In medium binding energies and Mott points for $d$, $t$, $^3$He and $\alpha$
clusters in low density nuclear matter have been determined at specific
combinations of temperature and density in low density nuclear matter produced
in collisions of 47$A$ MeV $^{40}$Ar and $^{64}$Zn projectiles with $^{112}$Sn
and $^{124}$Sn target nuclei. The experimentally derived values of the in
medium modified binding energies are in good agreement with recent theoretical
predictions based upon the implementation of Pauli blocking effects in a
quantum statistical approach.Comment: 5 pages, 3 figure

A. Bonasera

B. Hale

C. Bottosso

D. Fabris

G. Nebbia

G. Prete

G. Röpke

G. Viesti

H. Zheng

J. B. Natowitz

J. Wang

K. Hagel

K. Schmidt

L. Qin

M. Barbui

M. Cinausero

M. Huang

M. Lunardon

M. R. D. Rodrigues

M. E. Fisher

R. Wada

S. Kowalski

S. Moretto

S. Pesente

S. Shlomo

S. Typel

T. Keutgen

V. Rizzi

Y. El Masri

Z. Chen

Z. Majka

English

arXiv

Crossref

Physical Review Letters

Experimental Determination of In-Medium Cluster Binding Energies and Mott Points in Nuclear Matter

In medium binding energies and Mott points for d, t, 3He and α clusters in low density nuclear matter have been determined at specific combinations of temperature and density in low density nuclear matter produced in collisions of 47A MeV $^{40}Ar$ and $^{64}Zn$ projectiles with $^{112}Sn$ and $^{124}Sn$ target nuclei. The experimentally derived values of the in medium modified binding energies are in good agreement with recent theoretical predictions based upon the implementation of Pauli blocking effects in a quantum statistical approach

Hagel, K.

Wada, R.

Wang, J.

Zheng, H.

Kowalski, S.

Bottosso, C.

Barbui, M.

Rodrigues, M. R. D.

Schmidt, K.

Fabris, D.

Lunardon, M.

Moretto, S.

Qin, L.

Nebbia, G.

Pesente, S.

Rizzi, V.

Viesti, G.

Cinausero, M.

Prete, G.

Keutgen, T.

El Masri, Y.

Majka, Z.

Natowitz, J. B.

Shlomo, S.

Bonasera, A.

Röpke, G.

Typel, S.

Chen, Z.

Huang, M.

GSI Repository

Texas A&M University

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Experimental Determination of In-Medium Cluster Binding Energies and Mott Points
in Nuclear Matter
K. Hagel,1 R. Wada,2, 1 L. Qin,1 J. B. Natowitz,1 S. Shlomo,1 A. Bonasera,1,3 G. Ro¨pke,4 S.
Typel,5 Z. Chen,2 M. Huang,2 J. Wang,2 H. Zheng,1 S. Kowalski,6 C. Bottosso,1 M. Barbui,1 M.
R. D. Rodrigues,1 K. Schmidt,1 D. Fabris,7 M. Lunardon,7 S. Moretto,7 G. Nebbia,7 S. Pesente,7
V. Rizzi,7 G. Viesti,7 M. Cinausero,8 G. Prete,8 T. Keutgen,9 Y. El Masri,9 and Z. Majka10
1Cyclotron Institute, Texas A&M University, College Station, Texas 77843
2Institute of Modern Physics HIRFL, Chinese Academy of Sciences, Lanzhou, 730000,China.
3Laboratori Nazionali del Sud, INFN, via Santa Sofia, 62, 95123 Catania, Italy
4University of Rostock, FB Physik, Rostock, Germany
5GSI Helmholtzzentrum fu¨r Schwerionenforschung GmbH,
Theorie, Planckstrae 1, D-64291 Darmstadt, Germany
6Institute of Physics, Silesia University, Katowice, Poland.
7Dipartamento di Fisica dell Universita di Padova and INFN Sezione di Padova, Padova, Italy
8INFN Laboratori Nazionali di Legnaro, Legnaro, Italy
9FNRS and IPN, Universite´ Catholique de Louvain, B-1348 Louvain-Neuve, Belgium
10Smoluchowski Institute of Physics, Jagiellonian University, Krakow, Poland
(Dated: July 24, 2018)
In medium binding energies and Mott points for d, t, 3He and α clusters in low density nuclear
matter have been determined at specific combinations of temperature and density in low density
nuclear matter produced in collisions of 47A MeV 40Ar and 64Zn projectiles with 112Sn and 124Sn
target nuclei. The experimentally derived values of the in medium modified binding energies are in
good agreement with recent theoretical predictions based upon the implementation of Pauli blocking
effects in a quantum statistical approach.
PACS numbers: 25.70.Pq
Keywords: Intermediate Heavy ion reactions, chemical equilibrium, neutron and proton chemical poten-
tial,quantum statistical model calculations
I. INTRODUCTION
Simple nuclear statistical equilibrium models assume
that properties of the species in equilibrium are the same
as those of the isolated species. While this assumption
is tenable at very low density, it is untenable at higher
densities where in-medium effects lead to dissolution of
the clusters and a transition to cluster-free nuclear mat-
ter. To deal with this intermediate density range, Typel
et al. [1] have developed a quantum statistical approach
which includes cluster correlations in the medium and
interpolates between the exact low-density limit and the
very successful relativistic mean field (RMF) approaches
appropriate near the saturation density. The general-
ized RMF model developed attributes the decrease of
the cluster fractions at high densities to a reduction of
the cluster binding energies due to the Pauli blocking.
This leads to the Mott effect of vanishing binding [2].
Well-defined clusters appear only for densities below ap-
proximately 1/10 of the saturation density and get dis-
solved at higher densities. The maximum cluster density
is reached around the Mott density. Because of the pres-
ence of strong correlations in the scattering state contin-
uum that are effectively represented by one resonance,
there is a non-vanishing cluster fraction above the Mott
density [3]. We report here the first experimental deriva-
tion of temperature and density dependent binding ener-
gies of d, t, 3He and α clusters, directly from experimental
particle yields. Experimental values for Mott points for
d, t, 3He and α clusters are in good agreement with the
predictions made in reference [1].
EXPERIMENTAL TECHNIQUES
We reported in Refs. [4] and [5] that measurements of
nucleon and light cluster emission from the participant
matter which is produced in near Fermi energy heavy
ion collisions could be employed to probe the EOS at
low density and moderate temperatures where cluster-
ing is important. The NIMROD 4pi multi-detector at
Texas A&M University has now been used to extend our
measurements to higher densities. Cluster production
in collisions of 47A MeV 40Ar with 112,124Sn and 64Zn
with 112,124Sn was studied. NIMROD consists of a 166
segment charged particle array set inside a neutron ball
[6]. The charged particle array is arranged in 12 rings
of Si-CsI telescopes or single CsI detectors concentric
around the beam axis. The CsI detectors are 1-10 cm
thick Tl doped crystals read by photomultiplier tubes. A
pulse shape discrimination method is employed to iden-
tify light particles in the CsI detectors. Neutron mul-
tiplicity is measured with the 4pi neutron detector sur-
rounding the charged particle array. The combined neu-
2tron and charged particle multiplicities were employed
to select the most violent events for subsequent analysis.
Further details on the detection system, energy calibra-
tions and neutron ball efficiency may be found in refer-
ence 6.
ANALYSIS
The dynamics of the collision process allow us to probe
the nature of the intermediate velocity “nucleon-nucleon”
emission source [7–10]. Measurement of emission cross
sections of nucleons and light clusters together with suit-
able application of a coalescence ansatz [7] provides the
means to probe the properties and evolution of the inter-
action region. The techniques used have been detailed
in several previous publications [4, 5, 8–11] and are de-
scribed briefly below. A notable difference from refer-
ences 4 and 5 is the method of density extraction. This
is discussed more extensively in the following. We em-
phasize that the event selection is on the more violent
collisions. Cross section weighting favors mid-range im-
pact parameters.
An initial estimation of emission multiplicities at each
stage of the reaction was made by fitting the observed
light particle spectra assuming contributions from three
sources, a projectile-like fragment (PLF) source, an inter-
mediate velocity (IV) source, and a target-like fragment
(TLF) source. A reasonable reproduction of the observed
spectra is achieved. Except for the most forward detec-
tor rings the data are dominated by particles associated
with the IV and TLF sources. The IV source velocities
are very close to 50% of the beam velocity as seen in
many other studies ([7–10] and references therein). The
observed spectral slopes reflect the evolution dynamics of
the source [10, 12, 13]. For further analysis, this IV source
is most easily sampled at the intermediate angles where
contributions from the other sources are minimized. For
the analysis of the evolution of the source we have se-
lected the data in ring 9 of the NIMROD detector. This
ring covered an angular range in the laboratory of 38◦ to
52◦. The results of the three-source fit analyses, as well
as inspection of invariant velocity plots constructed for
each ejectile and each system, indicate that this selection
of angular range minimizes contributions from secondary
evaporative decay of projectile like or target like sources
[11].
We treat the IV source as a nascent fireball created
in the participant interaction zone. The expansion and
cooling of this zone leads to a correlated evolution of
density and temperature which we probe using particle
and cluster observables, yield, energy and angle. As in
the previous work [4, 5] we have employed double iso-
tope yield ratios [14, 15] to characterize the temperature
at a particular emission time. Model studies comparing
Albergo model temperatures and densities to the known
input values have shown the double isotope ratio temper-
atures to be relatively robust in this density range [16].
However the densities extracted using the Albergo model
are useful only at the very lowest densities [16]. Both of
these results are confirmed in the more extensive calcu-
lations of reference 1. In this study we have employed a
different means of density extraction, the thermal coales-
cence model of Mekjian [8, 10].
To determine the coalescence parameter P0, the radius
in momentum space, from our data we have followed the
Coulomb corrected coalescence model formalism of Awes
et al. [17] and previously employed by us in reference [7].
In the laboratory frame the derived relationship between
the observed cluster and proton differential cross sections
is
d2N(Z,N,EA)
dEAdΩ
= RNnp
A−1
N !Z!
(
4
3piP
3
0
[2m3(E−EC)]
1
2
)A−1
×
(
d2N(1,0,E)
dEdΩ
)A
(1)
where the double differential multiplicity for a cluster of
mass number A containing Z protons and N neutrons
and having a Coulomb-corrected energy EA, is related
to the proton double differential multiplicity at the same
Coulomb corrected energy per nucleon, E − EC , where
EC is the Coulomb barrier for proton emission. Rnp is
the neutron to proton ratio. Since within the framework
of the coalescence model the yield ratios of two isotopes
which differ by one neutron are determined by their bind-
ing energies and the n/p ratio in the coalescence volume,
we have used the observed triton to 3He yield ratio to
derive the n/p ratio used in this analysis.
In the Mekjian model thermal and chemical equilib-
rium determines coalescence yields of all species. Under
these assumptions there is a direct relationship between
the derived radius in momentum space and the volume
of the emitting system. In terms of the P0 derived from
Eq. (1) and assuming a spherical source
V =
((Z!N !A3
2A
)
(2s+ 1)e
E0
T
) 1
(A−1) 3h3
4piP 30
(2)
where h is Plancks constant and Z, N , and A are the
same as in Eq. (1), E0 is the binding energy and s the
spin of the emitted cluster and T is the temperature.
Thus the volume can be derived from the observed P0
and temperature values assuming a spherical shape in
terms of the P0 derived from Eq. (1).
Because our goal was to derive information on the den-
sity and temperature evolution of the emitting system,
our analysis was not limited to determining an average
P0 value. Instead, as in our previous studies [4, 5, 8],
results for d, t, 3He, and 4He, were derived as a function
of vsurf, the velocity of the emerging particle at the nu-
clear surface, prior to Coulomb acceleration [17]. From
3the relevant P0 values we then determined volumes using
Eq. (2). A comparison of these volumes indicated good
agreement for t, 3He and 4He. The volumes derived from
the deuteron data are typically somewhat smaller. This
appears to reflect the fragility of the deuteron and its
survival probability once formed [18]. For this reason we
have used average volumes derived from the A=3 and 4
clusters to calculate the densities. Given that mass is
removed from the system during the evolution, we deter-
mined the relevant masses for each volume by assuming
that the initial mass of the source was that determined
from the source fitting analysis and then determining the
mass remaining at a given vsurf from the observed energy
spectra. This is also an averaging process and ignores
fluctuations. Densities were determined by dividing re-
maining masses by volumes.
RESULTS
Temperatures and Densities
Inspection of the results for the four different sys-
tems studied revealed that the temperatures, densities
for all systems are the same within statistical uncertain-
ties. Therefore we have combined them to determine the
values reported in this paper.
We present, in Figure 1 the experimentally derived
density and temperature evolution of the IV source. Es-
timated errors on the temperatures are 10% below ρ =
0.01 fm−3increasing to 15% at ρ = 0.03 fm−3. Estimated
errors on the densities are 20%.
In a recently submitted paper we reported equilibrium
constants for α cluster formation as a function of temper-
ature and density [19]. These equilibrium constants were
then compared with those predicted by several different
astrophysical equation of state models. Specifically we
defined the equilibrium constants, Kc, for cluster forma-
tion in terms of density as
Kc(A,Z) = ρ(A,Z)/[(ρp)
Z(ρn)
N ] (3)
where ρ(A,Z) is the density of clusters of a specific mass
number A and atomic number Z, N is the neutron num-
ber in the cluster and ρp and ρn are, respectively, the
densities of free protons and neutrons. In the present
work, we employ the observed temperature and density
dependence of these equilibrium constants to extract the
in medium modifications of the cluster binding energies
and determine Mott points.
3
, nuc/fmρ
0 0.01 0.02 0.03 0.04
T,
 
M
eV
5
10
15
FIG. 1: Temperatures and densities sampled by the expand-
ing IV source.
DERIVATION OF IN MEDIUM BINDING
ENERGIES
Our departure point for extraction of the medium mod-
ified cluster binding energies is the chemical equilibrium
expression relating the density of a cluster of mass num-
ber A and atomic number Z to the densities of neutrons
and protons in the same volume V [12].
ρ(A,Z) =
N(A,Z)
V
=
A
3
2 λ
3(A−1)
T ω(A,Z)
(2sp+1)Z(2sn+1)A−Z
×ρZp ρ
A−Z
n exp
B(A,Z)
T
(4)
In this expression, λT =
h
(2pim0T )1/2
is the thermal
wavelength of a nucleon, sp and sn are the proton and
neutron spins, T is the temperature and B(A,Z) is the
cluster binding energy. The term ω(A,Z) is the internal
partition function of the cluster, taken here to 1 for the
Z = 1 and Z = 2 clusters considered.
Minich et al. [20, 21] used a related yield expression
to analyze intermediate mass fragment yields in multi-
fragmentation experiments but added both a mixing en-
tropy term and a surface entropy term, as initially pro-
posed by Fisher [22]. This latter term leads to a power
law behavior of the mass distribution at the critical point
and its ramifications have been widely explored [23–25].
Neither of these entropy contributions is explicitly in-
cluded in the Albergo formulation [12]. In the present
analysis of the experimental data we include a mixing
4entropy term in the free energy. This term has the form
∆F = T (Z ln(Z/A) +N ln(N/A)) (5)
where once again Z, N and A are those of the cluster
being formed [20]. As mixing is a spontaneous process the
free energy of mixing is negative and therefore favors the
cluster formation. We do not include a Fisher term. Our
reasoning for this is that, without additional corrections,
the Fisher term as normally formulated and applied to
larger clusters whose properties are very similar to the
bulk properties, is not applicable to the yields of the very
small clusters, A ≤ 4, which we are treating. We base this
conclusion on the results of molecular dynamics studies of
the cluster size dependence of the surface energy [26–28]
and the binding energies per nucleon of the competing
Z = 1, 2 species being significantly different from the
bulk.
Thus, rearranging Eq. (4) , substituting Kc from Eq.
(3) and taking the logarithm of each side we can write a
general expression for each cluster,
ln[Kc/C(T )] = B/T − Z ln(Z/A)−N ln(N/A) (6)
where C(T ) includes all terms on the right hand side of
Eq. (4) except the exponential term. Using the experi-
mentally determined equilibrium constants and tempera-
tures we then solve this expression to obtain the apparent
binding energies, B(ρ, T ), of the clusters for the different
temperatures and densities sampled in the experiments.
The binding energies extracted for d, t, 3He and α clus-
ters decrease monotonically with increasing density as
shown in Fig. 2.
MOTT POINTS
By definition, a Mott point corresponds to a combi-
nation of density and temperature at which a cluster
binding energy, B(ρ, T ), is zero with respect to the sur-
rounding medium. Since the observed temperatures and
densities are correlated in our experiment (see Figure 1)
each point in Figure 2 at which the experimentally de-
rived binding energy is zero corresponds to a particular
combination of density and temperature. Thus, with the
present data, we are able to extract a single Mott point
for each cluster. In Figure 3 we present the values of
the Mott temperatures and densities and compare them
with the loci of the values of medium modified binding
energies predicted by Typel et al. [1] using the thermody-
namic Green function method. Such a Mott line was also
calculated in reference [29] while the contribution of cor-
relations was considered without discriminating among
different clusters. This approach makes explicit use of
an effective nucleon-nucleon interaction to account for
medium effects on the cluster properties [30]. We see
that the agreement between the predictions and the ex-
perimental results is quite good.
3
, nuc/fmρ
0 0.005 0.01 0.015
,T
), M
eV
ρ
B
(
-5
0
5
10
15
20
25
30
35
B(T) d
B(T) t
He3B(T)  
He4B(T)  
FIG. 2: In medium binding energies derived from the exper-
iments as a function of density. T and ρ are changing in a
correlated fashion. (See text.)
3
, nuc/fmρ
0 0.005 0.01 0.015 0.02
T,
 
M
eV
0
5
10
15
Expt d
Expt t
He3Expt  
He4Expt  
QS d
QS t
He3QS  
He4QS  
FIG. 3: Comparison of experimentally derived Mott point
densities and temperatures with theoretical values. Symbols
represent the experimental data. Estimated errors on the
temperatures are 10% and on the densities 20%. Lines show
polynomial fits to the Mott points presented in reference [1].
5SUMMARY AND CONCLUSIONS
We have presented a first experimental determination
of in medium cluster binding energies and Mott points for
d, t, 3He and α clusters produced in low density nuclear
matter. Our results are in good agreement with those
predicted by a recent model which explicitly treats these
quantities. Inclusion of the in-medium effects in astro-
physical equations of state should improve the utility of
those for modeling astrophysically interesting events.
ACKNOWLEDGEMENTS
This work was supported by the United States Depart-
ment of Energy under Grant # DE-FG03- 93ER40773
and by The Robert A. Welch Foundation under Grant #
A0330.
[1] S. Typel, G. Ro¨pke, T. Kla¨hn, D. Blaschke and H. H.
Wolter, Phys. Rev. C 81, 015803 (2010).
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In-medium binding energies and Mott points for d, t, He-3 and alpha clusters in low-density nuclear matter have been determined at specific combinations of temperature and density in low-density nuclear matter produced in collisions of 47A MeV Ar-40 and Zn-64 projectiles with Sn-112 and Sn-124 target nuclei. The experimentally derived values of the in-medium modified binding energies are in good agreement with recent theoretical predictions based upon the implementation of Pauli blocking effects in a quantum statistical approach

G. Ropke

M. R. D.

Y. E. Masri

LUNARDON, MARCELLO

MORETTO, SANDRA

VIESTI, GIUSEPPE

Archivio istituzionale della ricerca - Università di Padova

R��pke, G.

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:1110.4972v2  [nucl-ex]  20 Mar 2012Experimental Determination of In-Medium Cluster Binding Energies and Mott Pointsin Nuclear MatterK. Hagel,1 R. Wada,2, 1 L. Qin,1 J. B. Natowitz,1 S. Shlomo,1 A. Bonasera,1,3 G. Ro¨pke,4 S.Typel,5 Z. Chen,2 M. Huang,2 J. Wang,2 H. Zheng,1 S. Kowalski,6 C. Bottosso,1 M. Barbui,1 M.R. D. Rodrigues,1 K. Schmidt,1 D. Fabris,7 M. Lunardon,7 S. Moretto,7 G. Nebbia,7 S. Pesente,7V. Rizzi,7 G. Viesti,7 M. Cinausero,8 G. Prete,8 T. Keutgen,9 Y. El Masri,9 and Z. Majka101Cyclotron Institute, Texas A&M University, College Station, Texas 778432Institute of Modern Physics HIRFL, Chinese Academy of Sciences, Lanzhou, 730000,China.3Laboratori Nazionali del Sud, INFN, via Santa Sofia, 62, 95123 Catania, Italy4University of Rostock, FB Physik, Rostock, Germany5GSI Helmholtzzentrum fu¨r Schwerionenforschung GmbH,Theorie, Planckstrae 1, D-64291 Darmstadt, Germany6Institute of Physics, Silesia University, Katowice, Poland.7Dipartamento di Fisica dell Universita di Padova and INFN Sezione di Padova, Padova, Italy8INFN Laboratori Nazionali di Legnaro, Legnaro, Italy9FNRS and IPN, Universite´ Catholique de Louvain, B-1348 Louvain-Neuve, Belgium10Smoluchowski Institute of Physics, Jagiellonian University, Krakow, Poland(Dated: July 24, 2018)In medium binding energies and Mott points for d, t, 3He and α clusters in low density nuclearmatter have been determined at specific combinations of temperature and density in low densitynuclear matter produced in collisions of 47A MeV 40Ar and 64Zn projectiles with 112Sn and 124Sntarget nuclei. The experimentally derived values of the in medium modified binding energies are ingood agreement with recent theoretical predictions based upon the implementation of Pauli blockingeffects in a quantum statistical approach.PACS numbers: 25.70.PqKeywords: Intermediate Heavy ion reactions, chemical equilibrium, neutron and proton chemical poten-tial,quantum statistical model calculationsI. INTRODUCTIONSimple nuclear statistical equilibrium models assumethat properties of the species in equilibrium are the sameas those of the isolated species. While this assumptionis tenable at very low density, it is untenable at higherdensities where in-medium effects lead to dissolution ofthe clusters and a transition to cluster-free nuclear mat-ter. To deal with this intermediate density range, Typelet al. [1] have developed a quantum statistical approachwhich includes cluster correlations in the medium andinterpolates between the exact low-density limit and thevery successful relativistic mean field (RMF) approachesappropriate near the saturation density. The general-ized RMF model developed attributes the decrease ofthe cluster fractions at high densities to a reduction ofthe cluster binding energies due to the Pauli blocking.This leads to the Mott effect of vanishing binding [2].Well-defined clusters appear only for densities below ap-proximately 1/10 of the saturation density and get dis-solved at higher densities. The maximum cluster densityis reached around the Mott density. Because of the pres-ence of strong correlations in the scattering state contin-uum that are effectively represented by one resonance,there is a non-vanishing cluster fraction above the Mottdensity [3]. We report here the first experimental deriva-tion of temperature and density dependent binding ener-gies of d, t, 3He and α clusters, directly from experimentalparticle yields. Experimental values for Mott points ford, t, 3He and α clusters are in good agreement with thepredictions made in reference [1].EXPERIMENTAL TECHNIQUESWe reported in Refs. [4] and [5] that measurements ofnucleon and light cluster emission from the participantmatter which is produced in near Fermi energy heavyion collisions could be employed to probe the EOS atlow density and moderate temperatures where cluster-ing is important. The NIMROD 4pi multi-detector atTexas A&M University has now been used to extend ourmeasurements to higher densities. Cluster productionin collisions of 47A MeV 40Ar with 112,124Sn and 64Znwith 112,124Sn was studied. NIMROD consists of a 166segment charged particle array set inside a neutron ball[6]. The charged particle array is arranged in 12 ringsof Si-CsI telescopes or single CsI detectors concentricaround the beam axis. The CsI detectors are 1-10 cmthick Tl doped crystals read by photomultiplier tubes. Apulse shape discrimination method is employed to iden-tify light particles in the CsI detectors. Neutron mul-tiplicity is measured with the 4pi neutron detector sur-rounding the charged particle array. The combined neu-2tron and charged particle multiplicities were employedto select the most violent events for subsequent analysis.Further details on the detection system, energy calibra-tions and neutron ball efficiency may be found in refer-ence 6.ANALYSISThe dynamics of the collision process allow us to probethe nature of the intermediate velocity “nucleon-nucleon”emission source [7–10]. Measurement of emission crosssections of nucleons and light clusters together with suit-able application of a coalescence ansatz [7] provides themeans to probe the properties and evolution of the inter-action region. The techniques used have been detailedin several previous publications [4, 5, 8–11] and are de-scribed briefly below. A notable difference from refer-ences 4 and 5 is the method of density extraction. Thisis discussed more extensively in the following. We em-phasize that the event selection is on the more violentcollisions. Cross section weighting favors mid-range im-pact parameters.An initial estimation of emission multiplicities at eachstage of the reaction was made by fitting the observedlight particle spectra assuming contributions from threesources, a projectile-like fragment (PLF) source, an inter-mediate velocity (IV) source, and a target-like fragment(TLF) source. A reasonable reproduction of the observedspectra is achieved. Except for the most forward detec-tor rings the data are dominated by particles associatedwith the IV and TLF sources. The IV source velocitiesare very close to 50% of the beam velocity as seen inmany other studies ([7–10] and references therein). Theobserved spectral slopes reflect the evolution dynamics ofthe source [10, 12, 13]. For further analysis, this IV sourceis most easily sampled at the intermediate angles wherecontributions from the other sources are minimized. Forthe analysis of the evolution of the source we have se-lected the data in ring 9 of the NIMROD detector. Thisring covered an angular range in the laboratory of 38◦ to52◦. The results of the three-source fit analyses, as wellas inspection of invariant velocity plots constructed foreach ejectile and each system, indicate that this selectionof angular range minimizes contributions from secondaryevaporative decay of projectile like or target like sources[11].We treat the IV source as a nascent fireball createdin the participant interaction zone. The expansion andcooling of this zone leads to a correlated evolution ofdensity and temperature which we probe using particleand cluster observables, yield, energy and angle. As inthe previous work [4, 5] we have employed double iso-tope yield ratios [14, 15] to characterize the temperatureat a particular emission time. Model studies comparingAlbergo model temperatures and densities to the knowninput values have shown the double isotope ratio temper-atures to be relatively robust in this density range [16].However the densities extracted using the Albergo modelare useful only at the very lowest densities [16]. Both ofthese results are confirmed in the more extensive calcu-lations of reference 1. In this study we have employed adifferent means of density extraction, the thermal coales-cence model of Mekjian [8, 10].To determine the coalescence parameter P0, the radiusin momentum space, from our data we have followed theCoulomb corrected coalescence model formalism of Aweset al. [17] and previously employed by us in reference [7].In the laboratory frame the derived relationship betweenthe observed cluster and proton differential cross sectionsisd2N(Z,N,EA)dEAdΩ= RNnpA−1N !Z!(43piP30[2m3(E−EC)]12)A−1×(d2N(1,0,E)dEdΩ)A(1)where the double differential multiplicity for a cluster ofmass number A containing Z protons and N neutronsand having a Coulomb-corrected energy EA, is relatedto the proton double differential multiplicity at the sameCoulomb corrected energy per nucleon, E − EC , whereEC is the Coulomb barrier for proton emission. Rnp isthe neutron to proton ratio. Since within the frameworkof the coalescence model the yield ratios of two isotopeswhich differ by one neutron are determined by their bind-ing energies and the n/p ratio in the coalescence volume,we have used the observed triton to 3He yield ratio toderive the n/p ratio used in this analysis.In the Mekjian model thermal and chemical equilib-rium determines coalescence yields of all species. Underthese assumptions there is a direct relationship betweenthe derived radius in momentum space and the volumeof the emitting system. In terms of the P0 derived fromEq. (1) and assuming a spherical sourceV =((Z!N !A32A)(2s+ 1)eE0T) 1(A−1) 3h34piP 30(2)where h is Plancks constant and Z, N , and A are thesame as in Eq. (1), E0 is the binding energy and s thespin of the emitted cluster and T is the temperature.Thus the volume can be derived from the observed P0and temperature values assuming a spherical shape interms of the P0 derived from Eq. (1).Because our goal was to derive information on the den-sity and temperature evolution of the emitting system,our analysis was not limited to determining an averageP0 value. Instead, as in our previous studies [4, 5, 8],results for d, t, 3He, and 4He, were derived as a functionof vsurf, the velocity of the emerging particle at the nu-clear surface, prior to Coulomb acceleration [17]. From3the relevant P0 values we then determined volumes usingEq. (2). A comparison of these volumes indicated goodagreement for t, 3He and 4He. The volumes derived fromthe deuteron data are typically somewhat smaller. Thisappears to reflect the fragility of the deuteron and itssurvival probability once formed [18]. For this reason wehave used average volumes derived from the A=3 and 4clusters to calculate the densities. Given that mass isremoved from the system during the evolution, we deter-mined the relevant masses for each volume by assumingthat the initial mass of the source was that determinedfrom the source fitting analysis and then determining themass remaining at a given vsurf from the observed energyspectra. This is also an averaging process and ignoresfluctuations. Densities were determined by dividing re-maining masses by volumes.RESULTSTemperatures and DensitiesInspection of the results for the four different sys-tems studied revealed that the temperatures, densitiesfor all systems are the same within statistical uncertain-ties. Therefore we have combined them to determine thevalues reported in this paper.We present, in Figure 1 the experimentally deriveddensity and temperature evolution of the IV source. Es-timated errors on the temperatures are 10% below ρ =0.01 fm−3increasing to 15% at ρ = 0.03 fm−3. Estimatederrors on the densities are 20%.In a recently submitted paper we reported equilibriumconstants for α cluster formation as a function of temper-ature and density [19]. These equilibrium constants werethen compared with those predicted by several differentastrophysical equation of state models. Specifically wedefined the equilibrium constants, Kc, for cluster forma-tion in terms of density asKc(A,Z) = ρ(A,Z)/[(ρp)Z(ρn)N ] (3)where ρ(A,Z) is the density of clusters of a specific massnumber A and atomic number Z, N is the neutron num-ber in the cluster and ρp and ρn are, respectively, thedensities of free protons and neutrons. In the presentwork, we employ the observed temperature and densitydependence of these equilibrium constants to extract thein medium modifications of the cluster binding energiesand determine Mott points.3, nuc/fmρ0 0.01 0.02 0.03 0.04T, MeV51015FIG. 1: Temperatures and densities sampled by the expand-ing IV source.DERIVATION OF IN MEDIUM BINDINGENERGIESOur departure point for extraction of the medium mod-ified cluster binding energies is the chemical equilibriumexpression relating the density of a cluster of mass num-ber A and atomic number Z to the densities of neutronsand protons in the same volume V [12].ρ(A,Z) =N(A,Z)V=A32 λ3(A−1)T ω(A,Z)(2sp+1)Z(2sn+1)A−Z×ρZp ρA−Zn expB(A,Z)T(4)In this expression, λT =h(2pim0T )1/2is the thermalwavelength of a nucleon, sp and sn are the proton andneutron spins, T is the temperature and B(A,Z) is thecluster binding energy. The term ω(A,Z) is the internalpartition function of the cluster, taken here to 1 for theZ = 1 and Z = 2 clusters considered.Minich et al. [20, 21] used a related yield expressionto analyze intermediate mass fragment yields in multi-fragmentation experiments but added both a mixing en-tropy term and a surface entropy term, as initially pro-posed by Fisher [22]. This latter term leads to a powerlaw behavior of the mass distribution at the critical pointand its ramifications have been widely explored [23–25].Neither of these entropy contributions is explicitly in-cluded in the Albergo formulation [12]. In the presentanalysis of the experimental data we include a mixing4entropy term in the free energy. This term has the form∆F = T (Z ln(Z/A) +N ln(N/A)) (5)where once again Z, N and A are those of the clusterbeing formed [20]. As mixing is a spontaneous process thefree energy of mixing is negative and therefore favors thecluster formation. We do not include a Fisher term. Ourreasoning for this is that, without additional corrections,the Fisher term as normally formulated and applied tolarger clusters whose properties are very similar to thebulk properties, is not applicable to the yields of the verysmall clusters, A ≤ 4, which we are treating. We base thisconclusion on the results of molecular dynamics studies ofthe cluster size dependence of the surface energy [26–28]and the binding energies per nucleon of the competingZ = 1, 2 species being significantly different from thebulk.Thus, rearranging Eq. (4) , substituting Kc from Eq.(3) and taking the logarithm of each side we can write ageneral expression for each cluster,ln[Kc/C(T )] = B/T − Z ln(Z/A)−N ln(N/A) (6)where C(T ) includes all terms on the right hand side ofEq. (4) except the exponential term. Using the experi-mentally determined equilibrium constants and tempera-tures we then solve this expression to obtain the apparentbinding energies, B(ρ, T ), of the clusters for the differenttemperatures and densities sampled in the experiments.The binding energies extracted for d, t, 3He and α clus-ters decrease monotonically with increasing density asshown in Fig. 2.MOTT POINTSBy definition, a Mott point corresponds to a combi-nation of density and temperature at which a clusterbinding energy, B(ρ, T ), is zero with respect to the sur-rounding medium. Since the observed temperatures anddensities are correlated in our experiment (see Figure 1)each point in Figure 2 at which the experimentally de-rived binding energy is zero corresponds to a particularcombination of density and temperature. Thus, with thepresent data, we are able to extract a single Mott pointfor each cluster. In Figure 3 we present the values ofthe Mott temperatures and densities and compare themwith the loci of the values of medium modified bindingenergies predicted by Typel et al. [1] using the thermody-namic Green function method. Such a Mott line was alsocalculated in reference [29] while the contribution of cor-relations was considered without discriminating amongdifferent clusters. This approach makes explicit use ofan effective nucleon-nucleon interaction to account formedium effects on the cluster properties [30]. We seethat the agreement between the predictions and the ex-perimental results is quite good.3, nuc/fmρ0 0.005 0.01 0.015,T), MeVρB(-505101520253035B(T) dB(T) tHe3B(T)  He4B(T)  FIG. 2: In medium binding energies derived from the exper-iments as a function of density. T and ρ are changing in acorrelated fashion. (See text.)3, nuc/fmρ0 0.005 0.01 0.015 0.02T, MeV051015Expt dExpt tHe3Expt  He4Expt  QS dQS tHe3QS  He4QS  FIG. 3: Comparison of experimentally derived Mott pointdensities and temperatures with theoretical values. Symbolsrepresent the experimental data. Estimated errors on thetemperatures are 10% and on the densities 20%. Lines showpolynomial fits to the Mott points presented in reference [1].5SUMMARY AND CONCLUSIONSWe have presented a first experimental determinationof in medium cluster binding energies and Mott points ford, t, 3He and α clusters produced in low density nuclearmatter. Our results are in good agreement with thosepredicted by a recent model which explicitly treats thesequantities. Inclusion of the in-medium effects in astro-physical equations of state should improve the utility ofthose for modeling astrophysically interesting events.ACKNOWLEDGEMENTSThis work was supported by the United States Depart-ment of Energy under Grant # DE-FG03- 93ER40773and by The Robert A. Welch Foundation under Grant #A0330.[1] S. Typel, G. Ro¨pke, T. Kla¨hn, D. Blaschke and H. H.Wolter, Phys. Rev. C 81, 015803 (2010).[2] G. Ro¨pke, et al., Nucl. Phys. A379, 536 (1982).[3] M. Schmidt et al., Ann. Phys. 202, 57 (1990).[4] S. Kowalski et al., Phys. Rev. C 75, 014601 (2007).[5] J. B. Natowitz et al. Phys. Rev. Lett. 104, 202501(2010).[6] S. Wuenschel et al., Nucl. Instr. Meth. Phys. Res. A 604,578 (2009).[7] A. Z. Mekjian, Phys. Rev. C 17, 1051 (1978); Phys. Rev.Lett. 38, 640 (1977).[8] K. Hagel et al., Phys. Rev. C 62, 034607 (2000).[9] R. Wada et al., Phys. Rev. C 39, 497 (1989).[10] R. Wada et al., Phys.Rev. C 69, 044610 (2004).[11] L. Qin, Thesis, Texas A&M University (2008).[12] S. Albergo et al., Nuovo Cimento A89, 1 (1985).[13] A. Kolomiets, V. M. Kolomietz, and S. Shlomo, Phys.Rev. C 55, 1376 (1997).[14] W. Bauer, Phys. Rev. C 51, 803 (1995).[15] H. Zheng et al. Phys. Lett. B696,178 (2011).[16] S. Shlomo et al., Phys. Rev. C 79, 034604 (2009).[17] T. C. Awes et al., Phys. Rev. C 24, 89(1981).[18] I. Cervesato, et al., Phys. Rev. C 45, 2369 (1992).[19] L. Qin et al., arxiv 1110.3345, submitted to Phys. Rev.lett., October, 2011.[20] R. W. Minich, et al., Phys. Lett. B 118, 458 (1982).[21] A. S. Hirsch et al., Nucl. Phys. A418, 267 (1984).[22] M. E. Fisher, Physics, 3, 225 (1967); Report on Prog.Phys. 30, 615 (1967).[23] J. B. Elliott et al., Phys. Rev. C 67, 024609 (2003).[24] J. B. Elliott et al., Phys. Rev. Lett. 85, 1194 (2000).[25] M. Huang et al., Phys. Rev. C 82, 054602 (2010).[26] B. Hale, Aust. J. Phys. 49, 425 (1996).[27] J. Kiefer and B. N. Hale, AIP Conference Proceedings,534, 260 (2000).[28] M. W. Curtin et al., Phys. Lett. B123, 289 (1983).[29] H. Stein et al., Z. Phys. A351, 295 (1995).[30] G. Ro¨pke, et al., Nucl. Phys. A399, 587 (1983).

Experimental Determination of In-Medium Cluster Binding Energies and
  Mott Points in Nuclear Matter

Experimental Determination of In-Medium Cluster Binding Energies and Mott Points in Nuclear Matter

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