and helpful discussions, and
to D. Allan Bromley, his predecessor, for his support in the early years. I must, o f course,
thank the administrative staff, Lisa Close, Rita Bonito, Karen DeFelice, and Mary Anne
Schulz, who keep the lab running, and Sara Batter and Jean Belfonti who were
instrumental in dealing with the bureaucracy. The WNSL support staff, especially John
Baris, Tom Barker, Joe Cimino, Al Jeddry, Tom Leonard, and Dick Wagner were very
patient and helpful to me during "the xenon years," as were the craftsmen at the Gibbs
Shop, who taught me much about creativity.
The friends that I have made here over the years have been an important part of
my life. Jon Gilligan and Steve Klepper were valuable members of our study group, who
taught me a great deal during our marathon study sessions. I would like to thank Bernard
Phlips, Lisa Close, Pat Ennis, and Craig Levin for their friendship and for many hours of
stimulating conversation. I am happy to count Dan Blumenthal among my closest friends.
His companionship and endless diversions have helped make graduate school much more
interesting. I am much indebted to Vicki Greene who taught me many lessons about life,
as well as work. Without her help my switch to this experiment might not have been
successful. I thank Vicki for inspiring me, for teaching me, and for being such a
wonderful friend.
I am extremely grateful to my family for their constant love and support. Their
encouragement has helped me more than I can say. I especially thank my parents, Joe and
Marilyn Germani, for instilling in me the desire to continually learn and better myself.
Finally, to my wife Maureen Wylie Germani, who kept our lives in order throughout this
process, I owe my deepest gratitude. Maureen, your love and encouragement have gotten
me through so many trying times, for this I am forever grateful.
C o n t e n t s
Acknowledgments ii
List of Figures vii
List of Tables ix
1 Introduction 1
1.1. Background....................................................................................................2
1.2. Coalescence at the AGS............................................................................... 9
1.3. Variables.........................................................................................................10
2 Experiment and Apparatus 13
2.1. Experiment 814..............................................................................................13
2.2. Beam Definition.............................................................................................15
2.3. Event Characterization..................................................................................17
2.3.1. Multiplicity Array............................................................................18
2.3.2. Target Calorimeter..........................................................................18
2.4. Forward Spectrometer...................................................................................19
2.4.1. Tracking Chambers........................................................................ 20
2.4.2. Scintillator Hodoscope...................................................................22
2.4.3. Uranium Calorimeter...................................................................... 23
2.4.4. Pattern Recognition....................................................................... 25
2.5. Time-of-Flight Trigger................................................................................. 27
2.6. Data Acquisition System.............................................................................. 32
iv
2.7. The Data........................................................................................................ 33
3 Analysis 34
3.1. Pass 1..............................................................................................................35
3.1.1. Beam Cut........................................................................................35
3.1.2. Tracking.........................................................................................36
3.2. Pass 2..............................................................................................................39
3.2.1. Tracking Cut...................................................................................39
3.2.2. Charge Selection............................................................................40
3.2.3. Particle ID ..................................................................................... 43
3.3. Efficiencies.....................................................................................................48
3.3.1. Tracking.........................................................................................48
3.3.2. Charge Identification......................................................................52
3.3.3. Geometrical Acceptance............................................................... 55
3.3.4. Total Efficiency.............................................................................. 57
4 Results 58
4.1. Cross Sections................................................................................................58
4.2. Centrality.......................................................................................................60
4.3. Extrapolations to Pt = 0................................................................................ 63
5 Discussion 74
5.1. Proton Spectra.............................................................................................. 75
5.2. Coalescence Model.......................................................................................78
5.3. Source Size....................................................................................................88
5.3.1. Thermodynamic Model................................................................. 88
5.3.2. Improved Coalescence Model...................................................... 96
5.4. Coalescence with ARC................................................................................. 99
6 Conclusion 108
A Photomultipliers and Electronics 111
v
B Invariant Yield Plots and Coalescence Fits
Bibliography
L i s t o f F i g u r e s
2.1 A schematic of E814 apparatus................................................................................. 14
2.2 The beam scintillator (BSCI) telescope.................................................................... 16
2.3 The silicon multiplicity array..................................................................................... 16
2.4 Schematic views of the DCII and DCIII chambers..................................................21
2.5 Forward scintillator (FSCI) slat and uranium calorimeter (UCAL) module 24
2.6 Block diagram of the time-of-flight trigger logic.................................................... 29
2.7 The active region of forward scintillator wall for the TOF trigger........................ 31
3.1 BSCI pulse height distributions for TOF triggered events.......................................37
3.2 Forward scintillator pulse height distribution showing charge cuts........................ 41
3.3 Drift chamber pad plane pulse height distributions with DCQ cut......................... 42
3.4 Particle identification plot for charge 1 tracks..........................................................44
3.5 Particle identification plot for charge 2 tracks..........................................................45
3.6 Mass distribution plots.............................................................................................. 47
3.7 Detection efficiency of forward scintillator slats......................................................49
3.8 FSCI pulse height spectra used for determining charge cut efficiencies................. 53
3.9 Acceptance plots....................................................................................................... 56
4.1 Charged particle multiplicity spectra for Pb targets................................................. 62
4.2 Deuteron invariant yield vs. pt for Si + Pb............................................................... 66
4.3 Triton invariant yield vs. pt for Si + Pb.....................................................................67
4.4 3He invariant yield vs. pt for Si + Pb........................................................................ 68
vii
4.5 Invariant yields at pt = 0 vs. rapidity for minimum bias Si + Pb collisions 69
4.6 Invariant yields at pt = 0 vs. rapidity for Si + Pb collisions at 55% of ogeom........70
4.7 Invariant yields at pt = 0 vs. rapidity for central Si + Pb collisions.........................71
4.8 Invariant yields at pt = 0 vs. rapidity for Si + Cu collisions at 55% of ageom........72
4.9 Invariant yields at pt = 0 vs. rapidity for Si + Al collisions at 55% of ageom........ 73
5.1 Charged particle multiplicity spectrum from ARC Si + Pb collisions....................77
5.2 Fits to ARC proton mt spectra..................................................................................79
5.3 Invariant yield at pt = 0 of ARC protons vs. rapidity for Si + Pb........................... 80
5.4 Coalescence scaling coefficient, BA, vs. rapidity for minimum bias Si + Pb 83
5.5 Coalescence scaling coefficient, BA, vs. centrality for Si + Pb.............................. 85
5.6 Coalescence scaling coefficient, BA, vs. incident beam momentum.......................87
5.7 Thermodynamic model source radius vs. centrality for Si + Pb.............................. 94
5.8 Improved coalescence model source radius vs. centrality for Si + Pb.................... 98
5.9 Comparison of ARC coalescence calculation with measurements.......................... 102
5 .10 Time of last interaction and source radius for protons from RQMD.....................105
5.11 Mean proton source radius vs. impact parameter from RQMD............................ 106
B. 1 Invariant yields and BA vs. rapidity for Si + Pb collisions at 89% of ageom 114
B.2 Invariant yields and BA vs. rapidity for Si + Pb collisions at 55% of ageom 115
B.3 Invariant yields and BA vs. rapidity for Si + Pb collisions at 8% of ogeom 116
B.4 Invariant yields and BA vs. rapidity for Si + Pb collisions at 8-15% of ogeom 117
B.5 Invariant yields and BA vs. rapidity for Si + Pb collisions at 15-30% of ageom... 118
B.6 Invariant yields and BA vs. rapidity for Si + Pb collisions at 30-44% of ageom... 119
B.7 Invariant yields and BA vs. rapidity for Si + Pb collisions at 44-64% of ageom... 120
B.8 Invariant yields and BA vs. rapidity for Si + Pb collisions at 64-89% of ageom... 121
B.9 Invariant yields and BA vs. rapidity for Si + Cu collisions at 55% of ageom 122
B. 10 Invariant yields and BA vs. rapidity for Si + Al collisions at 55% of crgeom........123
viii
L i s t o f T a b l e s
2.1 Pre-trigger thresholds................................................................................................30
3.1 Detector efficiencies..................................................................................................50
3.2 Forward scintillator efficiencies.................................................................................51
3.3 Cut efficiencies.......................................................................................................... 57
4.1 Geometric cross sections, target thicknesses, and incident beam............................60
4.2 Pre-trigger multiplicity thresholds............................................................................ 63
4.3 Number of identified particles for the targets used..................................................64
5.1 Coalescence scaling coefficients and radii.................................................................82
5.2 Thermodynamic model radii......................................................................................92
5.3 Improved coalescence model radii............................................................................ 97
5.4 Parameters for ARC coalescence calculation........................................................... 100
A. 1 Photomultipliers and signal processing electronics.................................................. 111
B. 1 Coalescence scale factors, BA, for all centrality cuts used.....................................113
C h a p t e r 1
I n t r o d u c t i o n
One of the goals of relativistic heavy ion physics is to study nuclear matter under the
extreme conditions of high temperatures and densities in the hope of discovering the
transition of nuclear matter into quark-gluon plasma (QGP). This new phase of nuclear
matter, which has yet to be observed, is expected to be formed when nuclear matter is
subjected to extreme compression and/or temperatures, such that the individual nucleons
overlap and the constituent quarks are no longer bound into hadrons. An important step
in the process of searching for, and eventually understanding, the QGP is the development
of techniques for studying the properties and space-time evolution of the systems of hot,
compressed nuclear matter created in relativistic heavy ion collisions.
Light nuclei emitted with velocities near the center-of-mass velocity of the system
(mid rapidity) probe the later stages of the hot, dense nuclear matter formed in relativistic
heavy ion collisions. It is unlikely that such nuclei are pre-formed, or simply fragments
broken off of the target or projectile, since in order to be slowed down to near the center-
of-mass velocity from the projectile velocity, or sped up from the target velocity, they
would have to endure collisions that would easily dissociate them into their constituent
1
2
nucleons. Thus, these nuclei must be emitted from the region of highly excited nuclear
matter and cannot be simply fragments from one of the initial nuclei.
The relative yields of such light nuclei contain information about the environment
in which the nuclei were formed. Various models relate these yields to the volume or
density of the system at the time the nuclei are formed. We will explore these models and
assess the potential of using measurements of light nuclei yields to determine the size of
the emitting source.
The measurements presented in this dissertation comprise a study of the
production of light nuclei from the highly excited region created in the center of mass of
nucleus-nucleus collisions. First, a discussion of the history of the measurements of light
nucleus production will be given, along with a description of several theories that have
been developed in order to interpret the measurements of light nucleus yields and shed
light on the state of the excited nuclear matter in which the nuclei are formed.
1.1. Background
The first measurements of the production of deuterons in high energy collisions were
made in 1960. An experiment was performed at the newly commissioned CERN proton
synchrotron with a 25 GeV proton beam incident on Al and Pt targets [1]. Two
interesting observations were made about the production of deuterons. First, the
measurements, which were made at 15.9° to the beam axis, found that deuterons were
copiously produced. Second, the ratio of deuterons to protons was independent of
momentum. This led the experimenters to conclude that the production mechanism was
something other than a pick-up process. The pick-up process corresponds to the incident
proton "snatching" a neutron from a particular energy state within the target nucleus. The
resulting deuteron energy spectrum thus shows discrete energy states, as does the angular
3
distribution [2], This process is clearly not consistent with the data. Subsequent
measurements at Brookhaven's Alternating Gradient Synchrotron (AGS) using 30 GeV
proton beams incident on Al and Be measured the production of deuterons and tritons out
to angles as large as 90° [3], They too concluded that the deuteron production was not
consistent with a pick-up process. It was then proposed that the deuterons could be
directly produced in elementary nucleon-nucleon collisions [4], This, however, was ruled
out because it predicted a strong dependence on energy and production angle, which was
not seen in the data. In fact, deuterons were observed at angles well beyond those which
are kinematically allowed.
It was in 1963 that Butler and Pearson [5] first proposed a very different
mechanism for deuteron production. In their model deuterons are produced by the
coalescence of a neutron-proton pair in the cascade, or shower, of secondary nucleons
that develops in the target nucleus during the collision. This process allows a neutron-
proton pair to form a deuteron if its relative momentum is small. The recoil momentum is
absorbed by the nuclear optical potential. Their calculation predicts a relationship
between the momentum spectrum of the deuteron and the square of the proton momentum
distribution. The proportionality constant is related to the depth of the optical potential.
This model adequately reproduced all available data on deuteron production.
Soon after, the Butler-Pearson model was generalized by Schwarzschild and
Zupancic [6]. Instead of relying on the optical potential of the target nucleus, they
phrased the theory in terms of a phenomenological parameter, the coalescence radius, p0.
This parameter characterizes the maximum amount of relative momentum that the neutron
and proton can have and still coalesce. Deuteron formation was seen to be "governed by
the probability of finding a neutron within a small sphere of radius p around the point
representing any given proton in momentum space" [6], This simpler model also fit data
quite well, and turned out to be more reasonable to apply to relativistic nucleus-nucleus
4
collisions. In such collisions one cannot rely on the optical potential of the nucleus since
the colliding nuclei cease to exist as individual entities.
It was not until the mid 1970's that the Schwarzschild and Zupancic model was
generalized to clusters of mass A and applied to heavy ion collisions [7,8], The argument
is as follows: The probability of finding a nucleon within a sphere of radius p0 centered
around momentum p is
where M is the average number of nucleons in the interaction region and (yd3N/dp3) is
the invariant momentum distribution of nucleons. The probability of finding A nucleons in
such a sphere is given by the binomial distribution
( 1 - 2 )
If the nucleon multiplicity is large and the number of nucleons per cluster is small, then
A«M and A ) !« M a. Also, if the mean number of nucleons in the sphere is
small, i.e., <P M«1, then (l - <P)M_A «1. This results in
<P(A) = ftT(M<P)A. (1.3)A!
By substituting Equation 1.1 in and setting this equal to the probability of finding a cluster
of mass A within a sphere in momentum space, we get a cluster momentum distribution of
where = Ap. Now, we need to take into account the fact that the cluster is made up of
Z protons and N neutrons. Since all protons are identical we must account for the number
of combinations of Z protons out of A nucleons, which leads to a factor of A!/(Z!N!). We
shall assume that the neutron and proton momentum distributions are the same, except for
the ratio of the total number of neutrons to protons in the projectile and target
5
(1.5)
Finally, we must take into account spin factors. This is done by multiplying by the spin
degeneracy of the cluster, 2sA+l, and dividing by the degeneracy of Z protons and N
neutrons
In order to express the equation in the terms of invariant yields (see §1.3.), which are
measured experimentally, we multiply by A m, where m is the mass of the nucleon, and
use the fact that EA = mAy = Amy to get the coalescence equation
used a different form of the coalescence radius that incorporated the spin factors into it.
The relationship between that radius, po, and the one used above is
The coalescence equation (1.7) implies some interesting properties. It shows the
power law relationship between the cluster momentum distribution and the proton
light nucleus production in relativistic nucleus-nucleus collisions reasonably well in a
plethora of experiments, most of which were performed at the Bevalac. An excellent
review of experimental and theoretical work done up until 1985 can be found in the paper
by Csemai and Kapusta [9], Perhaps the most striking example of the success of this
power law relationship is the measurement by Jacak, et al., [10] in which nuclei up to mass
14, produced in collisions of 137 MeV/nucleon Ar incident on Au, were successfully fit by
the coalescence equation. This success is, in some sense, surprising since the proton
(1.6)
(1.7)
where sA is the spin of the cluster. It should be noted that much of the earlier literature
(1.8)
distribution, hinted at by Butler and Pearson. This relationship has been found to describe
6
spectrum that appears in Equation 1.7 is actually the pre-clustering momentum
distribution, but experiments use the measured proton distributions. In fact, the agreement
is not as good when the proton distribution is corrected for depletion affects [11].
Another interesting aspect of this model is that it depends only on the intrinsic
properties of the cluster. The only parameter in the model is the coalescence radius, which
characterizes how close in momentum the nucleons must be in order to coalesce. It
therefore follows that the coalescence radius should be independent of the type of collision
or how the cluster was emitted. In fact, the coalescence radius has been found to be
independent of angle of emission and momentum of the cluster in many experiments [9],
However, some dependence on the target and projectile was seen, as well as a weak beam
energy dependence [11]. More serious discrepancies were seen when low impact
parameter collisions were selected [12],
Some of the inadequacies of the model lie in the fact that no allowance is made for
the dynamics of the system in which the clusters are formed. The sole parameter is related
only to properties of the cluster, and there is no prediction for the value of the coalescence
radius, or how it is related to properties of the environment in which the clusters are
produced. Thus, it does not allow one to extract any information about the properties of
hot and dense nuclear matter that is formed in heavy ion collisions.
In an attempt to give this empirical coalescence model a more dynamical basis
Bond, Johansen, Koonin, and Garpman [13] worked out an approximate treatment of the
many-body problem. By assuming that the A-nucleon phase space distribution could be
factored into A one-nucleon distribution functions, they reduced an almost impossible
calculation into a more manageable one. Under the presumption that the coalescence
happens on a time scale that is short compared to the inter-particle collision rate, the
sudden approximation of quantum mechanics can be used. The problem can then be put in
terms of density matrices and an overlap of the wave functions of A nucleons with the
cluster wave function. The wave function overlap can be neglected by assuming that the
7
nucleons are uniformly distributed over the spatial extent of the cluster. With the final
assumption that the nucleon phase space distribution is independent of position within
some volume, V, the calculation results in the equation
d3NA 2sa +1A-l
fd’N p ]
zf d X }
dpi 2A V l dpS J I dPn J
This reproduces the basic power law of the empirical coalescence model, however the
scale factor is related to the volume of the interaction region instead of a
phenomenological parameter. More recent calculations [14,15] attempt to treat the wave
function overlap in a more realistic way and result in similar, albeit more complicated,
forms.
This calculation does provide a more rigorous explanation for the coalescence
model, and, at the same time relates the coalescence radius to properties of the
environment. However, the proton distribution in Equation 1.9 is the pre-clustering
distribution, as in the empirical model. Thus, not all of the inadequacies are removed.
An alternative approach to coalescence, pioneered by Mekjian [16], is based on
thermodynamic equilibrium. In this model it is assumed that early in the collision a region
of high temperature and density is formed, in which the mean free path of the nucleons is
short compared to the size of the system. This leads to both thermal and chemical
equilibrium. In this equilibrium, composite particles are formed but are quickly dissociated
by collisions with other nucleons. Thus, a balance is achieved between the formation and
dissociation of composite particles. When the density decreases to the point where
collisions are infrequent, the composites can then survive. If the transition from the
equilibrium state to the non-interacting state happens quickly (freeze-out), the relative
yields of these clusters will reflect the equilibrium in which they were formed. This
framework is analogous to nucleosynthesis in the big bang and supemovae [17],
8
In this model the relationship between the cluster momentum distribution and the
proton distribution can be calculated. Under the assumption that an equilibrium is
established in a volume V, and that the phase space density is low, the following equation
can be derived [18]:
Here, Rnp is as defined in Equation 1.5, sA is the spin of the cluster, and the binding
energy and excited states of the cluster have been neglected. It is interesting to note that
this model exhibits the same power law structure as the coalescence model. However, in
this case, the proton momentum distribution corresponds to the observed distribution, not
the pre-clustering distribution implicit in the coalescence model. Also, we see the same
relationship between the scale factor and the interaction volume that results from the
many-body coalescence calculation.
The fact that the interaction volume appears in the scale factor permits the
calculation of the size of the system at the time that the composite particles are formed, or
freeze-out. This is done by measuring the ratio of the cluster cross section to the A111
power of the proton cross section. Several experiments have determined the radius of the
interaction volume, assuming a spherical volume. The resulting radii are of the same order
as the radii of the projectile nuclei [10,11], However, a systematically smaller radius is
found for more tightly bound clusters, which may be an indication that they are emitted by
a smaller source [19], and thus might be formed earlier in the evolution of the system.
Several other models have been developed to describe the emission of light nuclei
from relativistic heavy ion collisions. Fireball and firestreak models based on
thermalization of a region (or regions) of the interaction volume have been used with
varying success, as well as hydrodynamic models. These models are reviewed in Das
Gupta and Mekjian [18], Although they agree reasonably well with data, they have not
(1.10)
9
been as successful as the models that result in a simple power law. Finally, models based
on nuclear cascade calculations have also had a reasonable amount of success in
reproducing data. Some of the early work is reviewed in Csemai and Kapusta [9]. The
more recent work with cascade calculations is very promising [20], and will be discussed
in detail in Chapter 5.
1.2. Coalescence at the AGS
The data discussed in the previous section were all taken at Bevalac energies, i.e., 2.1
GeV/nucleon and lower. This dissertation will examine cluster production at the higher
energies achieved at the AGS. The systems created in collisions at 14.6 GeV/nucleon at
the AGS constitute a different environment from those made at lower energies. At
Bevalac energies the velocities of target and projectile fragments are not sufficiently
separated so as to clearly distinguish them from the clusters formed in the excited region
formed in the center-of-mass of the colliding nuclei [10]. However the target and
projectile velocities are very well separated at the AGS. The temperatures (as determined
by the transverse momentum spectra) of nucleons emitted at velocities near that of the
center-of-mass of the system are on the order of 150 MeV at the AGS, as compared to
about 50 MeV at the Bevalac. In addition, collisions at the AGS produce many more
pions and kaons, demonstrating that more degrees of freedom are available at higher
energies. Under these conditions one expects that densities will be higher and that the
system may experience a significant amount of expansion before freezing out. Recall that
model calculations of source sizes in Bevalac experiments resulted in sizes on the order of
the projectile nucleus. We shall see if this is the case at the AGS.
Theoretical predictions for the evolution of nuclear matter in phase space suggest
that peak densities achieved are 8 times that of normal nuclear matter [21], These
densities are predicted to last for several fin/c, implying some degree of equilibration. It is
important to study these phase space trajectories experimentally, both to test such
calculations and to search for phase changes.
As suggested in the thermodynamic model, the study of light nucleus production
can provide information on the size of the system, and thus the density at freeze-out. Such
measurements, in conjunction with measurements of the temperature of the system, will
furnish information on the state of nuclear matter under such extreme conditions.
Furthermore, if, as suggested by previous measurements, more tightly bound nuclei are
formed earlier in the collision, one might be able to study the evolution of nuclear matter
through phase space.
1.3. Variables
Before proceeding, it will be advantageous to discuss some of the variables that will be
used in the analysis. A cylindrical coordinate system will be used throughout this
dissertation. The only relevant axis is the beam axis, which will be defined as the z axis.
Since neither the target nor the projectile is polarized, there should be a symmetry about
the z axis. So variables will be expressed in terms of longitudinal and transverse
components. Due to cylindrical symmetry, the polar angle will usually be integrated out.
The motion along the beam axis is relativistic. The relativistic y (= ^ l / ( l -p 2)) of
the beam is approximately 15, so it is useful to express variables in Lorentz invariant form.
First, note that all transverse components are inherently Lorentz invariant. The
longitudinal momentum, p/, however, is not. In its place we will use rapidity, y, which is
defined as
11
where E is the total energy of the particle and c = 1. The rapidity, which can be thought
of as the longitudinal velocity, is a particularly useful quantity because of its
transformation properties. A Lorentz transformation along the beam axis simply
corresponds to a shift along the rapidity axis. Thus, distributions expressed in terms of
rapidity will retain their functional form when transformed.
There are two other quantities that are useful in describing the kinematics of a
particle. They are the transverse momentum, pt, and the transverse mass, which is defined
as
m, = V p? +m2 O -12)
where m is the rest mass of the particle. Since both pt and mt are transverse quantities,
they are invariant under Lorentz transformation.
In the experiment to be discussed, we will express the results in terms of a
Lorentz-invariant differential cross section, most commonly called the invariant cross
section:
d3a d2a— r = E -------dp 27tp;dp;dp,
In order to put this in terms of rapidity, first observe that by substituting the rapidity in
terms of E and p/ into the definition of cosh and sinh we obtain the relations:
sinh(y) = — (114)m,
cosh(y) = — . (1.15)m,
By differentiating Equation (1.14) and using Equation (1.15) we find
Edy = dp,. (116)
Finally, by observing that ptdpt = mtdmt, the invariant cross section can be expressed in
terms of the rapidity, and either pt or mt:
12
E d3q _ 1 d2o _ 1 d2qdp3 2x ptdp,dy 2tc mtdmtdy
One final definition is in order. It is often more convenient to think in terms of the
number of particles emitted per event of interest, rather than cross section. What is meant
by "event of interest" is a collision that is characterized as interesting in some way.
Usually, interesting events are characterized by their impact parameter. This will be
discussed in detail later. There is a certain probability that a random nucleus-nucleus
collision will be an event of interest, and that probability is characterized by the reaction
cross section, q0. We then define the invariant yield, Njnv, as the ratio of the invariant
cross section to the reaction cross section:
Nm = - -----U ^ - = — 1------ ^-2— (1.18)2 k p.dp.dy 27tq0 p.dp.dy
The invariant yield is just the number of particles, N, emitted at a particular rapidity and
transverse momentum per event of interest. Most results in this dissertation will be
expressed in terms of the invariant yield.
C h a p t e r 2
E x p e r i m e n t a n d A p p a r a t u s
2.1. Experiment 814
Experiment 814 was proposed in October of 1985 to study electromagnetic and nuclear
interactions of high energy heavy ion beams at Brookhaven National Laboratory's AGS.
A schematic representation of the apparatus is shown in Figure 2.1. Upstream of the
target is a set of beam defining-scintillators. Surrounding the target is a set of calorimeters
providing nearly 4n of coverage. These, along with the silicon multiplicity detector
located just downstream of the target, are used to measure global observables for event
characterization. Downstream of the target calorimeters is the forward spectrometer,
which measures reaction products within a rectangular aperture centered around the beam
axis. The spectrometer provides measurements of the momentum, charge, velocity, and
energy of particles entering it. It is designed to have full acceptance for the reaction
products of large impact parameter, or peripheral, interactions, for studying the
dissociation of nuclear projectiles in the Coulomb field of the target nucleus, as well as
beam velocity fragments from nuclear collisions. In addition, the spectrometer allows the
13
Figure 2.1:
A schematic
of the Experim
ent 814 apparatus.
15
measurement of a small sample of the particles with center-of-mass velocity produced in
small impact parameter, or central, collisions.
The E814 apparatus has proven to be a flexible tool. The E814 collaboration has
performed a diverse series of studies with this apparatus. Measurements of the
electromagnetic dissociation of projectile nuclei [22] have shown that it is possible to
measure the properties of excited heavy ions with sufficient resolution to probe their
structure. Energy flow [23] and baryon distributions [24,25,26] have been studied to
determine the amount of nuclear stopping and thermalization. Also, searches for such
exotica as pineuts [25] and strangelets [27,28] have helped to constrain parameters of
QCD, and a study of the production of antiprotons [29,30] has increased our knowledge
of the collision environment and formation times. The measurements in this dissertation
were some of the last to be made on this apparatus with silicon beams. The apparatus will
continue to be used to study interactions with Au beam, recently made available at the
AGS.
The following sections describe the detectors which make up the E814 apparatus.
A list of specific electronics and photomultipliers used can be found in Appendix A.
2.2. Beam Definition
The beam scintillator telescope (BSCI) accomplishes the dual task of measuring the
charge of the beam particle entering the apparatus and providing a time reference for the
experiment. The telescope consists of four scintillators as shown in Figure 2.2. The
beam passes through two thin scintillator disks, S2 and S4. Each disk is viewed by two
photomultiplier tubes. A coincidence between these two scintillators serves to define the
beam. In addition, there are two thick annular scintillators (SI, S3) that are used as
vetoes. These veto scintillators are each viewed by four photomultiplier tubes. The
16
S I S3• 0.6 cm
15.0 cm
I " cm
S2• 0.0S cm
1.9 cm
S4
II0.0S cm
0.7 cm
■ 0.6 cm
IS.O cm
[ 1.0 cm
target
177.8 cm--------------- 203.2 cm---
627.4 cm--- 652.8 cm---
Figure 2.2: The beam scintillator (BSCI) telescope.
«--------------------- — — 8.17 cm------------------ *
«-------3.37 cm
Detector 1 Detector 2
Figure 2.3: The silicon multiplicity array.
17
photomultipliers for the S4 scintillator have better timing qualities than the others, and are
used to determine tp, or the start time for the experiment. The pulse height information
from all four scintillators and the timing information from S4 are used in the trigger to
determine when a beam particle has entered the apparatus.
The pulse height and time information from all beam scintillators is digitized and
written to tape so that the beam definition may be refined in the off-line analysis. In
addition to the beam scintillators, the upstream silicon detector, located between the beam
scintillators and the target, can be used to help identify the beam. This detector has
excellent charge resolution, however, it is not available for use in the trigger. The pulse
height information from the upstream silicon detector is written to tape and used in the
off-line analysis.
2.3. Event Characterization
It is desirable to know something about the topology of the heavy ion collisions. Ideally,
we would like to know the impact parameter of the two colliding nuclei. Since it is not
possible to directly measure the impact parameter, we are forced to characterize the events
by various global observables, like the transverse energy (Ej) and the number of charged
particles (Nc) emitted from the nucleus-nucleus interactions. While these variables do not
tell us the impact parameter, they are a gauge of the number of nucleons participating in
the collision, which is correlated inversely with the impact parameter.
In Experiment 814 there are three sets of detectors that measure global
observables. First, the silicon multiplicity array (MULT) measures the number of charged
particles emitted in the forward hemisphere. Second, the target calorimeter (TCAL)
measures the transverse energy over most of the solid angle around the target. A third
detector, the participant calorimeter (PCAL), measures the transverse energy for more
18
forward angles. The PCAL was not used in this measurement, and thus will not be
discussed here. Some details about the TCAL and multiplicity array are discussed below.
2.3.1. Multiplicity Array ( M U L T )
The charged particle multiplicity array [31] (Figure 2.3) consists of two annular silicon
pad detectors. Each detector is 300 pm thick and 38 mm in radius. The active area,
which extends to a radius of 34 mm, is divided into 512 pads. The detectors are located at
a position downstream of the target such that they cover an angular range of 2.4° to 45.3°
in the laboratory frame. This corresponds to a range of pseudorapidity (q) from 3.86 to
0.88.
Each pad in the detector is connected to a preamplifier and a discriminator. In
order to be sensitive to minimum ionizing particles, the discriminator thresholds were, set
to values corresponding to one half of the most probable energy loss that a minimum
ionizing particle would deposit in the silicon at normal incidence. Only the pattern of
which pads fired is recorded, hence it is not possible to determine how many charged
particles hit a pad that fired. This insensitivity to multiple occupancy results in a measured
multiplicity that is lower than the true value. The magnitude of this effect is less than 20
percent and can be corrected for off-line.
In addition to the pattern of hit pads, an analog sum of hit pads is formed. This
sum provides an estimate of the total multiplicity of each event, and is available to be used
in the trigger.
2.3.2. Target Calorimeter ( T C A L )
For the purpose of measuring the transverse energy of an event, the target is surrounded
by a calorimeter of 992 Nal(Tl) crystals. Each crystal is 13.8 cm in length, corresponding
19
to 5.3 radiation lengths and 0.33 hadronic interaction lengths. The crystals are arranged in
an approximately projective geometry, consisting of five walls, forming an open box
around the target. Four of the walls are parallel to the beam axis and equidistant from the
target. The fifth wall is upstream of the target and has a hole for the beam to go through.
The TCAL covers a pseudorapidity range of -2.0<r|<0.8. Details about the testing,
calibration, and operation of the TCAL can be found in reference [32],
The TCAL signals are read out using vacuum photodiodes that were mounted on
the face of each crystal. After pulse-shaping, the signals from the diodes are digitized and
available for recording by the data acquisition system.
Lining the four walls parallel to the beam axis, is an array of 52 scintillator paddles
(TPAD). Each paddle lies along a row of crystals. The signals are summed together and
discriminated in order to provide a crude multiplicity measurement for the trigger system.
2.4. Forward Spectrometer
Our experiment is designed to study distributions of particles with low transverse
momenta. Thus, we need to identify all particles that are emitted in a small solid angle
around the beam axis. This solid angle is defined by the opening in the participant
calorimeter, which for this measurement was 38 mrad in the x direction by 24 mrad in y
and centered around the beam axis. The identification of particles that pass through this
opening is accomplished by the forward spectrometer, located just downstream of the
PCAL.
The spectrometer consists of a set of tracking chambers, a scintillator hodoscope,
and a calorimeter. These detectors are described in detail below. The procedure for
identifying particles is as follows. First, tracks are reconstructed from position
information provided by the tracking chambers, the hodoscope, and calorimeter. From the
20
curvature of a given track in the magnetic field and the charge, as determined by the
scintillators, the momentum can be calculated. The identity of each track is determined by
momentum, time of flight, and charge measurements.
2.4.1. Tracking Chambers
There are three chambers in the tracking system. They were designed to handle high
multiplicities and have a large dynamic range, since they must detect beam fragments as
well as minimum ionizing particles. They are equipped with a moveable reduced gain
section since the beam passes directly through them. Further, they were designed to
minimize the amount of mass in the particle trajectories in order to reduce the amount of
multiple scattering and secondary hadronic interactions. The design and performance of
the tracking chambers has been described in [33,34,35],
The first chamber, DCI, and is shown schematically in Figure 2.4 (a). It is located
4 meters from the target, and between spectrometer magnets D8 and D9. The active area
is 16 cm vertically (y) and 26 cm in the bend plane of the magnets (x). The chamber
consists of a single plane of wires oriented horizontally, with 4 mm anode wire spacing.
One cathode plane is made of printed circuit board, the surface of which is segmented into
0.8 mm by 2 mm pads. The other cathode plane is provided by the aluminized mylar
window. Field-shaping wires are located between each anode wire to improve the
uniformity of the electric field around the anode wires. The pads have a resistive coating,
so that resistive charge division can be employed to determine a particle's position along
the wires. The vertical position is determined solely by which wire was struck. The
vertical resolution is thus given by the wire spacing. The horizontal position resolution is
120 pm.
The two remaining tracking chambers, DCII and DCIII, are similar, differing
significantly only in their size. DCII has an active area of 80 cm in x by 30 cm in_y, and is
21
(a)
WINDOW■ • neijD WIRE r . ANOOC WIRE-~-e— ■ pad
X---- - CUARO S W
WINOOW
cuaro snap resistm; snap
FIELD WIRE ANOOC WIRE
PAD
(b ) Bean View Chevron Pad Plane
(Detail)
Chevron Pad Plone
Typical Drift Cell
125 Micron Field Wires17 Micron Anode Vires
Drift C h amber Plan View
0.3 cn DCII 0.6 c n DC I I I
Bean DirectionFigure 2.4: (a) Schematic views of the DCI chamber, (b) Schematic views of DCII and DCIII.
22
located 6.9 meters downstream of the target. D C m is located 11.6 meters from the target
and has an active area of200 cm in x by 50 cm in y . Both of these chambers, represented
schematically in Figure 2.4 (b), have a drift section, consisting of 6 wire planes, followed
by a pad plane made of chevron shaped pads. Geometric charge division is used to
determine vertical positions. The wire planes in the drift section consist of anode wires,
separated by 6 mm and 12 mm in D C II and D C m , respectively. Field-shaping wires are
located between each anode wire. In order to facilitate track reconstruction, the positions
of the sense and field wires are staggered on alternate planes. Each wire plane is
separated by aluminized mylar cathodes. Positions in the drift section are determined from
the time it takes ionization electrons to drift to the wires. This yields a resolution (a) of
120 pm in D C II, and 140 pm in D C III. The wires are parallel to the y-axis, giving the
best position resolution in the bend plane of the spectrometer.
2.4.2. Scintillator Hodoscope
The forward scintillator hodoscope (FSCI) is used to measure the charges and times of
flight of charged particles in the forward spectrometer. It is crucial for particle
identification, as it provides two of the three quantities needed, the third being the rigidity.
Additionally, the FSCI supplies x and y information for use in tracking.
The FSCI consists of three separate walls, each located approximately 5 meters
upstream from a calorimeter section. Only the large downstream wall, located 31.32
meters from the target, was used for this measurement. This wall contains 44 individual
scintillator slats, each being 10 cm wide, 120 cm long, and 1 cm thick, as shown in Figure
2.5 (a). Both ends of each slat are optically coupled to photomultipliers through acrylic
light guides.
Both time and pulse height information are obtained from each tube. The anode
signal from a given tube is split in two. One output sent to an ADC and is recorded for
23
use in determining the charge off-line. The other output is discriminated using a dual
output discriminator and is fed into both a TDC and a Fast Encoding Readout TDC
(FERET). The TDC information is recorded for time of flight determination, and the
FERET output is available for use in the trigger. In addition, the dynode signals from the
top and bottom tubes of each slat are combined in an analog sum, and digitized by Fast
Encoding Readout ADC (FERA). This pulse height information may also be used in the
trigger.
2.4.3. Uranium Calorimeter
The final detector in the forward spectrometer is the UCAL, which is a
uranium/copper/scintillator sampling calorimeter. It is divided into three sections, as
shown in Figure 2.1, however, only the downstream wall, located 36.31 m from the target,
was used in these measurements. The wall consists of 20 individual modules, each being
120 cm high, 20 cm wide, and 75 cm deep. A module, shown in Figure 2.5 (b), is
composed of 41 sections, each containing a 5 mm copper plate and two 3 mm depleted
uranium plates. These absorber layers alternate with sheets of 2.5 mm thick plastic
scintillator. The scintillator sheets are divided into 12 optically decoupled towers, each 10
cm high. The light from each tower is collected by wave shifter bars mounted along both
sides. Each wave shifter extends the entire length of the module, and is read out the back
with a photomultiplier. There are a total of 24 photomultiplier tubes per module.
The signal from each photomultiplier is split in two parts. One is digitized and
recorded by the data acquisition system. The other part goes to an analog summing unit
which sums the outputs of the 24 photomultipliers of each calorimeter module. This sum
signal is sent to both a discriminator and a FERA. The discriminator output is used for a
Figure 2.5:
(a) A
forward scintillator
(FSCI) slat,
(b) A
uranium calorim
eter (U
CA
L) m
odule.
(a )
PHOTOMULTIPLIERTUBES
( b )K>4k
25
time measurement by sending it to a TDC, while the FERA information is available for use
in the trigger. For measurements in this work, only the individual photomultiplier ADC
values were used.
The UCAL measures the energy of particles incident on it. The energy resolution
has been measured to be [28]
oe/E = 0.55/VE ©0.14. (2.1)
In addition to the energy, x and y positions can be determined. The position resolution
(o), as measured with 12 GeV hadrons, is 1.3 cm in x and 1.8 cm in y [36].
2.4.4. Pattern Recognition
The information recorded to tape from all of the detectors in the forward spectrometer
must be correlated in order to determine the trajectories of the particles that passed
through them. After calibration, the data for an event consists of drift times for individual
wires in the drift chambers, deposited charge for each channel in the pad planes, time and
pulse height for each phototube in the forward scintillators, and the energy measured in
each tower of the calorimeter. The tracking code first correlates hits within a given
detection plane, since the passage of a particle often leaves signals in several channels of a
detector, as in the pad planes. Then hits in different planes are matched up to identify
track candidates. Finally, each potential track is traced back through the magnets to see if
it could have come from the target.
Each detection plane, or pseudoplane, has position information on each hit it
registered. The pad planes measure both x and y positions, as do the FSCI and UCAL.
For the pad planes and the UCAL, the signals in channels that are above threshold are
sorted into groups called clusters. A cluster is a set of contiguous channels whose signals
are most likely due to a single particle striking the detector. The clustering algorithm
works by finding the channel with the largest signal and assigning a certain number of its
26
neighboring channels to it. Once these channels are removed from the list of hit channels,
the next largest signal is found, and the process is repeated. The scintillators in the FSCI
supply the x position simply by which slat is hit, while the y position is determined from
the difference in time between the top and bottom phototubes.
The drift sections of DCII and DCm are each treated as a single pseudoplane.
They give information on x position, but also supply directional information, since they
each contain six separate wire planes. Each wire that fired has associated with it a drift
time, which is converted into a distance from the wire. It is not possible to determine
which side of the wire the particle passed, so a hit on both sides is assumed. This left-right
ambiguity is reduced by the fact that the positions of the sense wires are staggered on
alternate wire planes In order to sort these hits into groups, called elements, associated
with a single particle, a "tree" algorithm is used. From each hit on the first plane, all
possible links are formed to hits on the next plane. Any link whose slope exceeds a certain
value is discarded. This is repeated for all six wire planes, thus forming all possible paths.
Each acceptable path, or element, is saved for further consideration. To overcome
detector inefficiencies, the algorithm allows the formation of elements with as few as three
hits. Some elements, however, may share wire hits, forcing a decision to be made on
which most probably reflects the correct path. In these cases, the hit positions that
compose each element are fit to a line and the element whose fit has the smallest x2 is
chosen. This can cause a valid track to be discarded if two tracks are veiy close together,
however since the mean number of tracks is low (approximately 2) and the number of
wires in the drift planes is large (128 in DCII and 160 in DCIII), the probability of such an
occurrence is less than 1%.
Next, the clusters and elements in the pseudo planes downstream of the magnets
are grouped together to form segments of tracks. The same sort of tree algorithm that is
used to form elements is employed here. In this case, however, for a link to be acceptable
it must be trackable, i.e., it must be possible to trace the link through the opening in the
27
PCAL and to the target, in both x and y. At this stage, tracking through the magnets, in
the bend plane, is accomplished through a single-bend model with small-angle
approximations. For the elements, which have directional information, they may either be
trackable or simply point at each other, in the sense that the projection of each
measurement on the other agrees within uncertainty.
Finally, each segment is traced back through the magnets, using a more detailed
magnet model. Initially, each candidate track is constrained to originate from the center of
the target. The x and y positions at DCI are then calculated and compared to all clusters
found. If a DCI cluster is found to match within uncertainties, the candidate is
appropriately flagged. The final values of the track parameters are then calculated. These
parameters include the position and angle at the target, and the rigidity. DCI information
is used if it is available. In cases where candidates share clusters or elements, the
candidates with associated DCI clusters are chosen over those without them. The
remaining ambiguities dealt with by choosing the "set of maximum compatibles", i.e., the
set of solutions that yields the maximum number of uniquely defined tracks. In cases
where the set of maximum compatibles is still not unique the set with the smallest average
deviation of the hits from the track is chosen. The above discussion does not apply to the
FSCI clusters. Tracks were allowed to share hits in the FSCI since the slats are quite
wide. Each successful candidate track is stored, along with calculated parameters,
associated uncertainties, and references to the individual segments, clusters, and elements
that comprise it. For a more detailed description of the pattern recognition see [25,29],
2.5. Time-of-Flight Trigger
The TOF trigger is designed to accept events in which at least one interaction product
enters the forward spectrometer and has a velocity near that of the center of mass. This is
28
accomplished by placing a requirement on the time of flight as measured in the FSCI, and
having the field in the spectrometer tuned such that mid rapidity particles of interest will
hit the scintillator wall in the region instrumented with trigger electronics. In the
implementation of this trigger, the task is divided up into four stages, beam trigger, pre
trigger, Level 1, and Level 2. The trigger logic is schematically shown in Figure 2.6.
The beam trigger is necessary to signal the passage of a beam particle into the
apparatus. A good beam signal is defined by a coincidence between the two beam
scintillators of the BSCI, the thresholds of which are set for a Z=14 particle. In addition,
the signal in the annular veto scintillators must be below that expected for a minimum
ionizing particle. Thus, a beam trigger is generated by the following logical requirement:
BEAM = S, • S2 • Sj • S4 • busy
where the busy signal is generated once the pre-trigger is satisfied. This ensures that a
beam trigger is generated only when the system is free to process the event. Additionally,
a beam trigger is inhibited if any of the BSCI scintillators fired within a window of 1
microsecond before that trigger. This prevents cases where a beam particle that does not
cause a pre-trigger to be generated enters the apparatus immediately prior to one that does
cause a trigger.
The main function of the pre-trigger is to determine if a nuclear interaction
occurred. This is accomplished through the measurement of the multiplicity of charged
particles. A threshold on the number of particles in the silicon multiplicity detector, in
conjunction with a certain number of the TP AD scintillators firing, make up the pre
trigger requirement. The values for these thresholds were target dependent, and were set
as close to minimum bias as the trigger rate limitations would allow. Table 2.1 contains
the pre-trigger thresholds used for each target.
If the pre-trigger requirements are satisfied, a gate is generated to strobe the ADCs
and provide an appropriate start or stop signal for the TDCs. At this point, the Level 1
trigger begins. This part of the trigger is a late beam veto. If none of the BSCI
29
BCAM-f \busy-J— ) l_-
PULSER h — \ r—BUSY-4— '
GATES TO ADC. TOC BUSY
4-ClCAR
START
< 3
NO LEVEL YES
/— NO
START
DAQ
I END
Figure 2.6: A block diagram of the time-of-flight trigger logic.
30
Target MULT Threshold TP AD Threshold2% Pb 2% Cu 2% Al 4% Pb 4% Cu 4% Al
303030453530
3334 4 4
Table 2.1: Pre-trigger thresholds for targets used. The MULT threshold is the number of charged particles in the silicon multiplicity detector, and the TP AD threshold is the number of target paddle scintillators that fired.
The Level 2 trigger determines if there was a late particle in the spectrometer by
examining the time measurements in each FSCI slat in the trigger region, shown in Figure
2.7. The scintillators in this region are numbered 15-38, with slat number 15 being the
furthest from the neutral line. The time measured in a given slat is calculated from the
FERET signals. An on-line slewing correction is made to these times using the FERA
information. This correction removes the pulse height dependence of the timing signals
and yields an on-line timing resolution of 600 ps. Each signal also has a value
corresponding to zero time subtracted from it to correct for differences in flight path and
cable delay.
The time signal from each slat is then compared to the average time it would take
for a particle traveling at v = c to reach it. The Level 2 trigger is satisfied if any FSCI slat
in the trigger region measures a time of flight between 2 and 30 ns. This corresponds to a
rapidity range of 1.0 to 2.3. Note that the nucleon-nucleon center of mass rapidity is 1.7.
scintillators detect a particle within 1 ps of the trigger particle, then the Level 1 trigger is
satisfied.
31
N E U T R A L
n e t e r s
Figure 2.7: The active region of forward scintillator wall for the time-of-flight trigger. The slats in this region are numbered 15-38, with slat number 15 being the furthest from the neutral line.
32
The reason for the cutoff at 30 ns is to avoid erroneous signals due to albedo, the spray of
back-scattered particles, from the calorimeters.
An event for which all four sub triggers, beam, pre, Level 1, and Level 2, are
satisfied, is thus deemed a valid TOF trigger. The data acquisition is then signaled to
record the data from all detectors to tape, and label the event as a data trigger. In addition
to TOF triggered events, a small number of other triggers are written to tape. These
include beam, pre, and empty triggers. An empty trigger is one in which there was no
beam in the apparatus. These extra events are useful in determining ADC pedestals (the
value that an ADC registers when no signal is present) and for use in calibration studies.
2.6. Data Acquisition System
The signals from most detector systems are read out and digitized by Fastbus electronics.
The FSCI and MULT detectors, as well as the trigger, are handled by CAMAC
electronics. Each Fastbus crate is controlled by a SLAC Scanner-processor (SSP). They,
in turn, are controlled by an SSP in the master Fastbus crate, which handles signal flow
and event management. When the second level trigger signals that an event is to be
recorded, the master SSP generates a busy signal to inhibit the trigger system, while
signaling the lower level SSPs to begin readout of the modules in each crate. The master
SSP coordinates the transfer of data from each crate, CAMAC and Fastbus, to a 4 Mbyte
memory module, also located in the master crate. Once all of the data is in memory, the
busy signal is cleared and the system is ready for another event. At the end of an AGS
spill, the memory module is read out by a Microvax III, which then writes the data to one
of the two high speed 9 track tape drives available. User interface to the data acquisition
is provided by VAXONLINE [37], a Fermilab software package, which also furnishes
events to an on-line monitoring system.
33
The data discussed in this work were taken over a two week period in February of 1991 at
Brookhaven National Laboratory's AGS. The AGS provided a beam of 28Si nuclei in
"spills" of approximately 1 second in duration. The time between spills was about 3
seconds. The beam momentum was 14.6 GeV/c per nucleon. The intensity of the beam
was approximately 105 particles per spill. Of this intensity only 40% satisfied the
requirements of the beam trigger. The target materials used were Pb, Cu, and Al. For
each material, two different target thicknesses were used, one corresponding to
approximately 2% of an interaction length, the other was about 4%.
Initially, the thinner targets were used to reduce the probability of secondary
reactions the target. This affords better event characterization. Half way through the run,
however, it was decided to switch to the thicker targets to increase the rate at which the
light nuclei of interest were produced. Increasing the beam intensity was not an option
because of the limitations of the amount of beam that the drift chambers could handle.
The loss in event characterization was minimal. Thus, where applicable, results from the
two thicknesses of a given target material are combined. Details of the targets used can be
found in Table 4.1.
2 .7 . T h e D a t a
C h a p t e r 3
A n a l y s i s
The purpose of the first stage of the analysis is to reduce the number of events that must
be analyzed by sorting out erroneous triggers. The trigger was designed to determine
which events were interesting enough to record. In this case, an event with at least one
particle moving at or near center-of-mass velocity is considered interesting. Such an event
is distinguished by requiring a late hit in the scintillator hodoscope, as described in §2.5.
However, there are several ways that the trigger can be erroneously satisfied. The beam
itself has a non-negligible amount of contamination. If a low Z particle, such as a proton,
were to enter the apparatus a short time after the passage of a beam particle, it would go
undetected in the beam scintillators, its signal dwarfed by the signal from the silicon
nucleus. Such a particle could cause a late time signal in the forward scintillators. A
similar situation occurs when a second beam particle enters the apparatus within the beam
gate. This double beam effect is rejected to a high degree in the trigger, but the rejection
is not perfect due to misalignments and gaps in coverage of the beam scintillators. In
addition to these effects, late hits can result from reaction products that decay in flight or
interact with intervening air or detector material.
34
35
Once events with erroneous triggers have been eliminated, tracks must be
reconstructed out of the information from individual detectors. Events which are known
to have valid late hits in the forward scintillators also contain other uninteresting tracks.
Some of the most common particles that enter the spectrometer after a collision are beam
rapidity protons and pions. Thus, the tracks must be sifted through to find the ones which
are potentially interesting. Those late tracks can then be identified by calculating the mass
and charge associated with the track. Finally, the quantities of interest, like rapidity and
transverse momentum, can be calculated.
The above process is performed in two steps. In the first step, the trigger
requirements on the integrity of the beam are reinforced by the beam cut. This is where
most erroneous triggers are eliminated. Events that pass the beam cut are then analyzed
by the track reconstruction routine, Quanah. Each track is then subjected to another set of
requirements, which will be discussed below, to ensure that it did originate from the target
and satisfy the time-of-flight requirement. At this point, all useful quantities associated
with each remaining track are stored on disk in ntuple format, a data structure provided in
the CERN program libraries. In the second step of this analysis the tracks are identified.
Once a track is associated with a particular type of particle, momentum and rapidity can be
calculated. Then histograms are constructed for each type of particle and cross sections
are calculated.
3.1. Pass 1
3.1.1. B e a m Cut
The first cut placed on the data is meant to ensure that there was indeed a single well-
defined beam particle entering the apparatus. Even though the trigger quite effectively
36
vetoes double beam events, it is prudent to refine this off-line. Figure 3.1 shows the pulse
height in a BSCI veto scintillator (a), and a beam-defining scintillator (b), for events
associated with TOF triggers. Recall that the beam-defining scintillators are disks through
which the beam passes, while the veto scintillators are annuli surrounding them (see Figure
2.2). The veto scintillator shows values above pedestal, and a double beam signal, at
twice the pulse height of the silicon peak, can be seen in the beam scintillator. Clearly
some unwanted events were not rejected by the trigger. Thus, an event is not analyzed if
either of the veto scintillators, SI or S3 , has a signal above pedestal. Further, the beam-
defining scintillators, S2 and S4, must have a pulse height below the double beam peak.
Both the veto pedestal values and the Z=14 pulse height values are determined by
examining events in which only the beam trigger was used.
A further requirement on the charge of the beam particle was available off-line that
was not available at the trigger level. The upstream silicon detector, located upstream of
the target, but downstream of the BSCI telescope, provides this information. The final
step of the beam cut is the requirement that this silicon detector measures a pulse height
within 3a of the Z=14 peak. This value was again determined by analyzing events with
only a beam trigger.
3.1.2. Tracking
Once it has been determined that the event had a valid beam signal, the track
reconstruction routine, Quanah, is called. As described in §2.4.4., Quanah reconstructs
tracks out of the hits registered in a series of detectors, or pseudo-planes. The routine has
the flexibility to allow the user to determine which of the available pseudo-planes to
consider. In this case all pseudo-planes were used. This includes both DCII and DCIII
wire planes, which measure only x (magnet bend plane) positions, DCII and DCIII pad
planes, FSCI and UCAL planes, all of which provide both x and y information. In
Counts
Counts
37
Pulse Height
Pulse Height
Figure 3.1: Pulse height distributions for events with TOF triggers from (a) beam veto scintillator and (b) beam defining scintillator.
38
addition, each track is required to originate from the target. The information from DCI is
not used in the tracking at this point, however, tracks that did have DCI clusters
associated with them are flagged.
When all tracks in an event have been reconstructed, the tracking cut is applied.
There are two requirements that a track must pass. The first is that the time of flight to
the FSCI must be between 2 and 30 ns. This corresponds to rapidities between 1 and 2.3.
The cutoff at 30 ns eliminates tracks that could have erroneous time measurements due to
albedo (the spray of back-scattered particles from the calorimeters). This part of the cut
selects mid rapidity particles. The second part simply requires that the track strike the
FSCI wall in a region that is well separated from the beam. At this stage a separation of 2
slats from the slat that received the beam was used. This is restricted further in Pass 2.
Each track that passed both the time of flight and slat number requirements was
accepted for further analysis. The rigidity and velocity of the particle are calculated from
the parameters of the track and the time of flight. Since at this point neither the charge
nor the mass of the particle is known, it is not possible to calculate momentum or rapidity.
So a mass is calculated assuming a charge of 1. Because the mass is linear in the charge it
can be corrected later, once the charge assignment has been made. Further, the
longitudinal and transverse momenta are calculated assuming charge 1. This information,
along with FSCI pulse height (for later charge determination), total charged particle
multiplicity (from the silicon multiplicity detector), and other diagnostic information are
written to disk in ntuple format. This structure, provided in the CERN program libraries,
is convenient for making the final cuts and particle identification, and for calculating
quantities of interest.
39
3.2. Pass 2
The ntuples that result from Pass 1 contain a list of well defined late tracks (along with all
relevant information associated with the tracks) that originate from a clearly identified
beam particle striking the target. In Pass 2, the ntuples are scanned to identify the tracks,
sort them by particle, and histogram them in the y-pt plane. This is done with the CERN
analysis program, Physics Analysis Workstation, or PAW.
3.2.1. Tracking Cut
The Pass 2 tracking cut is essentially the same as the Pass 1 cut, except that the
requirements are more restrictive. After studying the results of Pass 1, several sources of
background were identified. The following restrictions were found to significantly reduce
these backgrounds.
The track reconstruction routine used in the analysis does not require the track to
match up with a hit in DCI, the tracking chamber located between the two spectrometer
magnets. When there is a corresponding DCI hit, it is used in defining the track, and a flag
is set signifying so. It is often possible to define a trajectory through the magnets and back
to the target for spurious tracks downstream of the magnets and also for particles that
were produced (either by decay or collision) before the second magnet. Thus having a
match up with a hit between the magnets can have a powerful effect on eliminating this
background. All tracks used in the final analysis are required to have a DCI cluster.
This cut also restricts the acceptable region of the FSCI wall. The tracks that hit
the FSCI in slats 30-37 are dominated by protons with rapidities at or near that of the
beam . The light nuclei of interest that hit these scintillators have times of flight such that
most are excluded by the TOF cut. The particles that survive this cut are very difficult to
40
distinguish from the proton background. Therefore the acceptable region of the FSCI wall
was restricted to slats 15-29.
3.2.2. Charge Selection
The charge of a track is assigned by examining the pulse height in the FSCI slat associated
with the track. Unfortunately, the pulse heights for different charged particles are not well
separated. The energy loss fluctuations of charged particles passing through the
scintillators can be quite large, producing long tails. The spectrum of charge 2 particles
sits on top of a small amount of the charge 1 tail. Furthermore, peaks due to more than
one charged particle hitting the same slat can be seen to fall rather close to the single
charged particle peaks. The presence of more than one particle in a scintillator is
unacceptable because erroneous time and position measurements almost certainly result.
Thus, it is important to accept tracks that have pulse heights in a range that is clearly
consistent with only one charge. Tracks with FSCI pulse heights below the two charge 1
threshold are assigned a charge of 1. Those tracks with pulse heights between the charge
2 threshold and the two charge 2 threshold are assigned a charge of 2. These cuts are
shown in figure 3.2.
In attempting to identify alpha particles, this charge cut is insufficient. Since the
charge 2 pulse height spectrum is somewhat contaminated by the tail of the charge 1
spectrum, it is inevitable that some of the copious charge 1 particles will be misidentified
as charge 2. This problem is exacerbated for alpha particles by the fact that deuterons
have the same rigidity, and thus will be indistinguishable from alpha particles in the
spectrometer if they are erroneously assigned a charge of 2. To reduce this deuteron
contamination the energy deposited in the pad planes of DCII and DCIII is examined.
Figure 3.3 is a plot of the energy measured in the DCII pad plane versus that found
in DCIII, for tracks that have already been identified as charge 2 by the forward
Counts
41
104
103
102
10
1 0 1 2 3 4 5 6 7 8 9 10Pulse height
Figure 3.2: Forward scintillator pulse height distribution. The shaded areas show the pulse height ranges corresponding to the charge cuts.
i i i i I 1 II I | i i i rj III I | I I 11 | I I I I | I I I I | I I I I | I I I I | I I I I| I I I I | I I I I | I I I I |
q = 1 Cut
q = 2 Cut
HHMIHMIIIHHIIHIMHIHIIMHllHlMllilHIHHIimHHIHHHIHHIIIHHlMHIMIMIHMMIIMMHHIHMIHHltMIIIMIHIMMItllUIHIIHIMllllMIUIIImiuuuiMiimitnMMii(HHMIIIIKHIIIMIIIMHIMIIHMtHllHttlMlllllH
IIMMIlMtliHHIIIMhlti)
IIMllMliHMHIIiOllllliltMiHHiillliHtlliMtHHiM'lMl'ii
* I < t I I 11 I I I I I I I I
42
o* 12000 0)^ 10000
3 8000oQ 6000
40002000
0
. \ | r 1 t i i
fnT 0 : o“ <b : o
r 0 0<b °= <t°
~ o,
| i i - r i | i i i i | i i i i | i i i ■_ o —
IV :<S> ° 0 3
o_ O 0 _ ° „ o O o 0 o$ ° ° o J
«8>o 0 o oQ
~ I- 1 1 1 1 1 1
o ^o&© o o oo 0 o I I I1 i i i i 1 i i i i 1 i i i i 1 i i i i-
0 1000 2000 3000 4000 DCIII Pulse height
DCII Pulse Height DCIII Pulse Height
Figure 3.3: The drift chamber pad plane charge cut (DCQ cut). Top: Pad plane pulse height; DCII vs. DCIII for tracks satisfying FSCI charge 2 cut. Lines represent threshold of cut. Regions I, II, and III contain charge 1 tracks. Region IV contains charge 2 tracks. Bottom: Projections on DCII and DCIII pulse height axes, respectively.
43
scintillator. This plot shows a clear charge 1 signal (region I) despite the charge
requirement already placed on the data. This plot also nicely demonstrates the problem
with charge identification, and the power of multiple charge measurements. One can
clearly see the large pulse height tail of the charge 1 particles that would be mistakenly
labeled as charge 2, if only one detector were used. These are located in the regions
labeled II and m. The lines indicate where the cuts are placed on each pad plane. Both
pad planes are required to have pulse heights above their respective cuts to pass the pad
plane charge cut (region IV).
3.2.3. Particle ID
The final identification of a track is determined by the charge, momentum, and time of
flight. When the reciprocal of momentum is plotted versus time of flight for tracks with a
certain charge, bands of particles with different mass emerge. Figures 3.4 and 3.5 show
such plots for charge 1 and charge 2 tracks, respectively. This structure is a result of the
relationship between the momentum and P, which is determined by the time of flight:
P = my p (3.1)
where m is the mass of the particle, y = l /^ /l -p 2, and P = v/c. Here v is the velocity of
the particle and c, of course, is the speed of light. Since the time of flight is measured with
respect to the time it takes for a v = c particle to reach a given FSCI slat via an average
trajectory, (tc), the true time associated with a track is TOF + (tc). The value of (tc) also
takes into account the differences in cable delay between different slats. Then
P =-----(“-TT (3.2)TOF+(t,)
where tc is the time for a v = c particle to follow the same trajectory as the track in
question.
44
Time of Flight (ns)
Figure 3.4: Particle identification plot for charge 1 particles. Mass lines of protons, deuterons, and tritons can be seen. The boxes represent the gates used for each particle. The vertical line represents the time-of-flight trigger. The points with times of flight lower than the trigger threshold are pions and beam velocity protons that accompanied a trigger particle.
1 /Mome
ntum
(GeV/c)
45
Time of Flight (ns)
Figure 3.5: Particle identification plot for charge 2 particles. Mass lines of 3He, and alpha particles can be seen. The boxes represent the gates used for each particle. The vertical line represents the time-of-flight trigger.
46
The identification of tracks is accomplished by gating on a region around the
expected band for a given particle. The gates used are shown in Figures 3.4 and 3.5. In
order to determine the efficiency of applying such a cut, the mass is calculated from
Equations (3.1) and (3.2) for all tracks within the gates and compared to the true value.
The mass resolution is a function of both the momentum resolution, Op, and the TOF
resolution, Ojof- K can be written in the following form, where momentum is measured
in GeV/c and c = 1:
The momentum resolution was experimentally determined to be 0.005xp2. The difference
between the calculated mass and the true mass for all tracks within the gates is found to be
within 3 om. This indicates that the losses from this cut are less than 99%.
The calculated mass is shown in Figure 3.6. Plot (a), which contains q=l tracks,
clearly shows peaks at the deuteron and triton masses. The q=2 tracks are displayed in (b)
and (c). Note that (b) contains tracks that have their charge determined by the FSCI pulse
height only. When the pad plane charge cut is applied, as in (c), most of the background is
removed, as well as most of the mass 4 peak. This illustrates the power of the multiple
charge measurement in rejecting the deuteron background.
Counts
Counts
47
Moss (GeV)
Mass (GeV)
Figure 3.6: (a) Mass distribution for identified charge 1 tracks, (b) Mass distribution for identified charge 2 tracks, without DCQ cut; and (c) with DCQ cut.
48
3.3. Efficiencies
3.3.1. Tracking
The combined efficiency of detection and track reconstruction is evaluated by examining a
sample of pre-trigger events. These events correspond to minimum bias interactions.
Most of the particles that enter the forward spectrometer in these events are beam rapidity
protons. The sample was further restricted to have only one track when analyzed with
Quanah using only the wire planes of DCII and DCIII. This yields a data set of events
with a high probability of having only one track, and with no trigger bias. The wire planes
are assigned an efficiency of 100%. This can be seen to be a reasonable approximation by
noting that a track can be identified if as few as three of the six wire planes register a hit.
Thus, if the individual wire plane efficiency is 90%, the efficiency for the entire drift
section is over 99%.
Using this set of events, the efficiency of an individual pseudoplane is given by
N-e = — **- (3.4)out
where Nin is the number of tracks found when the given pseudoplane was required in the
tracking, and Nout is the number found with that detector excluded. In both cases all other
pseudoplanes are used in the tracking. The efficiency determined by this method is a
convolution of the detection and track reconstruction efficiencies. The efficiencies for all
pseudoplanes, except FSCI, are given in Table 3.1. The FSCI efficiency is determined
separately.
The efficiency of the forward scintillators was determined by a slightly different
method so that the efficiency for each individual slat could be found. This was necessaiy
because the detection efficiency varied significantly from slat to slat. Figure 3.7
Effici
ency
49
FSCI Slat Number
Figure 3.7: Detection efficiency of forward scintillator slats.
50
Detector____________EfficiencyDCI Pads 0.87DCn Pads 0.98DCm Pads 0.93
UCAL_______________ 0.95Total 0.75
Table 3.1: The measured efficiencies for DCI, DCII, and DCIII pads, and for the UCAL.
demonstrates the problem. The detection efficiency for each slat is shown. The
efficiencies are determined by tracking the sample of events without requiring the FSCI
pseudoplane. For each track, the position along the FSCI wall is calculated from the track
parameters. The corresponding slat is then examined for a valid pulse. Since this position
as determined by the tracking has finite error, if a pulse is not found in the expected slat,
its nearest neighbors are queried. The total number of tracks that are expected in a given
slat corresponds to Nout in equation 3.4, while Njn is given by the number that have valid
signals in the expected slat. Scintillators 26-32 show a variation in the detection efficiency
that is too great to allow the use of an average value. Thus, the FSCI efficiency
corrections must be done on an event by event basis so that a different correction can be
made for each slat.
The track reconstruction efficiency must also be included in the corrections, along
with the detection efficiency. By assuming that the tracking efficiency is the same for each
slat, it can be determined in the following manner. First, the total efficiency for a given
slat (tracking and detection combined) is
8to, = 8<ta„ 'Et*„ (3-5)
where n refers to the slat number. By averaging over the entire region of slats used, one
obtains
( 8 tot) = e tik ‘ (8det) (3 .6 )
51
FSCI Slat Number Total Efficiency15 0.9516 0.9517 0.9318 0.9519 0.9420 0.9421 0.9622 0.9723 0.9424 0.9825 0.9526 0.7527 0.8928 0.8829 0.94
Table 3.2: Forward Scintillator total efficiencies, including detection and trackreconstruction efficiencies.
where the brackets indicate the average over slats. The value of (ew) is determined by
exactly the same process as the other tracking chambers, as described in the beginning of
this section. This yields the track reconstruction efficiency, which can then be used in
Equation 3.5, along with the individual slat detection efficiencies (Figure 3.7). Table 3.2
contains the total efficiencies for each slat.
The data is corrected for the total tracking and detection inefficiencies in two
steps. Due to the slat to slat variation in the FSCI of the efficiency, this correction is made
on an event by event basis, while projecting the ntuples on to the y-p, plane. The rest of
the inefficiencies are accounted for in the overall normalization when calculating cross
sections from raw counts.
52
3.3.2. Charge Identification
In order to assign a charge to each track, fairly restrictive cuts were placed on the FSCI
pulse height. To evaluate the efficiency of those cuts, a sample of events with a single
particle of known charge would be ideal. The efficiency would be determined from the
ratio of the integral over the region of the cut, to the total integral. In an attempt to
approximate this ideal case, two subsets of the Pass 1 data set were created.
It is relatively straight forward to obtain a sample of protons since they are
produced copiously. Without knowledge of the charge of a particle, one can only
calculate the ratio of mass to charge. The particle with the mass to charge ratio closest to
that of the proton is 3He, with m/q = 1.4. By restricting the time of flight to be greater
than 3 ns, the mass resolution is better than 6%. The proton sample is then obtained by
collecting all tracks whose mass to charge ratio is less than 0.98, which is over 3a from
that of 3He.
For a sample of charge 2 particles, pulse height information from DCII and DCIII
pad detectors was used. A requirement on the pulse height in both detectors served to
sort out a sample of tracks rich in charge 2 particles. This is equivalent to the pad plane
charge cut described in §3.2.2. The thresholds used, however, were significantly higher
than those used in the main analysis cut. This does not completely eliminate charge 1
particles from the sample, but it does greatly reduce their number.
An additional cut was applied to both data sets to reduce the number of
measurements with more than one charged particle in a given scintillator. Each track in
the above data sets is taken from events which may have more than one track. If any of
the other tracks in a given event share the same FSCI slat as one of the tracks that passed
the above requirements, both were eliminated. This, however, does not eliminate all
multiple charge measurements, since there are often particles that are not tracked.
53
FSCI Pulse Height
w F— i— i— i— i— |— i— i— i— i— |— i— i— i— i— |— i— i— i— i— |— i— i— i— i— =
FSCI Pulse Height
Figure 3.8: Forward scintillator pulse height spectra used for determining (a) charge 2 and (b) charge 2 cut efficiencies. The shaded areas show the pulse height ranges corresponding to the charge cuts. The lines are fits to an energy Toss straggling model
54
The FSCI pulse height spectra for both data sets are shown in Figure 3.8. The
shaded regions correspond to the range of the charge cuts. The curves are fits of a model
of energy loss straggling [38], The model is used only to give an indication of the amount
of multiple charge contamination, which in the case of the charge 1 spectrum is obviously
not a problem. The charge 2 spectrum has a significant two charge 2 signal. The
difference between the data and the fit in the region above 7.5 is subtracted from the total
integral of the data, to get the proper normalization for the efficiency calculation. This
corresponded to 2.4% of the total. Note also, the apparent lack of large pulse height
counts as compared to the fits. This deficit corresponds to 3% of the total for charge 1,
and 1.4% for charge 2. The total integrals were not corrected for this since such
straggling models can be inaccurate far from the peak. This does give an indication of the
uncertainty in the efficiency measurement.
The charge cut efficiencies are extracted from these spectra as the ratio of integrals
over the cut region to the total. The integrals are based solely on the data, except for the
two charge 2 correction that is estimated from the fit. The efficiencies thus measured are
0.96 for the charge 1 cut, and 0.88 for charge 2 cut. The corrections were applied to the
normalization in the cross section calculations.
The additional charge 2 requirement of appropriate pulse height in the DCII and
DCIII pad chambers (DCQ cut) was assumed to have approximately 100% efficiency.
This can be seen from the plots at the bottom of Figure 3.3. The vertical lines represent
the minimum pulse height required by the DCQ cut. By noting that the pulse height
spectra fall off very rapidly toward the low pulse height side of the peak (see Figure 3.8)
one can see that the losses from this cut are negligible.
55
3.3.3. Geometrical Acceptance
The geometrical acceptance is an estimate of the probability that a particle emitted at the
target with a given rapidity and transverse momentum will reach designated region of the
FSCI wall within the time-of-flight cut. For a certain range of y and pt, namely central
rapidity and pt = 0, virtually all particles will satisfy these requirements. However, due to
the geometry of the spectrometer, especially the rectangular shape of the opening in the
PCAL, there is a range of these parameters for which not all particles will be accepted.
The measured number of particles must therefore be corrected so as to account for this
loss.
The acceptance was determined from a Monte Carlo simulation of the E814
apparatus. In the simulation, a number of particles are generated at the position of the
target, with a uniform distribution in transverse momentum and rapidity. The trajectory of
each particle through the apparatus was calculated. From this, the position at the FSCI
and time of flight were determined. Additionally, the time of flight was smeared to
account for the uncertainty in the measurement. This was done using a gaussian with
mean equal to the calculated time of flight, and standard deviation corresponding the time
resolution. Particles are accepted if their time of flight and position at the FSCI are within
the range of the cuts applied to the data. This calculation is performed separately for each
of the particle species measured in the experiment.
The results of this calculation are shown in Figure 3.9. The number of accepted
particles with a given pt and y is plotted. The cutoff at high rapidity is due to the time of
flight restriction. The opening in the PCAL restricts the acceptance at high pt. Note that
the pt acceptance is strongly rapidity dependent. At lower rapidities, the particles get bent
out of the spectrometer and miss the FSCI. The pointed shape at low rapidity results
when a particle is moving at a velocity such that it is just barely bent out of the
p,(GeV/c)
p,(GeV/c)
56
y
y
y
Figure 3.9: Acceptance plots for deuterons, tritons, 3He, and alpha particles. The size of the boxes indicate the acceptance in each y-pt bin. The largest boxes correspond to 100% acceptance.
57
Cut EfficiencyTrackingCharge:
0.69
Q = 1 Q = 2
Total:
0.960.88
Q =1 0 = 2
0.660.61
Table 3.3: Summary of cut efficiencies.
spectrometer if it has zero pt, yet a small amount of transverse momentum is enough to
keep it within the acceptance.
To facilitate correcting the data, the acceptance plots are binned in the same units
as the data, in the y-pt plane. There are 105 randomly distributed particles generated in
each y-pt cell. The acceptance is simply the ratio of the number of particles accepted
within each cell to the number generated. Cells with less than 10% acceptance were not
used. The correction is applied to the data in each cell when cross sections are calculated.
3.3.4. Total Efficiency
To correct the data for all the inefficiencies of the system, the number of counts in each y-
pt bin is divided by the total efficiency. This value is the product of the efficiencies of each
cut. The values for the efficiencies are summarized in Table 3.3. Note that the tracking
efficiency contains the average FSCI efficiency. It is included here only to demonstrate
the overall efficiency. In practice, the FSCI efficiency is corrected for on an event by
event basis.
C h a p t e r 4
R e s u l t s
The analysis described in the previous chapter results in a set of fully reconstructed tracks,
each of which has been identified to be either a deuteron, triton, 3He, or alpha particle. In
this chapter the production cross sections for each species are calculated. The method for
sorting the data into different centrality bins is also discussed. Finally, the data is
projected to zero transverse momentum. The rapidity spectra of the invariant yields at
pt = 0 are presented for various targets and centralities.
4.1. Cross Sections
The calculation of cross sections from the raw number of counts measured follows from
the equation for the total number of scattered particles, Ntotaj, from an interaction with
cross section a:
Ntouj = aFaN8x (4.1)
Here, F is the flux of incident particles, a is the cross-sectional area of the beam (or target,
which ever is smaller), N is the density of scattering sites in the target, and 8x is the
58
59
thickness of the target in the beam direction [39], I f the beam is smaller than the target,
F a —>ninc, the total number o f beam particles incident on the target. The density of
scattering sites is
N = £ ! ^ a (42)A
where p is the density o f the target material, t is the target thickness, N A is Avogadro's
number, and A is the mass number of the target. Finally, given the fact that the number of
particles experimentally measured, N meas, is the number scattered times the efficiency for
detection, e, the cross section follows:
a = ( i K ^
N taP/N A
Values for the target thickness and incident beam are given in Table 4.1.
A useful way to represent the cross section is in the Lorentz invariant form. This
is given by
^ d3a 1 d2a tA ^o inv = E - T = - -----— — (4.4)
dp 27tp, dp.dy
In order to express the invariant cross section in terms of the yield per interaction, it is
necessary to divide by the interaction cross section. The invariant yield, N jnv, is
particularly useful when comparing with other experiments, since it is relatively insensitive
to small variations in the centrality, thus allowing comparison of measurements at slightly
different centralities.
As will be discussed in the next section, the centrality determination is made in
reference to the geometric cross section, which is an approximation of the total nuclear
interaction cross section. The geometric cross section is defined as:
< « )
60
Target Geometric Cross Thickness (g/cm2) Incident Beam Section (mb)______________________________________Pb 3630
2% 2.29 6.291 xlO84% 4.52 6.372xl08
Cu 22402% 1.18 1.280xl084% 2.26 1.526xl08
Al 16502% 0.65 4.717xl084% 1.30 3.103xl08
Table 4.1: Geometric cross sections, target thicknesses, and number of incident beam particles for targets used.
where Rp and Rt are the radii of the target and projectile, respectively. Using R = R0A X,
where Rq « 1.2 fm, the geometric cross section can be evaluated. Table 4.1 contains the
values for the geometric cross section for 28Si incident on Pb, Cu, and Al targets.
4 .2 . Centrality
Centrality refers to the degree of overlap of the two colliding nuclei. It is often interesting
to examine phenomena at different centralities, since the level o f centrality is related to the
number of nucleons that participate in the collision, and thus, how hot and dense the
system gets. Centrality is measured most directly by the impact parameter, b, which is
obviously out of the experimenter's reach. In Experiment 814, there are three measurable
quantities which are related, albeit somewhat imprecisely, to the impact parameter. Since
at higher centralities the collisions are more violent, the total amount of transverse energy,
as well as the number of charged particles emitted, offer measures of the centrality.
Transverse energy is measured by the TCAL and PCAL, while the multiplicity of charged
61
particles is measured by the silicon multiplicity detector (M ULT). Also, the amount of
energy measured at zero degrees to the beam axis is a measure o f the number of nucleons
that did not interact in the target, and therefore can be related to the number of
participating nucleons, a third measure of centrality.
In this work, the only measure of centrality used is the multiplicity o f charged
particles, N c. Figure 4.1 shows the spectrum in the multiplicity detector for two Pb
targets, with thicknesses corresponding to approximately 2% and 4% o f an interaction
length. The events used are those which satisfy only the pre-trigger requirements. The
shape is characteristic of the geometry of the collision. I f one considers the area
associated with a range of impact parameters (from b to b + 5b), i.e., 27tb5b, then the most
probable events are those with large impact parameters. Such events yield few charged
particles since they are peripheral collisions, hence the peak at low multiplicity. For
smaller impact parameters more nucleons participate in the interaction, hence more
particles are emitted. However, smaller impact parameter collisions are less likely,
because they correspond to a smaller area. Ultimately, at zero impact parameter the cross
section falls off abruptly. A gradual decrease and rapid fall off can be seen in Figure 4.1.
The difference between the two targets at high Nc is due to the fact that the raw
multiplicity is used. No corrections are made for the effects of 7t° conversion and 5-ray
production in the different targets [31]. This is not a problem, since the spectra from
different targets are not directly compared. These spectra are used as a guide in sorting
the data into centrality bins, in a manner that is relatively insensitive to such variations. A
centrality bin is defined by the range of Nc that corresponds to a particular percent of the
geometric cross section. The value of Nc which corresponds to a given centrality is
determined by the integral
(4.6)
da
/dN
e (m
barn
)
62
0 40 80 120 160 200 240
Number of charged particles (Ne)
Figure 4.1: Charged particle multiplicity spectra for two Pb targets of differentthicknesses.
63
Target Multiplicity Threshold Percent o f Geometric Cross Section
4% Pb 2% Pb 4% Cu 2% Cu 4% Al 2% Al
301520201515
648962645458
Table 4.2: Pre-trigger multiplicity thresholds for targets used.
where a(N c) corresponds to the desired percentage of the geometric cross section. The
value of N c for a given centrality varies for different targets and thickness.
At low multiplicity, the spectra also exhibit different shapes. This is just a
manifestation of the different pre-trigger thresholds used for each target. As a result of
trigger rate limitations, the minimum multiplicity required in the pre-trigger was higher for
the thicker targets. This afforded a greater sensitivity to higher centrality events, at the
cost of the least central ones. The minimum multiplicity for each target is given in Table
4.2. For centrality bins that are above threshold in both thicknesses of the same target
material, the results of the two targets are averaged.
4 .3 . E xtrapolations to Pt = 0
The E814 forward spectrometer has a very small acceptance in transverse momentum.
Further, the current measurement is limited to a subset of the total angular acceptance of
the spectrometer, due to the limited region of the FSCI that was instrumented for the time
of flight trigger. The measurements in this work, therefore, constitute a study of
production rates at pt = 0.
64
Target Deuteron Triton 3He AlphaNumber Rate Number Rate Number Rate Number Rate
4 % Pb 9 2 3 7 3X10-4 3 0 2 lxlO"5 2 1 8 8x10-6 6 2xl0"7
2 % Pb 4 5 9 7 3x10-4 174 lx lO -5 117 8x10-6 0 0
4 % Cu 1706 2x10-4 45 8X10-6 4 6 6x10-6 0 0
2 % Cu 1294 2x10-4 3 6 8x10-6 31 6x10-6 0 0
4 % Al 1411 1x10-4 6 0 3x10-6 4 4 3x10-6 0 0
2 % Al 6 0 4 1x10-4 15 3x10-6 21 3x10-6 0 0
Table 4.3: Number of identified deuterons, tritons, 3He, and alpha particles for the targets used. Production rates are the ratio of the number identified and the number of incident beam particles times the interaction length of each target.
It is well known that the invariant cross section of nucleons produced in relativistic
heavy ion collisions is described quite well by a Boltzmann function in mt:
= A (y)m.e (4-7)
The behavior of the light nuclei studied here is expected to be similar to that of the
nucleons. In fact, such a relationship has been seen for deuterons [40], For a typical value
of the temperature parameter for nucleons, 150 MeV, the production cross section varies
little over the pt range accessible to E814. Heavier clusters are expected to have higher
temperature parameters, which would exhibit an even flatter response. In fact, the
expected variation of O jnv over the range of pt being considered (0-200 M eV) is
approximately 6% for deuterons, and 4% for tritons and ^He. This is borne out in the
data, as can be seen in Figures 4.2-4.4, which show the invariant yield of deuterons,
tritons, and 3He as a function of pt for various rapidity bins. The data can therefore be
combined in a weighted average over pt, in each rapidity bin, to obtain the value of the
pt = 0 intercept. The average is taken over only the range of pt where the acceptance is
greater than 10%. The lines in Figures 4.2-4.4 represent these average values. Figures
4.5-4.9 show the resulting values of the invariant yield at pt = 0, as a function of rapidity,
for the three target materials used for both low and high centrality.
65
A summary o f the number o f identified particles is contained in Table 4.3. In order
to get a sense of the production rate per interaction we can normalize the number of
identified particles to the number of incident beam particles times the interaction length o f
each target. These values are also shown in Table 4.3. One can see the production rates
drop rapidly with increasing mass. This illustrates the difficulty o f measuring heavier
nuclei, since the rate decreases by more than an order o f magnitude with each unit of
mass. In fact, it is obvious that mass 4 is the limit of sensitivity of our measurement since
only 6 alpha particles were found.
1/(
2-n
pt)
d2cr
/dp,
dy
(mba
rns/
GeV
2/c
2)
66
p, (G e V /c )
Figure 4.2: Invariant yield o f deuterons plotted as a function o f pt, in rapidity intervals o f0.1 unit, for Si + Pb collisions. The top curve corresponds to y = 1.4-1.5. Distributionsare divided by successive factors o f 10 for clarity. Uncertainties are statistical only.
1 /(2
7vpt
) d2
cr/d
p,dy
(m
barn
s/G
eV
2/c
2)
67
-210
-310
-410
-510
-610
-710
-810
r ------ <J)-— - <©•
H l -
: -{3-
— e|i— _
- 4,-
■ * —
— ® — T
0 . 1 5 0.2
p, (GeV/c)
Figure 4.3: Invariant yield o f tritons plotted as a function o f pt, in rapidity intervals o f 0.1unit, for Si + Pb collisions. The top curve corresponds to y = 1.2-1.3. Distributions aredivided by successive factors o f 10 for clarity. Uncertainties are statistical only.
1 /(2
7TPi
) d2
a/dp
,dy
(mba
rns/
GeV
2/
c2)
68
-210
-310
T ■®*-----
-410 - Q = -
-510
-A— — A —
-610 — 9 - -
~ 9 —i---------l -------- ■---------,--------- 1 —
■ - 9 -— 0 —
-710
I « » I 1 1 1 I I I I I0 0.05 0.1 0.15 0.2
p, (G eV /c)
Figure 4.4: Invariant yield o f 3He plotted as a function o f pt, in rapidity intervals o f 0.1unit, for Si + Pb collisions. The top curve corresponds to y = 1.6-1.7. Distributions aredivided by successive factors o f 10 for clarity. Uncertainties are statistical only.
69
S i + P b M in im u m B i a s
o
><do
>NXCl.X
CNX
CLtcCN
-110
-2 10 r
-310
- 410
-£33— E3--E 3 -
-E3-
- A -
; , 4 -
I I I I
- o -
1.5
□ Deuteron
A Triton _
O "He
_|____|____I____I2.5
Rapidity
Figure 4.5: Invariant yield at pj = 0 o f deuterons, tritons, and 3He plotted as a function o frapidity for Si + Pb collisions. Multiplicity cuts corresponding to 89% o f ogeom(approximately minimum bias collisions) are used. Uncertainties are statistical only.
70
S i + P b 5 5 % a geom
0(N1S '0) -13 10 >x■oCL
x>
x>
CLCN
-210
- 310
- 410
- 510
“T I I" I i i i r
a -C3- -C3-CD-■ Q - . q .
“ 4s-
4*-=$=.
- 0 -
J I I L. _l I I L.1.5
n 1-----1-----r
□ Deuteron"
A Triton _ O ’He O Alpha
J I I L_2.5
Rapidity
Figure 4.6: Invariant yield at pt = 0 of deuterons, tritons, 3He, and alpha particles plotted as a function of rapidity for Si + Pb collisions. Multiplicity cuts corresponding to 55% of °geom are used- Uncertainties are statistical only.
71
S i + P b C e n t r a l
o-i 1---------1--------- r n 1-----1-----r t 1--------- 1--------- r
>CD
O
>\X>Cl-o
-110
«NT>
Cl^ -2
10
□ Deuteron'
A Triton _
O sHe
- 310
-A-A -
- A ' -a*7 x
- 410 J I I L _l____ I____ I____ L
1.5_1 I I l_
2.5
Rapidity
Figure 4.7: Invariant yield at pj = 0 o f deuterons, tritons, and 3He plotted as a function o frapidity for Si + Pb collisions. Multiplicity cuts corresponding to 8% o f o geom (centralcollisions) are used. Uncertainties are statistical only.
72
Si + C u 5 5 % a geom
OCNI>0)O
XXClX
-110
ZCNXCLtcCN -2
10
-310
-410
“i 1-----1-----r
-E 3-
. ■ ■ " f " . d>. 4 - i I - t - '
i 1-----1 r
□ Deuteron
A Triton J
O 3He
1 : -O- :
1.5 2.5
Rapidity
Figure 4.8: Invariant yield at p, = 0 o f deuterons, tritons, and 3He plotted as a function o frapidity for Si + Cu collisions. Multiplicity cuts corresponding to 55% o f ogeom are used.Uncertainties are statistical only.
1/2n
p,
d2N
/dp
tdy
(GeV
)-2
73
Si + Al 5 5 % a geom
Figure 4.9: Invariant yield at pt = 0 o f deuterons, tritons, and 3He plotted as a function o frapidity for Si + Al collisions. Multiplicity cuts corresponding to 55% o f ogeom are used.Uncertainties are statistical only.
C h a p t e r 5
D i s c u s s i o n
The previous chapter presents measurements o f cross sections at zero transverse
momentum for light nuclei up to mass 4. In order to extract physics out o f this
information we will explore several of the models of the production of light nuclei that
were discussed in Chapter 1. Since these models relate the relative yields of light nuclei to
properties of the system at the time the nuclei are formed, we can attempt to study the
environment that gave birth to the nuclei.
The first models that will be discussed in detail are an empirical coalescence model,
a thermodynamic model, and an improved coalescence model. These models describe the
production of light nuclei in terms of the spectra of nucleons from which they are formed.
Before considering these models, we will discuss the nucleon spectra that will be used.
The thermodynamic and improved coalescence models will be used to relate the
measurements to the size of the emitting source. Finally, a coalescence model based on a
cascade calculation will be discussed.
74
75
5 .1 . Proton S p ectra
The measurements discussed in this dissertation were taken with one setting of the
magnetic field in the spectrometer. The setting was chosen to maximize the acceptance
for particles of mass 2-4. As a result, the acceptance for protons was minimal. The
proton data obtained are limited to a narrow range of rapidities, which is insufficient for
performing the power law fits suggested by several models. Furthermore, proton data
measured on other runs of E814 consist of protons from a selected set o f central
collisions. In order to be able to study the relative yields of nuclei over a wide range of
centrality, we have chosen to use proton spectra generated by the ARC cascade
calculation to interpolate between the E814 central data and the E802 minimum bias
proton data. ARC reproduces both of these measurements well.
ARC[41] (A Relativistic Cascade) is a cascade calculation that treats nucleus-
nucleus collisions as a series of hadron-hadron collisions. Trajectories o f all initial and
produced particles are followed through the nuclear cascade until all interactions cease.
Individual hadron-hadron collisions are determined by measured cross sections (or
reasonable extrapolations where measurements are not available), and final state particles
are determined by branching ratios. Between collisions, the particles are assumed to
follow straight-line trajectories.
ARC has been very successful at reproducing many of the measurements of
relativistic heavy ion collisions at the AGS. Predicted spectra of protons, pions, kaons,
and antiprotons for Si-nucleus collisions at 14.6 GeV/nucleon have agreed quite well with
measurements from E814, E802, and E810. Preliminary results from the first Au beams
produced at the AGS are also well predicted by ARC [21,42],
The proton spectra used in this analysis were extracted from ARC (version 1.15)
output files [43], The files consisted of the results of 5x103 simulated Si + Pb collisions
76
and 104 simulated collisions o f Si with Cu and Al. For each collision a value for the
impact parameter was randomly generated between zero and Rmav. The value o f Rmav is
determined by the geometric cross section:
R ™ = r.(A ^ + A « ) (5.1)
where r0 = 1.2 fin and Ap and A j are the atomic masses for the projectile and target,
respectively. This value of Rmax corresponds to minimum bias interactions. For each
collision the output file contains a list of all particles and their respective four-momenta
after all collisions cease.
In order to use the ARC output for power law fits to the data presented here, two
things must be done. First, a measure of centrality analogous to the multiplicity
measurements in E814 must be constructed. Second, cross sections at pt = 0 must be
calculated as a function of rapidity.
The multiplicity of an ARC event is calculated in a manner similar to the E814
measurement. All particles in a given event are examined. To contribute to the
multiplicity the angle of a given particle with respect to the beam direction must be within
the angular acceptance of the silicon multiplicity detectors. The multiplicity, Nc, for an
event is the sum of all charged particles satisfying the angular requirement. A plot of the
multiplicity spectrum for minimum bias Si + Pb collisions is shown in Figure 5.1. Also
shown is the measured multiplicity spectra for Pb targets of different thicknesses. Since
corrections are not made to the data for 7t° conversion and 5 production in the target,
higher apparent multiplicities are observed for thicker targets. Since the ARC calculation
considers only a single target nucleus for each projectile nucleus, the ARC multiplicity
spectrum is analogous to that for a target of zero thickness. Allowing for the effect of
target thickness, the ARC multiplicity spectrum agrees well with the measured spectra.
Note that the ARC spectrum is not normalized to the data, but is normalized such that the
dcr/
dNc
(mba
rn)
77
Number of charged particles (Nc)
Figure 5.1: Charged particle multiplicity spectrum from ARC simulation o f minimum bias Si + Pb collisions. Also shown for comparison are measured spectra from two Pb targets of different thicknesses. Uncertainties are statistical only.
78
total cross section equals the geometric cross section. Events are sorted into centrality
bins in the same manner as the data, as described in §4.2.
For events within a given centrality bin, proton cross sections are calculated in the
following manner. The rapidity and transverse mass are calculated for each proton. The
invariant cross section per event is then calculated and histogrammed versus mt for each
rapidity bin. To determine the value of the cross section at pt = 0 Boltzmann fits are
performed. This is done by noting that
1 d2N -“«/N m = - - - , , = Am.e * (5.2)
27tm, dm,dy
where A and B are parameters of the Boltzmann function. Then N jnv/mt should be an
exponential in mt. Figure 5.2 shows an example of these spectra and the exponential fits
for several rapidity bins. The fits are quite good. The invariant yield at pt = 0, as
calculated from the fit parameters, is shown in Figure 5.3 for both central and minimum
bias centrality bins.
For comparison, measured proton spectra are plotted on these figures also. The
spectrum of protons produced in central collisions is from E814 studies o f central
collisions. Since no E814 data are available for minimum bias collisions, data from E802
are used. E802 does not measure particles at pt = 0 so extrapolations were made from
Boltzmann fits to proton pt spectra [40], In both cases the ARC spectra agree quite well
with the measured spectra.
5 .2 . C o a le s c e n c e M odel
Motivated by the success of the basic coalescence model in describing lower energy data
taken at the Bevalac, we will apply this model to our data. In the coalescence model, the
invariant cross section for production of a cluster with mass number A is related to the
79
1.1 < Y < 1.2m t— m m .-m
1. 3 < Y < 1.4
<j\>0o
1. 5 < Y < 1.6m t—m
1. 7 < Y < 1.8m ,—m
1. 9 < Y < 2 . 0m t— m
2 .0 < Y < 2 .1mt— m
Figure 5.2: A representative sample of exponential fits o f (invariant yield)/mt plotted as a function o f (mt-m) for protons generated in ARC simulations o f Si + Pb collisions. The maximum impact parameter was 11 fin, corresponding to minimum bias. Uncertainties are statistical only.
80
o
>a>O
>N*oexx>
«Nx>Q.
<N
Rapidity
20
> 1 7 .50)
S 15>s12.5
X>ex 10x>
7.5
T 1---- 1---- 1---- 1---- 1---- 1---- 1---- 1---- 1---- 1---1------ 1------- 1-1------1--1---- 1---- 1—
CENTRAL D, + ARC (Si+Au)
“ 7“
M b — - b
5 - —HE>-
& 2.5
<N 0 1 I I L _l I 1 L J I I L I I I I I_1.5 2.5 3
Rapidity
Figure 5.3: Invariant yield at pt = 0 of protons plotted as a function rapidity for ARC simulated Si + Pb collisions, for both (top) minimum bias and (bottom) central collisions. Central collisions are defined by multiplicities in the top 8% of Shown forcomparison are equivalent results from E802 for minimum bias and E814 for central collisions. Uncertainties are statistical only.
81
probability that A nucleons are emitted with relative momenta less than some value, p0
(spatial proximity is ignored). This furnishes a relation between the cross section for
emitting a light nuclear cluster and that for emitting nucleons. With the assumption that
the neutron and proton distributions differ only by a constant factor (Rnp), we obtained
Equation 1.7. This relation can be expressed in the form o f a power law as:
E d N * pA , 3 Adp;
< d3N p E p —
A
(5.3)dP I
where pA = App is the momentum of the cluster. The scale factor, BA, is given by
Ba = A ^ U - R J —— f — p3>| (5.4)A 2 N !Z ! (3m )
where Rnp, defined as in Equation 1.5, is the ratio of the total number of neutrons in the
target and projectile to the total number of protons. Also, sA is the spin of the cluster, m
is the mass of the proton, and p0 is the coalescence radius. The coalescence radius
characterizes the maximum amount of relative momenta that the nucleons can have and
still coalesce into a single nucleus. Obviously, p0 is related to the binding energy of a
particular nucleus since a more tightly bound nucleus can tolerate a broader momentum
distribution.
Once a particular species of nucleus is specified, BA should be constant, since it
depends only on parameters of the cluster (except for a slight target-projectile dependence
through Rnp). In fact, many experiments at the Bevalac [9] found that BA was not very
sensitive to beam energy, momentum of the cluster, or angle of emission. Some
dependence on the target and projectile was seen that is not explained by Rnp. With the
data presented here we can extend the study of BA to the higher energies at the AGS.
The procedure for extracting the BA values from the data presented in Chapter 4 is
as follows. For a given centrality bin, histograms of the invariant yield, N inv, versus
rapidity at pt = 0 are made for deuterons, tritons, and 3He. A corresponding histogram
System Cluster BA p0 p0Min-biasSi + Pb d (4.3±0.2)xl0*3 76 138
t (1.3±0.2)xl0*5 97 1333He (1.3±0.2)xl0-5 103 142
55% ogeomSi + Pb d (3.0±0.1)xl0*3 68 123
t (5.4±0.4)x 10"6 83 1153He (7.1±0.8)xl0-6 93 128
Si + Cu d (5.6±0.3)xl0-3 91 165t (5.4±1.0)x 10*5 134 184
3He (6.1±1.2)xl0-5 139 191Si + Al d (8.6±0.4)xl0-3 107 195
t (8.7±1.4)xl0*5 149 2043He (6.0±1.4)xl0*5 140 193
CentralSi + Pb d (1.3±0.1)xl0-3 51 92
t (1.4±0.3)xl0-6 67 923He (3.4±0.7)x 10'6 82 113
Table 5.1: Scaling coefficients, BA, and coalescence radii for deuterons, tritons, and 3He, for various targets and centralities. Min-bias corresponds to the top 89% of ogeom and central corresponds to the top 8% of ageom. The 55% ogeom bin was chosen because it was the least central data available for Cu and Al targets due to trigger biases.
for protons is calculated from an ARC output file, with the appropriate centrality cut. The
contents of the data histograms are divided, bin by bin, by the contents of the proton
histogram raised to the Atb power, where A is 2 for deuterons and 3 for tritons and 3He.
The resultant ratio histograms, which are generally flat in rapidity, are fit to a flat line in
order to determine the weighted average of the ratios. This ratio, averaged over rapidity,
is Ba . An example of the ratio plots can be seen in Figure 5.4. Table 5.1 contains the
values of BA and coalescence radii for Si + Pb, Si + Cu, and Si + Al collisions at various
centralities. Also included are values of the coalescence radius used in earlier literature,
Po, which is related to p0 as in Equation 1.8. Plots of N ,nv and BA versus rapidity for all
centrality cuts are contained in Appendix B.
Ba (G
ev2/
c2)
83
0.002 F-
0.006
0.004
“I 1-----1-----r
Deuteron
0
- 4 X 10
~ 0.275i 0.22
0.165 0.11
0.055 0
_l I I L. ' ' '____ L_ _l I I '
- 4 X 10
0.3260.2610.195
0.130.065
0
1.5 2.5
y
J I I L. J I I L_1.5
_l I I L- -I I I l_
J I I L2.5
y
i 1-----1-----1-----1-----1-----1-----1-----1-----1-----1-----1-----1-----r
3He
J I I L.1.5 2.5
y
Figure 5.4: Coalescence scaling coefficient, BA, plotted as a function of rapidity for deuterons, tritons, and 3He from minimum bias Si + Pb collisions. The lines are fits to a flat line. Uncertainties are statistical only.
84
The first thing to notice from Table 5.1 is that BA is target dependent. The values
for the Al target are significantly larger than those for the Pb target, and the Cu results fall
in between. This target dependence cannot be explained by the different values o f Rnp,
since this would tend to make BA larger for the heavier targets, while the opposite trend is
seen. It is worth noting that the Bevalac measurements also found larger BA values for
the lighter targets.
An even more striking departure from this coalescence model is found by
examining the centrality dependence of BA. This can be seen by comparing the values
from Si + Pb at the three different centralities in the table. We can examine the effect
more closely by dividing the data from the Pb target into finer centrality bins. Figure 5.5
shows Ba for several exclusive centrality bins. These centrality bins are defined by slices
of the multiplicity spectrum. They are meant to reflect non-overlapping ranges o f impact
parameter, similar to rings on a dart board. For example, the horizontal error bars of the
last bin in Figure 5.5 represent a range of multiplicities that corresponds to the top 89% of
the geometric cross section to the top 64%. The bin is expected to contain collisions that
range from the most peripheral to those of moderate overlap of the colliding nuclei. The
bins are called exclusive because they do not overlap in multiplicity, however, since a
given multiplicity corresponds to a range of impact parameter they are not necessarily
exclusive in impact parameter.
From this figure we see that there is a significant dependence of BA on the
centrality o f the collision. As the collisions progress from the most central to the most
peripheral the value of BA increases by more than a factor of 45 for deuterons and 104 for
mass 3 nuclei. This is a clear departure from what is expected in the basic coalescence
model. This implies that the coalescence of nucleons into clusters is related to the
dynamics of the collision, not just the intrinsic properties of the cluster. Although these
results constitute the first measurement of the coalescence scale factor as a function of
85
I<
0\1>
<CD
a / o geom
Figure 5.5: Coalescence scaling coefficient, BA, plotted as a function of centrality for deuterons, tritons, and 3He from Si + Pb. The values plotted are the average values of BA over the centrality range represented by the horizontal error bar. Uncertainties are statistical only.
86
centrality, earlier measurements at the Bevalac did imply that a single scale factor did not
describe both minimum bias and central data [12],
The Bevalac measurements provided no evidence for any dependence on beam
energy, for energies up to 2.1 GeV/nucleon. We can extend those studies to 14.6
GeV/nucleon in the hope of learning more about the role of collision dynamics in
coalescence. Figure 5.6 shows the values of BA plotted against the beam energy. The
points at the highest beam energy are the minimum bias values from Table 5.1. The
Bevalac data are for Ne + Pb at 0.4, 0.8, and 2.1 GeV/nucleon [19]. In order to justify the
comparison of Ne beam data to that with Si beam, note that the measurements for Ar + Pb
(from the same publication) yield the same value of Bd as for Ne + Pb, and values within
35% for the mass 3 nuclei.
The Bevalac data show no change in the coalescence scale factor with beam
energy, however, in extending the measurements to AGS energies, a decrease in the value
of Ba can be seen. A possible explanation for this lies in the consideration of the spatial
correlations of the coalescing nucleons. The coalescence model only requires that the
nucleons be close to each other in momentum space, but no account is taken of the
nucleons' spatial proximity to each other. In collisions at Bevalac energies, source sizes
are on the order of the size of the incident nucleus, i.e. 3-4 fm. This is comparable to the
size of the coalesced nuclei, thus spatial proximity is not a consideration. However, for
the higher energies at the AGS, it is reasonable to expect a significant amount of
expansion of the system before the densities are low enough to allow coalesced nuclei to
exist. One must also consider the DeBroglie wavelength, X = h/p, of the coalescing
nucleons. The lower energy nucleons in Bevalac collisions have longer wavelengths and
are less localized, making it easier for them to coalesce. The beam energy dependence of
the coalescence scale factor may be signaling the entrance into a
Ba (G
eV2/
c3)
87
-1- 10i<
-210
-310
- 4 10 r
-510
i i i—i i n i |----------1-1— i—r ■ i ■ i i i j----------------1----- 1
- □ Deuteron- O Triton
A ’He
-ft ftI I
Bevalac
0
1 O
A
□
AGS
AO
j i i i i i 11 j i i i i i 111______ i___ i10
-110
Ebeam/A
T—l"l II I.
J i ' i ' i
(GeV)
Figure 5.6: Coalescence scaling coefficient, BA, plotted as a function o f incident beam momentum. The points for the lowest three energies are for Si + 20Ne [19]; the high energy points are from this work.
88
regime where spatial correlations of the coalescing nuclei must be taken into account.
Furthermore, the centrality dependence of BA is consistent with this, since one would
expect peripheral collisions to have smaller source sizes than central collisions. This can
be seen from the fact that peripheral collisions involve fewer participant nucleons and
there is a negligible amount of target or projectile fragments in the rapidity range
examined.
It is difficult to imagine modifying the coalescence model to account for the spatial
correlation of the nucleons, since we have no information on the spatial distribution of
nucleons in the collisions. However, this sensitivity to the spatial distributions may
provide a method for determining the size of the emitting system. Thus, at this point it
will be instructive to explore some models that relate cluster production to the dynamics
of the collision instead of the intrinsic properties of the cluster, and provide a method for
determining source sizes.
5 .3 . S o u r c e S iz e
5 .3 .1. Thermodynamic Model
The thermodynamic model is based on assumptions about the region of the collision from
which the nuclear clusters are emitted. It is presumed that after the initial compression of
the nuclear matter, a region exists where thermal and chemical equilibrium are reached.
This equilibrium is characterized by high densities and temperatures and short mean free
paths. Thus, collisions are frequent and a balance is obtained between composite particle
formation and break-up. A further simplifying assumption is that this system expands until
a certain volume V is reached, at which point the density becomes so low that the particles
no longer interact. It is assumed that this process, or freeze-out, happens quickly (relative
to the expansion rate). The equilibrium properties at the time o f freeze-out determine the
relative yields o f the light nuclei.
The results of such a calculation yield a power law relationship between the
momentum density of the clusters and the A111 power of the proton density (shown
previously as Equation 1.10):
89
(5.5)d3N A n 2 sa + 1 (2ich)3A-l
dp3A 2 A V dP? Jwhere, as before, sA is the spin of the cluster, Rnp, the ratio of the number of neutrons to
protons in the target and projectile, is defined as in Equation 1.5, ti is Planck's constant,
and V is the volume over which the equilibrium is established. It should be noted that this
is a fundamentally non-relativistic calculation. We can put this in the same form as
Equation 5.3, the result of the coalescence model, by multiplying both sides o f Equation
5.5 by (Amy)A and using the fact that Ep = my and EA = Amy, where m is the nucleon
mass. Then we can identify a new formula for BA:
gThermo _ ^ SA + 1 {ZlOt)A np
A-l
(5.6)myVv /
Hence, this model results in a power law of the same form as the coalescence model,
however, the interpretation of the scale factor is very different.
Since the thermodynamic model results in this power law relationship, it is subject
to the same success in describing data that has already been discussed for the coalescence
model. In fact, the thermodynamic model is even more successful due to the different
interpretation of the scaling factor BA. Since B^10™0 is related to the properties of the
emitting system instead of intrinsic properties of the cluster, it is allowed to vary with
different initial conditions. This freedom can accommodate the cases where the
coalescence model fails.
90
The reciprocal relationship between B][he,mo and the volume o f the system provides
an explanation for the data's deviations from the coalescence model. For example,
consider the decrease in the value of BA seen between Bevalac energies and AGS energies
(Figure 5.6). It is reasonable to assume that in the higher energy collisions at the AGS,
the densities attained are higher, and there will be more expansion o f the system before
freeze-out occurs. A decrease in the value of BA with increasing beam energy is just a
result of the increase in the volume of the source region.
A similar argument explains the relationship between BA and centrality seen in
Figure 5.5. Consider first the fact that the higher the degree of overlap of the two nuclei
the more nucleons will participate in the interaction. Also, the more nucleons
participating, the larger the volume involved. By this simple argument we expect the
volume o f the participant region to increase with centrality. One would expect this effect
to be reinforced by the fact that the systems created in more central collisions reach higher
temperatures and densities than those in peripheral collisions, and thus would be expected
to undergo more expansion. Since we expect the volume to increase with increasing
centrality, it is not surprising that we see, experimentally, a decrease in the value o f B^ as
the centrality increases.
Armed with the fact that the thermodynamic model produces trends that would be
expected for a model related to the volume of the emitting system, we can use the
measurements of BA to calculate the size of the system. Before proceeding, however, it is
necessary to point out several caveats in applying such an interpretation. First, this model
is not a relativistic calculation, so it cannot be an accurate representation o f the collisions,
especially in the relativistic collisions at AGS energies. The non-relativistic
approximation, however, is not as bad as one might think. The relevant velocity in the
calculation is the velocity of the cluster with respect to the center of mass of the source.
In considering clusters with rapidities in the range of 1-2, this limits P of the cluster to be
less than 0.6 with respect to the center of mass, which corresponds to y < 1.3. Clearly, it
91
is necessary to take relativity into account, however the corrections are not expected to be
large. In the following calculation we will set y = 1.
Second, it is assumed that after the collision the participant region is a single,
spherically symmetric source in the center of mass, that expands uniformly until freeze-
out. We must make some assumption about the shape of the emitting region in order to
turn the volume in Equation 5.6 into a radius. By assuming a spherically symmetric source
we presuppose that the participating nucleons of the projectile completely stop in the
target. A high degree of stopping is seen in collisions at the AGS [25,26], so this
assumption is a reasonable first estimate. It would be more reasonable, however, to
assume an ellipsoid with different transverse and longitudinal radii to account for
additional longitudinal expansion due to the initial momentum. However, with no way of
knowing just how much longitudinal expansion to account for, the use of spherical
symmetry is the best guess at present. This assumption is reinforced by recent
measurements with pion interferometry [46] which see no evidence for oblateness in the
pion source.
With these caveats in mind, we can proceed with calculating a radius parameter.
From Equation 5.6 the radius parameter, RA, is
3 (2tr tf ' 2sa + i r ; ‘'/ A/A-l
4k m\
** A
2 Ba . 7
Table 5.2 contains the values of RA for several centralities and targets. The values
obtained from this equation appear to be reasonable estimates for the source size. I f we
estimate the radii of the target nuclei by r0x A \ where r0 = 1.2 fm, we find values of 7.1,
4.8, and 3.6 fm for Pb, Cu, and Al, respectively. In the context of this model one would
conclude that there is expansion of the system before freeze-out since the sizes are
92
Thermodynamic Model Radius Parameter (fin)Deuteron Triton 3He
Min-biasSi + Pb 6.3 ±0.1 5.5 ±0.1 5.2 ±0.1
55% cgeom Si + Pb 7.1 ±0.1 6.4 ±0.1 5.8 ±0.1Si + Cu 5.3 ±0.1 4.0 ±0.1 3.9 ±0.1Si + Al 4.5 ±0.1 3.6 ±0.1 3.8 ±0.1CentralSi + Pb 9.4 ± 0.3 8.1 ±0.3 6.5 ±0 .2
Table 5.2: Thermodynamic model radius parameter values for deuterons, tritons, and 3He, for various centralities and targets. Min-bias corresponds to the top 89% of ageom and central corresponds to the top 8% of ogeom. The 55% ageom bin was chosen because it was the least central data available for Cu and Al targets due to trigger biases (see §4.2.).
significantly larger than the projectile (Si has a radius almost identical to Al) for collisions
averaged over impact parameter, and larger than the target nucleus for central collisions.
These radii, however, appear to be larger than other estimates of source size. Two
particle interferometry, or Hanbury-Brown Twiss (HBT) [44] measurements attempt to
determine source sizes by studying correlations between like particles. Recent
measurements of two proton correlations result in radii of 4.75 ± 0.16 and 3.65 ± 0.14 for
central collisions of Si + Pb and Si + Al, respectively, at a rapidity of 1.3 [45], One should
note, however, that two particle interferometry measurements are not yet fully
understood, as evidenced by the fact that the results are dependent on the parametrization
chosen for the source distribution [46]. Furthermore, the two proton interferometry
results correspond to the size of the system at which protons no longer interact. Light
nuclei yield measurements are instead related to the size of the system when the coalesced
nuclei are no longer dissociated by collisions. These two conditions are not necessarily the
same.
93
Interestingly enough, we see evidence of different nuclei freezing out at different
times. The radius parameters calculated for deuterons are consistently higher than those
calculated for both mass 3 nuclei. This would imply that the mass 3 nuclei freeze out at
earlier times. Such a scenario makes intuitive sense when one considers the fact that the
mass 3 nuclei have much higher binding energies than deuterons. The binding energy of a
deuteron is 2.2 MeV, which makes it barely bound. A deuteron is not likely to survive
many collisions before being dissociated. Tritons and 3He, however, having binding
energies over 3 times higher, should be able to withstand more collisions and thus be
formed, on average, earlier in the evolution of the system.
We can also study the source size as a function o f centrality. In Figure 5.7 the
radius parameter is shown as a function of exclusive centrality bins. This gives different
information than simply comparing minimum bias collisions to central collisions. Recall
that minimum bias refers to an average over all impact parameters. The exclusive
centrality bins of Figure 5.7 represent small slices of the multiplicity spectrum, and thus
are an attempt to approximate non-overlapping ranges of impact parameter. In fact, the
radii corresponding to the most peripheral collisions are significantly smaller than the
minimum bias radii. This figure shows a strong dependence of the radius parameter on the
centrality. As one would expect, the most central collisions exhibit much larger radii than
the more peripheral ones. Also, we see the previously mentioned dependence on cluster
mass.
The radius parameter shows many intuitively satisfying trends, however in light of
the caveats, it is difficult to place much faith in the actual values. It is worth studying
further though, because this technique has the potential of being very useful. Whereas two
particle interferometry should be able to determine the source sizes for protons and
various mesons, high statistics are needed and the measurements are subject to subtle
corrections and interpretations. Radius measurements from cluster distributions, on the
other hand, do not depend so strongly on the statistics. The measurements presented here
94
^r/^geom
Figure 5.7: Thermodynamic model source radius plotted as a function of centrality for deuterons, tritons, and 3He from Si + Pb. The values plotted are the average values of RA over the centrality range represented by the horizontal error bar. Uncertainties are statistical only.
95
from the mass 3 distributions were made with approximately 250 particles o f each species,
measured within a single short run. Furthermore, the measurements of source radii for
various clusters probe different times within the evolution of the collision since the more
tightly bound cluster should freeze out earlier. It would thus be advantageous to
"calibrate" this method so one could know the relationship between the radius parameter
and the actual source size. One such opportunity for calibration may be afforded by
cascade calculations such as ARC. This idea will be discussed further in section 5.4.
Before leaving the topic of thermodynamics a brief discussion of entropy is in
order. Entropy is a potentially interesting quantity because it is expected to change
drastically if a phase change occurs. One expects that the relative yields of nuclear
clusters is related to the entropy of the system at freeze-out. I f the entropy is high, the
phase space density is low and cluster formation is suppressed. The entropy per baryon
(S/A) carried by protons, neutrons, and deuterons is related to the deuteron to proton
ratio, Rjp, as [9]:
y K = 3.945 - ln ( R j - l ^ R ^ l + R J (5.8)
The deuteron to proton ratio for central Si + Pb collisions is 1.2%, which yields an
entropy per baryon of 8.4. Measurements of R^p at Bevalac energies resulted in an
entropy per baryon around 6 [9], It is difficult, however, to interpret these results. In
collisions at the AGS, many more non-baryonic degrees o f freedom are available than at
lower energies, as can be seen from the higher yields of pions and other mesons. With a
significant amount of the entropy going into non-baryonic degrees of freedom one cannot
draw strong conclusions from the baryonic entropy alone.
96
5 .3 .2 . Improved Coalescence Model
The assumptions of thermal and chemical equilibrium implicit in the thermodynamic model
leave some doubt as to the validity of that model. The improved coalescence model put
forth by Sato and Yazaki [14], based on a density matrix formulation o f the many-body
problem, circumvents this shortcoming by relying on a dynamical picture o f coalescence.
It is assumed that after a fast process in the collision, a highly excited region is formed that
decays by emitting various particles. The momentum distribution of nucleons within this
region, as well as the emitted particles, are approximately given by density matrices.
In the center of mass of the system, the probability of proton emission is
represented by:
where ppn is the proton-neutron two-particle density matrix and \\i is the deuteron
internal wave function. I f we neglect p-n correlations we can replace the two-particle
density matrix with two one particle matrices, i.e.,
necessary to assume explicit forms for the wave function and D in order to complete the
where pp(r ,r ') is the proton density matrix. It then follows that the probability of
emitting a deuteron is
(5.10)
Ppn(r,,r2;r,',r2') = pp(r,,r;)pn(r2,r2'). (5.11)
Finally, the single-particle density matrix is written as
where Dp represents the spatial distribution of protons in the excited system. It is
97
Improved Coalescence Model Radius Parameter (fm)Deuteron Triton 3He
Min-bias Si + Pb 4.5 ±0.1 4.2 ±0.1 3.9 ±0.1
55% CJgeom Si + Pb 5.2 ±0.1 5.0 ±0.1 4.4 ±0.1Si + Cu 3.6 ±0.1 2.7 ±0.1 2.5 ±0.1Si + Al 2.9± 0.1 2.3 ±0.1 2.5 ±0 .2Central Si + Pb 7.1 ±0.2 6.5 ±0 .2 5.1 ±0 .2
Table 5.3: Improved coalescence model radius parameter values for deuterons, tritons, and 3He, for various centralities and targets. Min-bias corresponds to the top 89% of ageom central corresponds to the top 8% of ogeom. The 55% ageom bin was chosen because it was the least central data available for Cu and Al targets due to trigger biases (see §4.2.).
calculation. By choosing gaussians for these functions Equation 5.10 can be reduced to
the same form as the empirical coalescence model (see Equation 1.7) and one can relate
the coalescence radius, p0, to the size parameter of the excited region, v:
-ii(A-i)1
N !Z!
A-l
f p : i - a * 471- VaV(VA +v)
(5.13)
where vA are the parameters of the gaussian wave functions and the size parameter is
related to the rms radius of the excited region as
(5.14)
The values used for the wave function parameters are v2 = 0.20fm -2 and v3 = 0.36fm '2.
The results of this calculation are shown in Table 5.3 for several targets and
centralities. The source radii as calculated from this model are consistently smaller than
those from the thermodynamic model, however the same trends are seen. Figure 5.8
shows this radius as a function of exclusive centrality bins. Again, the results are
equivalent to those of the thermodynamic model with a 20-40% decrease in the
98
^ 10 E*4—
□T 9
8
7
6
5
4
3
2
1
00 0.25 0.5 0 .75 1
O'/ Oge0m
Figure 5.8: Improved coalescence model source radius plotted as a function o f centrality for deuterons, tritons, and 3He from Si + Pb. The values plotted are the average values of Ra over the centrality range represented by the horizontal error bar. Uncertainties are statistical only.
AA
*
□ Deuteron
A Triton
O ’He
I I I I I I I I I J I I I I I 1-----1-----1-----L_
99
normalization. Despite the smaller values produced by this model, one is still forced to
conclude that the system undergoes a significant amount o f expansion before freeze-out.
It is not surprising that these two models yield similar results as they both
reproduce the power law relationship between composite and nucleon cross sections that
is inherent in the empirical coalescence model. The improved coalescence model is the
most satisfying model of the three discussed so far. It supplies a dynamical basis for
coalescence without relying on assumptions of equilibrium. It does, however, require one
to make assumptions about the spatial distribution of nucleons within the source. Also, as
in the thermodynamic model, the calculation is not fully relativistic. It is obvious that
more work is needed in order to correctly calibrate this technique before one can
confidently interpret the calculated radii as actual source sizes. As was previously
mentioned, cascade calculations may provide some help in interpreting these models.
5 .4 . C o a le s c e n c e with A R C
A more realistic model than those discussed above is one based on a cascade calculation.
Such a model has been developed by Dover and Baltz [20] using the ARC cascade
calculation. A brief description of ARC and some of its successes have already been
discussed in §5.1. The advantage of using such a cascade calculation is that many of the
problems of the previously discussed models, such as relativistic considerations, are taken
into account in the dynamics of the cascade. Assumptions about the equilibrium
properties and shapes of the system are not necessary. Some assumptions and
approximations are inevitably present, but the widespread success that ARC has had in
reproducing baryon and meson distributions lends credibility to the calculation.
The Dover-Baltz coalescence calculation is based on a simple model. For nucleons
to coalesce they must be close to each other in both position and momentum after they
100
Nucleus Armav (fin)________ APmax (MeV)Deuteron 2.4 120
Triton, 3He 3.2 1604He 3.6 180
Table 5.4: Parameters for ARC coalescence calculation [20],
have stopped interacting. In some sense this is analogous to the basic coalescence model
previously discussed. The major differences are that cascade based coalescence is
considered on a microscopic level (i.e., individual nucleons from the cascade are
coalesced), and spatial separations are taken into account. The lack of consideration of
the spatial distributions of nucleons is a serious weakness in the original coalescence
model, which is rectified in the ARC calculation.
The procedure used for the ARC coalescence calculation is as follows. Deuterons
are formed by examining each neutron-proton pair after its last interaction with other
nucleons or mesons. I f the relative momentum, Ap, and position, Ar, in the two-body
center of mass frame satisfy
A p ^ A p ^ (5.15)
A r^ A r^ (5.16)
the pair is considered a deuteron. This is equivalent to requiring the neutron and proton to
fit within the wave function of a deuteron, where the deuteron wave function is
approximated by a square well. Improvements to this assumption are currently underway.
Larger clusters are formed in a similar manner, by first considering pairs of nucleons and
then adding nucleons, one at a time, to the cluster. For example, to form a triton, all n-p
pairs are considered. For those pairs that satisfy Equations 5.15 and 5.16, another neutron
is sought that also satisfies Equations 5.15 and 5.16, where Ar and Ap are calculated with
respect to the rest frame of the first n-p pair. In this manner any nucleus can be built up.
One should note, however, that all nuclei are formed after their constituent nucleons have
stopped interacting. Thus, all nuclei are formed at freeze-out.
101
A preliminary version o f this calculation has been performed for Si + Au and
Au + Au collisions at AGS energies. The values o f A r ,^ and Apn^ used depended on
the nucleus that was being formed. This was to allow for the fact that a more strongly
bound nucleus could tolerate a larger momentum spread o f the nucleons making it up than
could a more weakly bound one. The values used are in Table 5.4. The values were
chosen so as to make the results agree with all available data. The results compare
favorably with measurements o f deuteron dN/dy for y < 1.4 in central Si + Au collisions
from E802, and preliminary data on deuterons, tritons, 3He, and 4He at target rapidities in
minimum bias Au + Au collisions from E886. A comparison of the ARC coalescence
results with our data is presented in Figure 5.9. The calculation was performed for central
(highest 7% of the total cross section) Si + Au collisions. Cross sections at pt = 0 were
obtained by averaging over the lowest 400 MeV of the pt spectrum [47], Ideally the
average should be done over a much smaller range o f pt. The temperatures of the relevant
nuclei are expected to be slightly higher than those observed for protons, and at those
temperatures the cross section could drop by as much as 17% over a 400 M eV range.
This range was chosen simply due to limited statistics.
The agreement of the calculation with the data is reasonably good. The deuteron
cross section is somewhat over predicted, but the results for mass 3 nuclei agree quite
well. The alpha particle cross sections presented are for the 55% ogeom cut (even though
the ARC prediction is for a 7% ageom cut) and should thus be considered as
approximations for this comparison. One should note that no fits are performed to the
shapes of the spectra. The only adjustable parameters are A r ,^ and A p ,^ which tend
only to affect the normalization, and are constrained by data from E802 and E886, as well
as the data presented here. The shapes of the distributions result solely from coalescence
calculation and are not based on measured proton spectra, as are the previously discussed
102
o
>0O
>Nx>dlX>
«NX)
Q_CN
Rapidity
Figure 5.9: Results of the ARC coalescence calculation. The points are the measured invariant cross sections at pt = 0, for deuterons, tritons, and 3He for central (8% ogeom) Si + Pb collisions. Alpha particle cross sections are for 55% ageom and should be considered approximate in this comparison. The lines are the results from ARC coalescence for a 7% ageom centrality cut. Uncertainties are statistical only.
103
models. Even though the ARC coalescence calculation is similar in spirit to the
coalescence model, the methods are very different. The difference is characterized by the
fact that ARC coalescence provides much more information and insight into the dynamics
of cluster formation.
One of the most powerful aspects o f ARC coalescence is that it is a microscopic
calculation dealing with individual nucleons, the history o f which can be determined. It is
possible to trace a given cluster back to the time of its formation and study properties of
the system at that time, within the context of the cascade calculation. One obvious
question to answer is if there is a well-defined time at which the clusters are formed.
Many models assume some form of freeze-out phenomenon, but this is an approximation
of a process that most likely occurs over an extended period of time. A study of the times
of formation in ARC could help resolve whether freeze-out is reasonable concept. I f it
turns out that there is a well-defined freeze-out phenomenon in this model, one could
examine the spatial distribution of the clusters when they are formed. This would provide
information on the freeze-out radius, and could serve to normalize the radius parameter of
the thermodynamic and improved coalescence models. Such a normalization would be
very useful as the radius parameter would provide a relatively easy method of determining
source sizes.
Clearly, much can be learned from this model. However, the ARC coalescence
calculation is quite new, and such studies have not yet been done. The value of testing
this model further is obvious. Although the ARC coalescence calculations presented agree
reasonably well with our measurements, only a single case has been considered. The
measurements presented in this work comprise the first systematic study of the production
mass 2 and 3 clusters as a function of centrality, and thus can provide unique constraints
to this model. We look forward a more complete coalescence calculation with ARC in the
near future.
104
A simpler calculation that can give an indication o f the size o f the deuteron source
can be done by looking at the spatial distribution of protons immediately after their last
interaction. Our group has begun such a calculation using protons generated by the
Relativistic Quantum Molecular Dynamics model (RQMD). RQMD [48, 49, 50] is a
cascade calculation developed by Sorge, Stocker, and Greiner that incorporates quantum
effects such as particle decays and Pauli blocking into a Lorentz-invariant classical
description of hadron dynamics. RQMD has been as successful as ARC at reproducing
measurements of baryon and meson distributions at the AGS, and is used here because the
code has recently been made available. Figure 5.10 (a) shows the distribution of the times
of last interaction for protons within ±0.7 units of rapidity of the nucleon-nucleon center
of mass, for Si + Pb collisions at 14.6 GeV/nucleon. We see that the process of freeze-out
is predicted to occur over an extended time. While this is not unexpected, it is contrary to
the assumption of an instantaneous phenomenon inherent in the thermodynamic and
improved coalescence models.
The proton freeze-out radius is defined with respect to the center of the interaction
region in the nucleon-nucleon center-of-mass frame. This origin is given by averaging the
position vectors of all particles within ±0.7 units of rapidity of the nucleon-nucleon center
of mass. Figure 5.10 (b) is a plot of the distance o f all protons (in the given rapidity
range) from this origin. The upper curve includes all times shown in Figure 5.10 (a), while
the lower curve is for only those protons that had their last interaction within 3.5 fm/c of
the most probable time. These cases amount to two different definitions of the freeze-out
time. The resulting rms radii are 6.5 and 4.7, respectively. Figure 5.11 shows the these
radii as a function of impact parameter. The impact parameter bins correspond to the
same percentages of the total cross section as the exclusive centrality bins in Figures 5.7
and 5.8, however, since those are defined by ranges of multiplicity instead of impact
parameter, they cannot be directly compared. The radii that result from this calculation
are somewhat smaller than those calculated from the data, but a similar trend with
105
Time of last interaction ( fm /c )
Distonce from origin (fm )
Figure 5.10: Results o f preliminary calculation with RQMD. For protons in the rapidity range of 1-2.4, generated by RQMD for minimum bias Si + Pb collisions: (a) Distribution of the times of last interaction; (b) Distribution of distances from the origin of the interaction region. Solid curve has no restriction on time of last interaction; dashed curve is for times in the range o f 6-13 fm/c.
106
Preliminary
Impact Parameter (fm )
Figure 5.11: Results of preliminary calculation of mean radius of proton source plotted as a function of impact parameter, in Si + Pb collisions generated by RQMD. Protons are restricted to a rapidity range of 1-2.4. The edges o f the horizontal error bars represent range o f impact parameters used for each point. The top curve has no restriction on time of last interaction; bottom curve is for times in the range of 6-13 fm/c.
107
centrality is seen. It is difficult to draw strong conclusions from this preliminary
calculation since it represents our first attempt, and several refinements are in progress. It
does, however, show promise for giving insight into freeze-out radii extracted from the
data.
C h a p t e r 6
C o n c l u s i o n
The data presented in this dissertation are some o f the first measurements o f mass 2, 3 and
4 nuclei produced at mid rapidities in Si-Nucleus collisions at 14.6 GeV/nucleon. With the
E814 apparatus we have measured the invariant cross section for producing deuterons,
tritons, 3He, and alpha particles over the rapidity range of 1-2, and at zero transverse
momentum. The production of these nuclei has been studied as a function of centrality,
which has shed light on the feasibility of various models.
The power law production implicit in the empirical coalescence model was found
to adequately describe the data, however the interpretation of the model leaves something
to be desired. A strong dependence of the scale factor on the centrality of the collision, as
well as beam energy, implies that the coalescence radius is not a universal parameter,
dependent only on the type of cluster produced. This behavior signals the entrance into a
regime where the simple picture of coalescence in momentum space, with no regard for
physical proximity of the nucleons, is no longer adequate. More reasonable interpretations
are supplied by the thermodynamic and improved coalescence models, which also embody
power law production, but relate the scale factor to the volume of the emitting system.
108
109
These models furnish a proper interpretation o f the dependence of the scale factor on
centrality and beam energy.
By taking advantage of the dependence of these models on the volume, we have
determined the size of the emitting source from relative yield measurements o f the light
nuclei studied. The radii so measured indicate that the system undergoes a significant
amount of expansion before the light nuclei freeze out. Minimum bias collisions exhibit
source sizes only slightly smaller than the target nuclei, but significantly larger than the
incident Si nucleus. The sizes associated with central collisions are on the order of those
of target nuclei. This indicates that the system may remain coupled for much longer than
was previously expected. There is, however, a caveat to this interpretation. The
assumptions required to determine the source radius from the light nuclei yields leave
some doubt as to the validity of the interpretation of the calculated radius as the actual
size of the system. Although the radius shows the trends expected with centrality and
target nucleus, an independent confirmation, or "calibration," of these techniques is
needed.
A very promising technique for understanding the formation of light nuclei, and
possibly providing such a "calibration" of the source size calculation, is the coalescence
model based on the ARC cascade calculation. This model has had reasonable success at
reproducing the results from several experiments, as well as a small subset of the data
presented here. The work on this technique is still in its infancy, so the detailed
calculations of cluster production as a function of centrality, needed to fully compare to
the data presented in this dissertation, have not yet been done. We look forward to
further results from the ARC coalescence calculation, and the possible insights they can
give on source size and freeze-out phenomena.
Determining source size from light nuclei yields will prove to be a useful tool for
studying the hot and dense systems created in relativistic heavy ion collisions. The
measurement of light nuclei yields offer a much easier way to study source sizes than two-
110
particle correlation measurements. In the more severe environments expected in collisions
with Au beams, recently made available at the AGS, and in future experiments at RHIC,
light nuclei should be copiously produced, further simplifying the technique. Furthermore,
we expect that light nuclei with different binding energies will freeze out at different times.
This is evident in the radius measurements presented, since the tightly bound mass 3 nuclei
exhibit consistently smaller radii than deuterons, which are weakly bound. Thus, studying
the production of various nuclei can provide information about the system at several
stages of its evolution.
The prospects for future study of light nucleus production are very promising.
Experiment 864 at the AGS, planned for the near future, will greatly expand the range and
scope of the measurements presented here. Refinements of the ARC coalescence model
are already underway, and will help in the understanding and interpretation of light nucleus
production. With better understanding of how normal nuclear clusters are formed in hot,
dense nuclear matter, better predictions can be made o f the probability of producing more
exotic clusters. The ARC coalescence model has already made predictions for the
production of hypemuclei, strange dibaryons, and strangelets. Such particles, if they exist,
will provide insight into the structure of QCD. These studies, in conjunction with
measurements of the properties of the systems created in relativistic heavy ion collisions,
will provide useful information on the evolution and dynamics of nuclear matter under
extreme conditions.
A p p e n d i x A
P h o t o m u l t i p l i e r s a n d E l e c t r o n i c s
Detector Photomultiplier Signal Processing
Off-line Trigger
TCAL N/A LeCroy 1882 ADC LeCroy 4300B FERATP AD Thom EM I 9127B LeCroy 1882 ADC Philips 7106 discr.MULT N/A LeCroy PCOS 2735 LeCroy 4300B FERA
LeCroy PCOS 2732PCAL Thom EM I 9954ADRCH N/A LeCroy 1885 ADC N/A
LeCroy 1879 TDC N /A
FSCI Thom EM I 9954B LeCroy 1885 ADC LeCroy 4300B FERALeCroy 4290 TDC LeCroy 4303 FERET
UCAL Philips XP 2081 LeCroy 1882 ADC LeCroy 4300B FERALeCroy 4290 TDC
Table A. 1: Photomultipliers and Signal Processing Electronics.
I l l
A p p e n d i x B
I n v a r i a n t Y i e l d P l o t s a n d C o a l e s c e n c e F i t s
This appendix is a compendium of the data used in this dissertation. The invariant cross
sections at pt = 0 measured as a function of rapidity are presented for all centrality cuts
referred to in the text. Also included are plots of the coalescence scale factor, BA, as a
function of rapidity for the same centrality cuts. Equation 5.3 in the text defines BA,
The proton yield used was calculated using the ARC cascade code as described in §5.1.
Table B .l contains the values of BA averaged over rapidity, which correspond to the fits
shown in the figures.
The centrality bins are defined by cuts in multiplicity. They fall into two
categories. Inclusive centrality bins contain all events above a certain multiplicity. The
inclusive bins are labeled by the percent of the geometric cross section that the multiplicity
interval corresponds to. Exclusive centrality bins contain events within exclusive, or non-
which is the ratio of the measured invariant yield for a particle of mass A to the proton
invariant yield to the A111 power:
112
113
System Centrality
( % O f O pp nm )
Deuteron Triton 3He
Si+Pb Inclusive bins89% (4.3±0.2)xl0*3 (1.3±0.2)xl0*5 (1.3±0.2)xl0-555% (3.0±0.1)xl0*3 (S^rfcOAJxlO-6 (7.1±0.8)xl0-68% (1.3±0.1)xl0-3 (1.4±0.3)xl0-6 (S^lO.TJxlO-6
Si + Cu 55% (5.6±0.3)xl0*3 (5.4±1.0)xl0-5 (6.1±1.2)xl0*5Si + Al 55% (8.6±0.4)xl0-3 (8.7±1.4)xl0*5 (6.0±1.4)xl0-5
Si + Pb Exclusive bins
89-64%
64-44%
44-30%
30-15%
15-8%
(6.0±1.7)xl0*2 (2.5+1.2)x 10-2 (3.0±2.5)x 10-2(1.4±0.2)xl0*2 (9.5±3.6)xl0‘5 (l.llO .SJxIO -4(6.3±0.8)xl0-3 (5.1±1.3)xl0‘5 (4.0±1.2)xl0-5(2.7±0.2)xl0*3 (6.1±0.9)xl0-6 (8.7±1.7)xl0‘6(1.7±0.2)xlQ-3 (2.6±0.5)x 10-6 (S .e+M jxlQ -6
Table B .l: Coalescence scale factor, BA, averaged over rapidity for various targets and centralities.
overlapping, ranges of the multiplicity spectrum. This is equivalent to taking slices of the
multiplicity spectrum. The exclusive bins are labeled by the percent of the geometric cross
section that corresponds to each edge of the interval.
114
S i + P b 8 9 % a geom
- 4X 10
- 4 X 10
~ 0 .006 -ii i i | ii it| rrn:
Figure B .l: Top: Invariant yields at pt = 0 plotted as a function o f rapidity for Si + Pb collisions. Bottom: Coalescence scale factor, BA, plotted as a function o f rapidity. Multiplicity cuts corresponding to 89% of Ogeom are used. Uncertainties are statistical only.
115
Si 4- P b 5 5 % a geom
O' -1
100)
o
n 1-----1-----r 1--------- 1--------- 1--------- r
- -Q .ni - -D — D ■□ Deuteron A Triton o sHe
>N -2■q. 10CL"D
-3 -o 10Q.
CM - 410
x 1 0 2
'A'"--A.....- ©
-I 1____ 1____ L I I I L_ _l____ ' I I1.5
- 4 x 10
2.5
Rapidity- 4
X 10
cr 0.4
Figure B.2: Top: Invariant yields at pt = 0 plotted as a function of rapidity for Si + Pb collisions. Bottom: Coalescence scale factor, BA, plotted as a function of rapidity. Multiplicity cuts corresponding to 55% of ogeom are used. Uncertainties are statistical only.
116
S i + P b 8 % crgeom
oM' -1
10a>
>\ -2 ■o 10C lX>
« -3-o 10CL
CM - 410
-o- □ Deuteron A Triton o He
•a
-I I I L I I I L
-•<5
I I I___ L_1.5 2.5
Rapidity
x 1 0 5
j n i | r i T i | i i i i.
11 »11 » 1 I« I 1 U T1 1.5 2 2 .5
Triton
Figure B.3: Top: Invariant yields at pt = 0 plotted as a function of rapidity for Si + Pb collisions. Bottom: Coalescence scale factor, BA, plotted as a function of rapidity. Multiplicity cuts corresponding to 8% of ogeom are used. Uncertainties are statistical only.
117
S i + P b 8 % — 1 5 % a gcom
t— — i-----1 r
□ Deuteron 'A Triton o sHe
>x -2■q. 10Q.
"D •A- —A— i —A—’ -A-
«T -3 •a 10
••A"- “- . 6 - -
CL
CM - 410 _l I I L j i i i L J! I I___ L_
_ X 102
0.32•O° 0.28
> 0.240o
<CD
0.2 r
0.16
0.12
0.08
0.04
0
j 1111111111111_]
h
1.5
- 4 x 10
0.233
0.21 0.187
0.163 0.14
0.117
0.093 0.07
0.047
0.023 0
2.5
Rapidity- 4
x 10
-i 111111' 111111 hi
‘I i i i I i i i i I i i i r1 1.5 2 2.5
Triton
0.184
0.166 0.147
0.129
0.111 0.092 0.074
0.055 0.037
0.018 0
Li i i i | i rr i'| i r i~q
H IT1 1.5 2 2.5
’He y
Figure B.4: Top: Invariant yields at pt = 0 plotted as a function o f rapidity for Si + Pb collisions. Bottom: Coalescence scale factor, BA, plotted as a function of rapidity. Multiplicity cuts corresponding to 8-15% of Ogeom are used. Uncertainties are statistical only.
118
S i + P b 1 5 % - - 3 0 % a geom
oN' -1 S ' 10<U
O
>N -2■D 10CL
T>
«r -3"D 10CL£
CM - 410
□ Deuteron A Triton o 3He
-A- 0 ■ -A -- A - ; ' j• - -6 • -
I I —I__ I I I . .1 J- i -l— » i
x 1 0 2
1.5
- 4X 10
2.5
Rapidity- 4
x 10
Figure B.5: Top: Invariant yields at pt = 0 vs. rapidity for Si + Pb collisions. Bottom:Coalescence scale factor, BA, vs. rapidity. Multiplicity cuts corresponding to 15-30% o f°geom are used- Uncertainties are statistical only.
119
S i + P b 3 0 % — 4 4 % a geom
ott• -1 S ' 10a>
o
>N -2■D 10C lX
- -Q-- -Q
□ Deuteron A Triton o 3He
-••A-:
~ o '5x 10CL
CN - 410 ■J I I L.
I I •
-I I I L
-•0-I
_l I I L_
1.5 2.5
Rapidity
-3 x 10
i 0.014 r 0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
I I I 1 1 I 1 I I I I I IT"
Triton
~i i i < I i " i 1 i " i~1 1.5 2 2.5
>He y
Figure B.6: Top: Invariant yields at p, = 0 plotted as a function of rapidity for Si + Pbcollisions. Bottom: Coalescence scale factor, BA, plotted as a function of rapidity.Multiplicity cuts corresponding to 30-44% of ogeom are used. Uncertainties are statistical only.
120
S i + P b 4 4 % — 6 4 % a geom
OCM' -1
10a;
O
>N -2■q 10CLX>
n 1---------1---------r n 1-----1-----r "i 1---------1---------r
□ Deuteron A Triton o He
--Q- —-ta-- -G3- — DJ -- -Q- - -G3- -
—3 x 10
-I II I | II I I | I ITT
: . 1I I I I I I I I I I I I I 11 1.5 2 2.5
Triton
0.5
0.4
0.3
0.2
0.1
0
_i i i i I m m | i i i n
~i i 11 1111 11 11 11'1 1.5 2 2.5
•He y
Figure B.7: Top: Invariant yields at pt = 0 plotted as a function o f rapidity for Si + Pb collisions. Bottom: Coalescence scale factor, BA, plotted as a function of rapidity. Multiplicity cuts corresponding to 44-64% of are used. Uncertainties are statistical only.
121
o(N' -1
10<u
o
>N -2■D 10 CL■o
~ i n 3xi 10CL1=
\ " 4 \ 101
S i + P b 6 4 % — 8 9 % a
•A- " A-•(5
-I I I L
geom
□ Deuteron a Triton o He
-I I I L1.5 2.5
Rapidity
i<
oCM><1)CD
<CD
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
ht-i ii | ii ii | i ii
i i I i i i i
Deuteron
1 1.5 2 2.5
Triton
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
li r r r r r i i m i i i u
11 i i I i i I I 1 IJ-L.1 1.5 2 2.5
’ He y
Figure B.8: Top: Invariant yields at pt = 0 plotted as a function of rapidity for Si + Pb collisions. Bottom: Coalescence scale factor, BA, plotted as a function of rapidity. Multiplicity cuts corresponding to 64-89% of ogeom are used. Uncertainties are statistical only.
Si + Cu 5 5 % a geom122
• - iS ' 10a>
>N -2■o 10Q l
X)
C l
1=CN -4
10
-O. - -o - -Q -- -Q — "GJ - --Q-
.6 .■“o*** i - *0 • i
j t * — i i i i i
□ Deuteron A Triton o He
* I I t1.5 2.5
Rapidity
7 0 . 0 0 8
> 0 . 0 0 7u
> 0 . 0 0 6>O 0 . 0 0 5
□□ 0 . 0 0 4 r
Figure B.9: Top: Invariant yields at pt = 0 plotted as a function of rapidity for Si + Cu collisions. Bottom: Coalescence scale factor, BA, plotted as a function of rapidity. Multiplicity cuts corresponding to 55% of ogeom are used. Uncertainties are statistical only.
123Si + Al 5 5 % a geom
OM' -1
100
o
>n - 2■o 10Q."O
«r - 3 •o 10C l
CM -410
□ Deuteron A Triton O He
-D-. -D -- -D -- -GJ -- .q -- -Q -
. ...a ... i •••A—
i - a
■ i i i i ' I l I I i I 1------- 1-------L.
1.5
x 1 0 3
-i i il 11 ii n i t
■ 1111111 I I I 1 JJL
Deuteron
1 1.5 2 2.5
Triton
x 1 0 3
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
2.5
Rapidity
JIM |~ri II | II I C
*i I I I I I i ' ' I I I I I1 1.5 2 2.5
JHe '
Figure B.10: Top: Invariant yields at pt = 0 plotted as a function of rapidity for Si + Al collisions. Bottom: Coalescence scale factor, BA, plotted as a function of rapidity. Multiplicity cuts corresponding to 55% of OgQOm are used. Uncertainties are statistical only.
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