Using Light Charged Particles to Probe the Asymmetry Dependence of the Nuclear Caloric Curve

0.2 MB·English

by  Alan B. McIntosh

Using Light Charged Particles to Probe the Asymmetry Dependence of the Nuclear
Caloric Curve
A.B. McIntosh,1,∗ A. Bonasera,1,2 Z. Kohley,1,3,† P.J. Cammarata,1,3 K. Hagel,1 L. Heilborn,1,3 J. Mabiala,1 L.W.
May,1,3 P. Marini,1,‡ A. Raphelt,1,3 G.A. Souliotis,1,4 S. Wuenschel,1,3 A. Zarrella,1,3 and S.J. Yennello1,3
1Cyclotron Institute, Texas A&M University, College Station, Texas, 77843, USA
2Laboratori Nazionali del Sud, INFN, I-95123 Catania, Italy
3Chemistry Department, Texas A&M University, College Station, Texas, 77843, USA
4Laboratory of Physical Chemistry, Department of Chemistry,
National and Kapodistrian University of Athens, Athens GR-15771, Greece
(Dated: January 30, 2013)
Recently,weobservedacleardependenceofthenuclearcaloriccurveonneutron-protonasymme-
3 try N−Z through examination of fully reconstructed equilibrated quasi-projectile sources produced
A
1 inheavyioncollisionsatE/A=35MeV.Inthepresentwork,weextendouranalysisusingmultiple
0 light charged particle probes of the temperature. Temperatures are extracted with five distinct
2 probes using a kinetic thermometer approach. Additionally, temperatures are extracted using two
n probes within a chemical thermometer approach (Albergo method). All seven measurements show
a asignificant linear dependenceof thesourcetemperatureonthesourceasymmetry. Forthekinetic
J thermometer,thestrengthoftheasymmetrydependencevarieswiththeprobeparticlespeciesina
way which is consistent with an average emission-time ordering.
8
2
PACSnumbers: 21.65.Ef,25.70.Lm,25.70.Mn,25.70.Pq

]
x
e I. INTRODUCTION Ourrecentmeasurement[7]showeda clearandstrong
- dependence of nuclear temperatures on the asymmetry
l
c Theequationofstate(EoS)ofnucleiandnuclearmat- using protons as probes. It was shown that an accu-
u
n terdescribestheemergentpropertiesofanuclearsystem, rate reconstruction of the fragmenting quasi-projectile
[ i.e. the relation between the thermodynamic quantities, sourcewasessentialtoextractthisdependence. Thispa-
which arise from the microscopic interactions between per extends our previous analysis using charged clusters
1 the constituent nucleons. The nuclear EoS is of broad as probes of the temperature. All of these temperature
v
interest for its importance in nucleosynthesis, heavy ion probes,somekinetic andsomechemical,exhibitasignif-
5
0 collisions,supernovaedynamics, andneutronstars[1–4]. icant dependence of the temperature on asymmetry.
8 Toexaminethe EoS,oftenthe relationbetweenonlytwo Theoreticalcalculationsprovideconflictingpredictions
6 thermodynamic quantities is examined. These could be, ofthedependenceofthecaloriccurveontheasymmetry.
1. for example, energy and density, pressure and temper- Some theoretical approaches (the Thomas-Fermi model
0 ature, density and temperature, or temperature and en- [8] and the mononucleus model [9]) predict that critical
3 ergy;thislastiscommonlyreferredtoasthecaloriccurve temperatures or limiting temperatures should be higher
1 and has been measured for many finite nuclear systems. for neutron-poor systems. Other theoretical approaches
v: AprimeexampleofthisisshowninworkbyPochodzalla (the hot liquiddropmodel [10], the statisticalmultifrag-
i et al. [5] where an observed plateau in the caloric curve mentation model [11], and isospin-dependent quantum
X
wasinterpretedasasignatureofaphasetransition. More moleculardynamics[12])predicthighertemperaturesfor
r recently, systematic analysis has shown that the caloric neutron-richsystems. The treatmentofa “gas”phase in
a
curvedependsonthemassA=N+Z ofthefinitesystem amodelisexpectedtoimpacttheasymmetrydependence
[6]. of the temperature of the bulk system [9, 10], and thus
Currently,thelargestuncertaintyinthenuclearEoSis an observation of an asymmetry dependence may sup-
the asymmetry energy (also referred to as the symmetry port the physical picture of a nuclear liquid interacting
energy). Althoughsignificantongoingefforthasbeenput with its vapor [9]. Moreover, observation of an asym-
forthtoconstrainthis,verylittleofithasfocusedonthe metrydependenttemperaturemayallowinsightintothe
influenceoftheneutron-protonasymmetry, N−Z,onthe driving force of nuclear disassembly [13].
A
nuclear caloric curve. Experimental efforts to probe the asymmetry energy
in the nuclear equation of state, such as Refs. [14–16],
use isotopic fragment yields as an observable, and make
the assumption that the temperature is independent of
∗ CorrespondingAuthor: [email protected]
the asymmetry. Characterization of the asymmetry-
† Present Address: National Superconducting Cyclotron Labora-
dependence of the caloric curve would allow a refined
tory, Michigan State University, East Lansing, Michigan 48824,
USA interpretation of this fragment yield data (e.g. in the
‡ PresentAddress: GANIL,BdHenriBecquerel,BP55027-14076 statistical interpretation of isoscaling). Moreover, char-
CAENCedex05,France acterization of this asymmetry dependence may offer a
2
new opportunity to probe the asymmetry energy. degree of freedom is quite slow to equilibrate compared
Withoneexception[7],eachpreviousanalysisofexper- to the thermaldegree offreedom[25, 26]. SphericalQPs
imental data to investigate this phenomena has demon- are selected by a constraint on the longitudinal momen-
strated either a small asymmetry dependence [13, 17] or tum p and transverse momentum p of the fragments
z t
no discernible asymmetry dependence [18, 19]. However, comprising the QP: −0.3 ≤ log (Q ) ≤ 0.3 where
10 shape
an accurate selection of the asymmetry of the excited Q = Pp2z with the sums extending over all frag-
source is crucial to discerning the temperature depen- shape Pp2t
ments of the QP. In this way, we have obtained a cohort
dence [7, 19]. In the present work, we build on our
of quasi-projectiles, tightly selected on mass, with prop-
recent observation, now demonstrating for a variety of
erties consistent with thermal equilibration, and with
temperature probes that nuclear temperatures exhibit a
known neutron-proton asymmetry.
significant asymmetry dependence.
The uncertainty in the QP composition arises mainly
fromthe free neutronmeasurement. We employamodel
which deduces the number of free neutrons emitted by
II. EXPERIMENT, EVENT SELECTION AND
the QP from the measured number of total neutrons,
RECONSTRUCTION
background, and the efficiencies for measuring neutrons
producedfromQPandQTsources[18,19,23]. Wehave
To study the dependence of the nuclear caloric curve investigated the efficiency of the neutron ball detector
on neutron-proton asymmetry, collisions of 70Zn+70Zn, throughdetailedsimulations[27,28]. Theuncertaintyin
64Zn+64Zn,and64Ni+64NiatE/A=35MeVwerestud- thefreeneutronmeasurementhastwoeffects: theassign-
ied. For details of the experiment, see Refs. [20, 21]. mentofaQPtoanincorrectbininneutron-protonasym-
Charged particles and free neutrons produced in the re- metry, and an error in the calculation of the excitation
actions were measuredin the NIMROD-ISiS 4π detector energy. The uncertainty in the free neutron multiplicity
array [19, 22]. The energy resolution achieved allowed is due primarily to the efficiency of the neutron detector
excellent isotopic resolution of charged particles up to (≈70%), and to a lesser extent the yield of background
Z=17. For events in which all charged particles are iso- signals in the neutron detector. The efficiency causes
topicallyidentified,thequasi-projectile(QP,theprimary an increase in the width of the measured neutron distri-
excited fragment that exists momentarily after a non- bution. This width of the neutron distribution given a
central collision) was reconstructed using the charged fixed number of initial neutrons depends on the initial
particles and free neutrons. Thus, the reconstructionin- number of neutrons but is less than 2.1 for the multi-
cludesdeterminationoftheQPcomposition,bothAand plicities encountered in this analysis [27]. Subtracting
Z. thisinquadraturefromthewidthoftherawdistribution
To examine the caloric curve, it is desired to select (5.36) gives a “true” width of 4.97. Thus the increase
equilibrated quasi-projectile sources. To this end, three in the width of the neutron distribution due to the effi-
cuts are made on the particle and event characteris- ciency is at most 9% at the highest excitation energies.
tics. Similar cuts, which have been used previously The measured background multiplicity distribution has
[7, 16, 18, 19, 23, 24], have been shown to select events a width that is narrower than the raw distribution by a
with properties consistent with equilibrated QP sources. factor of more than three. Subtracting these widths in
Toexcludefragmentsthatclearlydonotoriginatefroma quadrature reveals that the increase in the width due to
QP,the fragmentvelocity inthe beamdirectionvz is re- the background is less than 6%. The combined effect of
strictedrelativetothevelocityoftheheavyresiduevz,res. the efficiency andthe background(added inquadrature)
Theheavyresidueisdefinedasthelargestfragmentmea- is thus on the order of 11%. Given this uncertainty, as-
sured in the event, and thus is the largestdecay product signment of a QP to an asymmetry bin is rather precise.
of the QP. The accepted window on vz is 1±0.65 for TheexcitationenergyoftheQPwasdeducedusingthe
vz,res
Z=1, 1±0.60 for Z=2, and 1±0.45 for Z≥3. This elim- measured free neutron multiplicity, the charged particle
inates fast pre-equilibrium fragments, fragments origi- kinetic energies in the transverse direction, and the Q-
nating from the quasi-target (QT), and some fragments value ofthe breakup. Excitationenergies aboveE∗/A =
whichmaybeproducednon-statisticallybetweentheQP 2MeVarewellmeasuredwiththepresentdetectorarray.
and QT. Selection of a narrow mass range is important The average kinetic energy per free neutron ascribed to
due tothe mass-dependenceofthe caloriccurve,andbe- the QP is given by 2.2+1.25(E /A ) where E is
CP CP CP
causealargerangeofsourcemassescouldintroducefluc- thetotalenergyofthechargedparticlesinthetransverse
tuationswhichimpactthemomentumquadrupolefluctu- directionandA isthemassoftheQPcalculatedusing
CP
ation thermometer. One would like to select a QP mass only charged particles. This relation is determined from
range not too far from the projectile mass, while still se- the experimental Coulomb-shifted transverse kinetic en-
lectinganensembleofeventswithsufficientstatisticsfor ergy spectra of protons. The aptness of the Coulomb
the analysis. The mass of the reconstructed QP is re- shift was seen by applying the shift to the triton (3H)
quired to be 48 ≤ A ≤ 52, satisfying the requirements. andhelion(3He)transversekineticenergyspectra,which
The selection of thermally equilibrated QPs is achieved completely overlap when the shift is applied. For an ex-
throughthe selectionofsphericalevents,since the shape citationE /A of6MeV,theaverageneutronkinetic
CP CP
3
energyascribedtotheQPis9.7MeVandthus,foraQP
16
with 50 nucleons where 5 become free neutrons, the free
neutrons contribute 0.97 MeV per nucleon to the total p
excitation energy per nucleon. An error of 11% on the 14 d
t
free neutron multiplicity corresponds to an error of 0.11
h
MeVpernucleon. Thiserrorissignificantlylessthanthe V) 12 α
spacing between even the closest caloric curves selected e
M
on asymmetry (see section III). We considered the pos- T ( 10
sibility of an auto-correlationbetween the portion of the
excitation energy due to the neutrons and the observed
8
dependence of the caloric curve. For any physically re-
alistic values of neutron kinetic energy and multiplicity,
theasymmetrydependenceofthecaloriccurveisrobust. 6
In summary, the uncertainties in the QP composition
0 1 2 3 4 5 6 7 8
and in the QP excitation energy due to the free neutron E* / A (MeV)
measurement are sufficiently small that they do not sig-
nificantly bias the results reported in this paper.
FIG. 1. (Color online) Momentum quadrupole fluctuation
The temperatures of the QPs are calculated using two temperatures for sources with mass 48 ≤ A ≤ 52 and all
different methods. The first method is the momentum asymmetries, extracted with light charged particle probes.
quadrupole fluctuation (MQF) thermometer [29], which
hasbeenpreviouslyusedtoexaminetemperaturesofnu-
clei [7, 18, 19, 30, 31]. The momentum quadrupole is secondarydecay (for details, see, for example, Ref. [33]).
defined as Twocombinationsofisotopicyieldswereusedto extract
temperatures. For R = Y(d)/Y(t), the constants are
Q =p2 −p2, (1) Y(h)/Y(α)
xy x y B = 14.32 MeV, a = 1.59, and lnκ = 0.0097 MeV−1;
B
using the transverse components px and py of the parti- forR= Y(6Li)/Y(7Li),theconstantsareB =13.32MeV,
cle’s momentum in the frame of the QP source. If the Y(h)/Y(α)
a=2.18, and lnκ =−0.0051 MeV−1.
approximationismadethattheemitted(probe)particles B
The two thermometers to be employed here operate
are described by a Maxwell-Boltzmann distribution, the
infundamentallydifferentways. TheMQFthermometer
variance of Q is related to the temperature by
xy
is a kinetic thermometer, using the momentum of emit-
hσ2 i=4m2T2, (2) tedparticlestodeterminethetemperature. TheAlbergo
xy
thermometer is a chemical thermometer which uses the
where m is the probe particle mass [18, 29]. Since the fragment yields. As such, they measure different appar-
longitudinal component, p , may be impacted by the ent temperatures. Differences in the apparent tempera-
z
velocity cut which was imposed to exclude fragments ture as measured by different thermometers has been a
which do not originate from an equilibrated QP source, focus of study for many years. It is understood that cer-
onlythe transversecomponentsareusedtocalculatethe tainkineticthermometersmaydifferfromchemicalther-
quadrupole. For this analysis, protons, deuterons, tri- mometers due to a variety of factors such as the Fermi
tons, helions (3He), and alpha particles are used sepa- motion of nucleons (see e.g. Refs. [34, 35]).
rately as probes (denoted p, d, t, h, and α respectively). Itisnotthepurposeofthispapertoexploredifferences
Since the thermal energy in the primary clusters is of between thermometers. Our purpose is to provide ex-
course significantly less than that in the QP, the width perimental evidence that regardless of the thermometer
of the momentum quadrupole distribution is dominated employed or the the value of the temperature extracted,
by the QP breakup. Thereforeit is expected thatthe ef- the extracted temperature depends on the composition
fects of secondary decay on this thermometer are small. of the source.
Temperatures of the excited QPs are also calculated
using yield ratios following the method of Albergo et al.
[32]. Here, the temperature is calculated as III. RESULTS AND DISCUSSION
B
T = , (3a) Figure 1 shows the temperature of the QP, as a func-
raw
ln(aR) tion of the excitation energy per nucleon (E∗/A) of the
1 QP,measuredwiththeMQFthermometerusingprotons,
T = , (3b)
1 − lnκ deuterons,tritons,helionsandαparticlesasprobeparti-
Traw B cles. Temperaturesarecalculatedfor1MeV-widebinsin
where R is a double ratio of isotopic yields, B is a dif- excitation energy per nucleon. The statistical errors are
ference of binding energies, a is a mass-weighted spin smaller than the size of the points. For all particle types
degeneracyfactor,and lnκ is a correctionaccountingfor examined, the temperature shows a monotonic increase
B
4
16 16 16 16 16
Proton Deuteron Triton Helion Alpha
14 14 14 14 14
V) 12 12 12 12 12
e
M
T ( 10 10 10 10 m Range 10
s
0.04 - 0.08
8 8 8 8 0.08 - 0.128
0.12 - 0.16
0.16 - 0.20
6 6 6 6 0.20 - 0.246
V) 1 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8
e 0.5 0.5 0.5 0.5 0.5
M 0 0 0 0 0
T (-0.5 -0.5 -0.5 -0.5 -0.5
∆ -1 -1 -1 -1 -1
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
E* / A (MeV) E* / A (MeV) E* / A (MeV) E* / A (MeV) E* / A (MeV)
FIG.2. (Coloronline)Top row: caloric curvesforlightcharged particles, selectedonsourceasymmetry(ms= NA−Z). Bottom
row: temperature difference between each caloric curve and the middle caloric curve (0.12 < ms ≤ 0.16) which is used as a
reference. Thehorizontallinescorrespondtotheaveragedifferenceovertheindicatedrangeinexcitation. Errorbarsrepresent
thestatistical uncertainties.
with E∗/A. At E∗/A = 2.5 MeV, the temperatures are Ni+Ni reactions at E/A = 35 MeV in conjunction with
between5.5and9.5MeV;byE∗/A=8.5MeV,thetem- a molecular dynamics simulation (CoMD) to extract the
peratures haverisento between 10.5and15.5MeV. The average emission order of the LCPs. If different par-
temperatures for the differentlight chargedparticles dif- ticles probe different space-time regions, then Coulomb
fer by up to 5 MeV. effects might slightly influence the results as well. The
There is an overall ordering to these caloric curves. persistenceoftheorderingatthehighestexcitationener-
Foragivenexcitationenergypernucleonabove5.5MeV gies (where very rapid breakupof the source may occur)
(where the bulk of the yield is), the α particles show the suggests that the time-ordering may not be a complete
lowest temperature, followed by the protons, deuterons, explanation. Alternately, the temperature ordering may
tritons, and, lastly, helions. Below 5.5 MeV, the order is be indicative of differences in the average density of the
the sameexceptthatthe αparticleandprotontempera- source from which the probe particles originate. Such
tures are inverted. The ordering of the temperatures for density fluctuations are not at odds with the selection
thedifferentLCPspeciesaresummarizedbytherelation of an equilibrated source. In fact, fragment production
wouldnotbepossibleintheabsenceofsuchfluctuations,
T ,T <T <T <T . (4)
α p d t h since fragments are essentially a high density region in a
This ordering is consistent with a previous analysis [18], low-density or zero-density background.
which used these same LCP probes to measure the tem- Aplateauinthecaloriccurvehasbeenobservedprevi-
peraturewiththeMQFmethodforreconstructedsources ously,andinterpretedasasignatureofaphasetransition
produced in reactions of Kr+Ni at E/A = 35 MeV. [5, 6, 35]. The lack of a plateau in the present data [7],
The ordering observed in Fig. 1 may be due to dif- however, is not unexpected. A plateau is not as well de-
ferent average emission times for the different particles fined for such small sources (A ≈ 50) as it is for heavier
[36–39]. The tritons and helions are “expensive” parti- sources [6]. Since the density is not constrained here, a
cles, as the energycost(Q-value)for their emissionfrom varying density may mask the plateau [31, 40]. Finally,
an excited source (Q ≈ 20 MeV) is high relative to pro- the plateau might lie above the excitation energies mea-
tons and α particles (Q ≈ 10 MeV). Expensive parti- sured in this experiment [6].
cles are most likely to be emitted in the early stages of Figure 2 (top row, second panel) shows the temper-
the de-excitation of the source when the available exci- ature as a function of excitation energy per nucleon
tation energy is high. Therefore, the temperature ex- (E∗/A) of the QP as determined with the MQF ther-
hibited by the expensive particles reflects the temper- mometer using deuterons as the probe particle. Data
ature of the source only at the early stages of the de- points are plotted for 1 MeV-wide bins in excitation en-
excitation, when the source is hottest. Less expensive ergy per nucleon. For clarity, the points are connected
particles can be emitted throughout the de-excitation. with lines to guide the eye. The error bars correspond
Hence, the average temperature probed by the less ex- to the statistical uncertainty and, where not visible, are
pensive particles is lower. This emission-time-order sce- smallerthanthepoints. Thetemperatureshowsamono-
nario is consistent with previous work [39], which used tonic increase with excitation energy. At E∗/A = 2.5
scaled transverse LCP flow in reactions of Zn+Zn and MeV, the temperatures are around 7 MeV; by E∗/A
5
= 8.5 MeV, the temperatures have risen to around 13
MeV. Each curve corresponds to a narrow selection in 4.2 H/He 4.2 Li/He
the asymmetry of the source, m = Ns−Zs, where N ,
Z ,andA aretheneutronnumbser,protAosnnumber,ansd 4 4
s s
mass number, respectively, of the source. The m dis-
s 3.8 3.8
tribution is peaked around 0.15 and is quite broad, hav-
V)
ing a standard deviation around 0.07. The distribution Me3.6 3.6
is divided into bins of equal width as indicated in the T (
legend. The average asymmetries for the selections are 3.4 3.4
0.0640, 0.0988, 0.1370, 0.1758, and 0.2145. The caloric
curve is observed to depend on the asymmetry of the 3.2 3.2 00..0048 << mms ≤≤ 00..0182
0.12 < ms ≤ 0.16
source. Increasing the neutron content of the QP source 3 3 0.16 < ms ≤ 0.20
shifts the caloric curve to lower temperatures. In fact, 0.20 < mss ≤ 0.24
the caloriccurvesforthe five sourceasymmetriesareap- 2.8 2.8
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
proximately parallel and equally spaced. An increase of E* / A (MeV) E* / A (MeV)
0.15 units in m corresponds to a decrease in the tem-
s
perature by about 1.3 MeV. The top left panel of Fig. 2
FIG. 3. (Color online) Albergo yield ratio temperatures ex-
shows the caloric curves extracted with the MQF ther- tracted with deuterons, tritons, helions and α particles (left
mometer using protons as the probe particles. This was panel), and extracted with 6Li, 7Li, helions and α particles
showinRef.[7],andisshownhereforcompleteness. The (right panel). The different curves correspond to narrow se-
three remaining panels of the top row of Fig. 2 show the lections on thesource asymmetry.
caloric curves extracted with the MQF thermometer us-
ing tritons, helions and α particles. Again, each curve
corresponds to a narrow selection in m . For all LCP
s of the initial system [7]. The temperature only slightly
probes, the caloric curve shifts to lower temperature for
changesas the asymmetry ofthe initial systemis varied,
increasingneutroncontent. Foranygivenprobeparticle,
but the temperature does show a significant variation
the curvesareapproximatelyparallelandequallyspaced
with the asymmetry of the QP. The method of asym-
within statistical uncertainties.
metry selection may account for other previous investi-
TheimpactoftheQPmassselection(48≤AQP ≤52) gation not observing an asymmetry-dependent tempera-
has been investigated. It was considered that a range ture[18,19]. AstheQPde-excites,eachparticleemission
of masses could “artificially” broaden the momentum changes its composition slightly. If it were possible, one
quadrupole distributions, making the extracted temper- would correlate the temperature probed by the emitted
atures appear higher than they actually are. It was ob- particles to the composition of the QP at the time the
servedthattheselectionofasinglemass(A=50)results particles were produced. Since this is not possible ex-
inthe sametemperaturesasshownhere,thoughthe sta- perimentally, our method determines the initial compo-
tistical uncertainties are of course larger. Therefore, the sitionoftheQP,andcorrelatesthiswiththetemperature
widthoftheQPmassselectionusedinthisanalysisdoes probed by the emitted particles, which are emitted over
not impact the extracted temperatures or their depen- arangeoftimes. Usingthe measuredinitialcomposition
dence on ms. insteadofthecompositionattheinstantofparticleemis-
The shift in temperature due to the asymmetry of the sion could mask the asymmetry-temperature correlation
sourceisexaminedmorecloselyinthelowerrowofFig.2. somewhat. Therefore,thetruedependenceofthenuclear
The central m bin (0.12<m ≤0.16) has been used as temperatureontheasymmetrymaybeevengreaterthan
s s
a reference. The difference in temperature between each the considerable dependence observed here.
caloriccurveandthe referenceisplotted asafunctionof Figure 3 shows caloric curves obtained with the Al-
the excitation energy per nucleon. For all LCP probes, bergothermometerusingd,t,h,αyields(leftpanel)and
thedifferencesarerelativelyconstantwithexcitationen- using6Li,7Li,h,α(rightpanel). Thesetemperaturesare
ergy. The average ∆T for each pair of ms bins is indi- lower than those obtained with the MQF method, rising
cated by the horizontal lines; the range used for averag- from around T = 3 MeV at E∗/A = 2 MeV to around
ingisindicatedbytheendpointsofthelines. Thoughthe T = 4 MeV at E∗/A = 8 MeV. The difference between
statisticalfluctuationsvaryforthedifferentLCPprobes, these temperatures and those extracted with the MQF
the same trends are observedin allcases. For the α par- methodisaresultofthe differentnatureofthe twother-
ticles, there may be a spreading of the caloric curves at mometers (chemical vs kinetic) as previously discussed.
low excitation energy; however, above E∗/A = 3 MeV, The many methods of extracting nuclear temperatures
the difference is approximately constant. The constancy are known to yield disparate values for a variety of rea-
showsthattheshiftinthecaloriccurveswithasymmetry sons(seee.g. Ref.[35]). Thequestionathandiswhether,
is independent of excitation energy. for a given thermometer, the temperature can be ob-
We have previously shownthe importance of selecting servedtodependontheasymmetry. Thedifferentcurves
the asymmetry of the QP rather than the asymmetry in Fig. 3 correspond to selections on the asymmetry of
6
the isotopically reconstructed QP. The temperatures ex- 0
tracted for neutron poor sources are consistently higher
-0.2
than for neutron rich sources. An increase of 0.15 units
in m corresponds to a decrease in the Albergo temper- -0.4
s
ature on the order of 0.15 MeV - small, but statistically
-0.6
significant. This trend is consistent with the results of
V)
the MQF method. e-0.8
M
calTooricexcaumrvineewinithmocrheadnegtinaigltshoeurtceemapseyrmatmureetrsyh,ifwteofptlhoet ∆T ( -1 Qupadrupole
d
∆T vs ∆m in Fig. 4. This is obtained in the follow- -1.2
s t Albergo
ing way. For each possible pairing (10 pairings total) of
-1.4 h H/He
the five caloric curves (one for each ms bin), the tem- α Li/He
perature difference as a function of excitation energy is -1.6
calculated. Thesedifferencesareapproximatelyconstant
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
as a function of excitation energy as seen in the lower ∆m
s
row of Fig. 2; the excitation-averaged ∆T is plotted in
Fig.4. Foreachpossiblepairingofthefivecaloriccurves,
FIG. 4. (Color online) Change in temperature ∆T with
∆ms correspondsto the difference in the meanasymme- changing asymmetry ms for the five different probes of the
tries of the pair of curves. The results obtained with the momentum quadrupole fluctuation temperatures and two
MQFthermometerusingprotons,deuterons,tritons,he- methods of extracting theAlbergo yield ratio temperature.
lions,andαparticlesareshownwiththecircles,squares,
up-triangles, down-triangles, and stars respectively; the
results from the two Albergo thermometers are shown changed somewhat due to preceding particle emissions.
with the open and closeddiamonds. The strength of the Thus, the correlation between the difference in the ex-
correlation for the Albergo method is weaker than it is tracted QP temperature (∆T) for a given difference in
fortheMQFmethod. Thismaybearesultofthesmaller initial QP asymmetry (∆m ) is least obscured for parti-
s
valueofthetemperaturefortheAlbergothermometeras cles emitted at the earliest times.
comparedto the MQF thermometer,and may as wellbe The slope of the correlation between ∆T and ∆m
s
due to the different nature of the thermometers (chem- maydependontheasymmetryenergycoefficientandthe
ical versus kinetic). For every method and probe, the Coulomb energy coefficient as suggested in Ref. [7]. The
negative correlationfollows a linear trend. The slopes of asymmetry energy and Coulomb energy will depend on
linear fits to the data are -7.3 ± 0.2, -9.2 ± 0.3, -9.4 ± these coefficients as well as the composition of both the
0.3, -10.6 ± 0.7, and -5.5 ± 0.1 MeV per unit change in QP source and the emitted particle. By varying these
ms for the p, d, t, h, and α particle probes. The slopes systematically,it may be possible to extract information
of the linear fits are -1.01 ± 0.07 and -0.94 ± 0.15 MeV leading to constraints on the asymmetry energy. This
per unit change in ms for the H/He and Li/He Albergo wouldbegreatlyaidedbyamoredetailedunderstanding
probes respectively. These linear fits describe the data ofthe emissiontime orderingofLCPs andits relationto
well over a broad range in source asymmetry. thetheorderingobservedinthetemperature(Fig.1)and
Thestrengthofthe∆T/∆ms correlationfortheMQF observed in the asymmetry dependence of the tempera-
thermometershowninFig.4dependsontheprobeparti- ture (Fig. 4). Extractionof information on the asymme-
cle. Interestingly,the slopesareorderedinthe sameway try energy would also benefit from an accurate estimate
by particle that the absolute values of the temperatures of the density felt by the probe particles.
are ordered for E∗/A > 5.5 MeV (Fig. 1). This region
in excitation is where the bulk of the yield is. The he-
lions, which exhibit the highest temperatures, show the
IV. CONCLUSIONS
strongest dependence on ∆m ; the protons and α par-
s
ticles, with the lowest temperatures, show the weakest
Experimental evidence for the dependence of nuclear
dependence on ∆m . The ordering of the slopes is sum-
s temperaturesontheneutron-protonasymmetryhasbeen
marized by the relation
presented. Our previous work [7] has shown a clear
t <t <t <t <t . (5) decrease in the temperature with the asymmetry using
α p d t h
themomentumquadrupolefluctuationthermometerwith
Thereasonfortheorderingoftheslopesisnotyetestab- protons as the probe particle.
lished. Itispossiblethattheorderingisadirectresultof In this work, the temperature is extracted with the
theemissionorder. Inthisscenario,theparticlesemitted MQF(kinetic)thermometerusingdeuterons,tritons,he-
earliest(helions, tritons) would be coming from a source lions, and alpha particles in addition to protons. All
whose compositionis experimentally well defined. Parti- these probes of the nuclear temperature exhibit a sim-
cles which are on average emitted later (alpha, protons) ilar dependence on asymmetry. The magnitude of the
wouldbe emergingfroma sourcewhose compositionhas dependence is on the order of 1 MeV for a change of
7
0.15 units in asymmetry. Moreover, this signal is not of the temperature.
limited to the MQF method. We have shown that two
Albergo(chemical)thermometers,H/HeandLi/He,pro-
vide temperaturesthatalsoclearlyexhibita dependence
the asymmetry on the order of 0.15 MeV for a change of
ACKNOWLEDGMENTS
0.15 units in asymmetry. For all seven ways of extract-
ing the temperature,the changeintemperature depends
linearly on the change in asymmetry. We thank the staff of the TAMU Cyclotron Institute
Additionally, the ordering of the LCP caloric curves for providing the high quality beams which made this
(MQF temperature) is consistent with an average time- experiment possible. This work was supported by the
orderingoftheLCPemission. Thissameorderingisalso Robert A. Welch Foundation (A-1266), and the U. S.
observed in the strength of the asymmetry dependence Department of Energy (DE-FG03-93ER-40773).
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