#  The Isospin Dependence Of The Nuclear Equation Of State Near The Critical
Point

M. Huang  Cyclotron Institute, Texas A$\&$M University, College Station, Texas
77843  Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou,
730000,China.  Graduate University of Chinese Academy of Sciences, Beijing,
100049, China.  A. Bonasera<EMAIL_ADDRESS>Cyclotron Institute, Texas
A$\&$M University, College Station, Texas 77843  Laboratori Nazionali del Sud,
INFN,via Santa Sofia, 62, 95123 Catania, Italy  Z. Chen  Cyclotron Institute,
Texas A$\&$M University, College Station, Texas 77843  Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou, 730000,China.  R. Wada
<EMAIL_ADDRESS>Cyclotron Institute, Texas A$\&$M University, College
Station, Texas 77843  K. Hagel  Cyclotron Institute, Texas A$\&$M University,
College Station, Texas 77843  J.B. Natowitz  Cyclotron Institute, Texas A$\&$M
University, College Station, Texas 77843  P.K. Sahu  Cyclotron Institute,
Texas A$\&$M University, College Station, Texas 77843  L. Qin  Cyclotron
Institute, Texas A$\&$M University, College Station, Texas 77843  T. Keutgen
FNRS and IPN, Université Catholique de Louvain, B-1348 Louvain-Neuve, Belgium.
S. Kowalski  Institute of Physics, Silesia University, Katowice, Poland.  T.
Materna  Cyclotron Institute, Texas A$\&$M University, College Station, Texas
77843  J. Wang  Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou, 730000,China.  M.Barbui  Cyclotron Institute, Texas A$\&$M
University, College Station, Texas 77843  C.Bottosso  Cyclotron Institute,
Texas A$\&$M University, College Station, Texas 77843  M.R.D.Rodrigues
Cyclotron Institute, Texas A$\&$M University, College Station, Texas 77843

###### Abstract

We discuss experimental evidence for a nuclear phase transition driven by the
different concentration of neutrons to protons. Different ratios of the
neutron to proton concentrations lead to different critical points for the
phase transition. This is analogous to the phase transitions occurring in
4He-3He liquid mixtures. We present experimental results which reveal the N/A
(or Z/A) dependence of the phase transition and discuss possible implications
of these observations in terms of the Landau Free Energy description of
critical phenomena.

Intermediate Heavy ion reactions, phase transition, Landau approach, symmetry
energy

###### pacs:

25.70Pq,21.65.Ef,24.10.-i

Nuclei are quantum Fermi systems which exhibit ma-
ny interesting features which depend on temperature and density. At zero
temperature and ground state density, nuclei are charged quantum drops, i.e.
they have a Fermi motion pres  due to their quantum nature, and nucleons
interact through a short range attractive force and the long range Coulomb
repulsions among the constituent protons. In the absence of the Coulomb force,
the Nuclear Hamiltonian is perfectly symmetric for exchange of protons and
neutrons apart from a small but NOT insignificant difference between the
proton and neutron masses. This symmetry is revealed by similar energy levels
in mirror nuclei, i.e. nuclei with the same mass number A but opposite numbers
of neutrons, N, and protons, Z. Of course this feature is observed for
relatively small systems because the Coulomb energy is small pres . Analogous
to the properties of mirror nuclei, we could expect that if we study nuclei at
finite temperatures, T, and low densities, $\rho$, then, if the Coulomb force
is not important, the invariance under exchange of protons to neutrons might
lead to important and interesting consequences. In fact, since the fundamental
Hamiltonian of nuclei is invariant under exchange of N with Z (apart from
Coulomb effects), we could expect that such an invariance should be manifested
only at high T (disordered state), while there is a spontaneous symmetry
breaking at lower T (ordered state). That means that in symmetric nuclear
matter at high T, the state with fragments having $N=Z$ defines the minimum of
the free energy, i.e. symmetric fragments such as deuterons and alphas would
be favored at low density huang ; land . On the other hand, there could be a
symmetry breaking favoring $N\neq Z$ at lower T. In this case fragments near a
(first order) phase transition might prefer either a neutron or proton rich
configuration. There might even be a more interesting situation, suggested by
the present data, the existence of a line of first order phase transitions
huang  which terminates in a tri-critical point. For such a line the free
energy has three equal minima: one with N=Z and the other two for $N\neq Z$.
Thus a phase transition is driven by the difference in isospin concentration
of the fragments $m=(N-Z)/A$. In this paper we will discuss data which clearly
demonstrate that m is an order parameter of the phase transition. Its
conjugate field huang  which we indicate with H, is due to the chemical
potential difference between protons and neutrons of the emitting source at
the density and temperature reached during a collision between heavy ions prl
; huang10_1 . We also note that the phase transition has a strong resemblance
to that observed in superfluid mixtures of liquid 4He-3He near the $\lambda$
point. In both systems, changing the concentration of one of the components of
the mixture, changes the characteristics of the Equation Of State (EOS) huang
; land .

In recent times a large body of experimental evidence has been interpreted as
demonstrating the occurrence of a phase transition in finite nuclei at
temperatures (T) of the order of 6 MeV and at densities, $\rho$, less than
half of the normal ground state nuclear density wci . Even though strong
signals for a first and a second-order phase transition have been found wci ;
bon00 , there remain a number of open questions regarding the Equation of
State of nuclear matter near the critical point. In particular the roles of
Coulomb, symmetry, pairing and shell effects have yet to be clearly
delineated.

Theoretical modeling indicates that a nucleus excited in a collision expands
nearly adiabatically until it is close to the instability region thus the
expansion is isentropic siem . At the last stage of the expansion the role of
the Coulomb force becomes very important. In fact, without the Coulomb force,
the system would require a much larger initial compression and/or temperature
in order to enter the instability region and fragment. The Coulomb force acts
as an external piston, giving the system an ‘extra push’ to finally fragment.
These features are clearly seen in Classical Molecular Dynamics (CMD)
simulations of expanding drops with and without a Coulomb field belk95 ; dorso
. The expansion with the Coulomb force included is very slow in the later
stage and nearly isothermal.

Even though at high T and small $\rho$ the nucleus behaves as a classical
fluid, the analogy to classical systems should not be overemphasized as, in
the (T,$\rho$) region of interest, the nucleus is still a strongly interacting
quantum system. In particular the ratio of T to the Fermi energy at the
(presumed) critical point is still smaller than 1 which suggests that the EOS
of a nuclear system is quite different from the classical one. To date this
expected difference has not been well explored mor ; wci ; dago ; mas ; nat ;
ala ; maria .

The paper is organized as follows: in the next section we discuss the
experimental setup in detail. This is followed by a description of the data
analysis and a discussion in terms of the Landau O($m^{6}$) free energy. We,
then derive some critical exponents and the EOS corresponding to possible
scenarios suggested by our data in terms of the Fisher model of fragmentation.
Finally we draw some conclusions and suggest possible future work.

##  I Experimental Details

The experiment was performed at the K-500 superconducting cyclotron facility
at Texas $A\&M$ University. 64,70Zn and 64Ni beams were incident on 58,64Ni,
112,124Sn, 197Au and 232Th targets at 40 A MeV. Intermediate mass fragments
(IMF) were detected by a detector telescope placed at 20o. The telescope
consisted of four Si detectors. Each Si detector had 5cm $\times$ 5cm area.
The thicknesses were 129, 300, 1000 and 1000 $\mu$m. All Si detectors were
segmented into four sections and each quadrant had a 5o acceptance in polar
and azimuthal angles. The fragments were detected at average angles of 17.5o
$\pm$ 2.5o and 22.5o $\pm$ 2.5o. Typically 6-8 isotopes were clearly
identified for a given Z up to Z=18 with an energy threshold of 4-10 A MeV,
using the $\Delta$E-E technique for any two consecutive detectors. The
$\Delta$E-E spectrum was linearized by an empirical code based on a range-
energy table. In the code, isotopes are identified by a parameter $Z_{Real}$.
For the isotopes with A=2Z, $Z_{Real}$ = Z is assigned and other isotopes are
identified by interpolating between them. The energy spectrum of each isotope
was extracted by gating on lines corresponding to the individual identified
isotopes. In order to compensate for the imperfectness of the linearization,
actual gates for isotopes were made on the 2D plot of $Z_{Real}$ versus
energy. The multiplicity of each isotope was evaluated from the extracted
energy spectra, using a moving source fit at the two given angles. Since the
energy spectra of some isotopes have very low statistics, the following
procedure was adopted for the fits. Using a single source with a smeared
source velocity around half of the beam velocity, the fit parameters were
first determined from the energy spectrum summed over all isotopes for a given
Z, assuming A=2Z. Then assuming that the shape of the velocity spectrum is the
same for all isotopes for a given Z. All parameters except the normalizing
multiplicity parameter were assumed to be the same as for the summed spectrum.
The multiplicity for a given isotope was then derived by normalizing the
standard spectrum to the observed spectrum for that isotope.

In order to evaluate the back ground contribution to the extracted
multiplicity a two Gaussian fit to each isotope peak was used with a linear
background. The second Gaussian (about 10$\%$ of the height of the first one)
was added to reproduce the valleys between isotopes. This component was
attributed to the reactions of the isotope in the Si detector. The centroid of
the main Gaussian was set to the value calculated from the range-energy table
within a small margin. The final multiplicity of an isotope with Z $>$ 2 was
obtained by correction of the multiplicity evaluated form the moving source
fit for the ratio between the sum of the two Gaussian yields and the linear
background.

The yields of light charged particles (Z $\leq$ 2) in coincidence with IMFs
were also measured using 16 single crystal CsI(Tl) detectors of 3cm thickness
set around the target. The light output from each detector was read by a photo
multiplier tube. The pulse shape discrimination method was used to identify p,
d, t, h and $\alpha$ particles. The energy calibration for these particles
were performed using Si detectors (50 -300 $\mu$m) in front of the CsI
detectors in separate runs. The yield of each isotope was evaluated, using a
moving source fit. Three sources (projectile-like(PLF), nucleon-nucleon-
like(NN) and target-like (TLF)) were used. The NN-like sources have source
velocities of about a half of the beam velocity. The parameters were searched
globally for all 16 angles. Detailed procedures of the data analysis are also
given in refs. huang10_2 ; chen10 .

Special care has been taken for 8He identification. All He isotopes are
identified in the Si telescope, using the $\Delta$E-E technique, in a narrow
energy range. When a proton and an $\alpha$ hit the same quadrant and when
both of them stop in the E detector, their $\Delta$E-E points overlap with
those of 8He. Since the multiplicities of protons and alphas are about three
orders of magnitude larger than that of 8He, the contribution of accidental
events becomes significant, especially for the reaction systems with lower
numbers of neutrons, in which 8He production is suppressed. Since Z=1
$\Delta$E-E spectra are not available in this experiment, $\Delta$E-E for Z
$\leq 3$ were measured in a separate run. Using the light charged particle
multiplicity extracted from the 16 CsI detectors, the accidental events were
simulated for each reaction for the observed $\alpha$ yield in the $\Delta$E-E
spectra in this experiment, the solid angle of the quadrant and the
multiplicity of Z=1 particles. In order to minimize the accidental events, the
runs with a low beam intensity were selected in each reaction. Typical
linearized $Z_{Real}$ spectra with these accidentals are shown in Fig.(1) for
70Zn + 232Th (N/Z $\sim$ 1.5) and 64Ni + 112Sn ( N/Z $\sim$ 1.25). As one can
see, the $Z_{real}$ values for the accidental events of proton and $\alpha$
pileup is nearly identical to that of 8He, while 6He is clearly identified.
The contributions from d + $\alpha$ and t + $\alpha$ are also reasonably
consistent with the observed background yields. A significant excess of 8He
yield beyond the accidentals is only observed for the reaction systems with
the 124Sn, 197Au and 232Th targets. After the correction of the accidental
contributions, the multiplicities of 6He and 8He were calculated using the
source fit parameters obtained for Li isotopes.

Figure 1: Typical $Z_{Real}$ spectra for He and Li isotopes. Accidental events
are generated only for p, d, t +$\alpha$ and shown separately by shaded
histograms as indicated.

##  II Data Analysis

The key factor in our analysis is the value $I=N-Z$ of the detected fragments.
A plot of the yield versus mass number when I=0 displays a power law behavior
with yields decreasing as $A^{-\tau}$ min ; prl . This is shown in Fig.(2) for
the 64Ni+124Sn case at 40 MeV/nucleon. In the figure we have made separate
fits for odd-odd (open symbols) or even-even(filled symbols) nuclei. As seen,
different exponents $\tau$ appear which suggests that pairing is playing a
role in the dynamics pres , leading to higher yields for even-even nuclei.

Figure 2: Mass distribution for the 64Ni+124Sn system at 40 MeV/nucleon for
I=0. The lines are power law fits with exponents $2.3\pm 0.02$ (odd-odd
nuclei, dashed line), and $3.4\pm 0.06$ (even-even nuclei, full line)
respectively.

The observation of the power law behavior suggests that the mass distributions
may be discussed in terms of a modified Fisher model bon00 ; min :

$Y=y_{0}A^{-\tau}e^{-\beta\Delta\mu A},$ (1)

where y0 is a normalization constant, $\tau=2.3$ is a critical exponent bon00
, $\beta$ is the inverse temperature and $\Delta\mu=F(I/A)$ is the free energy
per particle, $F$, near the critical point. Recall that in general, the free
energy is a function of the mass A (volume), $A^{2/3}$(surface) and the
chemical composition $m$ of the fragments and possibly pairing. The region we
are studying in this paper seems near the critical point for a liquid gas
phase transition (volume and surface equal to zero) but modified by
$m=\frac{I}{A}$. Because of this modification we can observe different
features of the transition such as a first order phase transition driven by
$m$, the order parameter.

We begin our analysis by noting that the Fisher free energy is usually written
in terms of the volume and the surface of a drop undergoing a (second order)
phase transition fisher . Our data indicate that those terms are not important
in the present case bon00  as we will show more in detail below. If they are
negligible this suggests that we are near the critical point for a liquid gas
phase transition. Because we have two different interacting fluids, neutrons
and protons, the transition becomes more complex and more interesting than in
a single component liquid. Experiments at different energies might display a
free energy which depends on all these factors. If we accept that $F$ is
dominated by the symmetry energy we can make the approximation that
$F(I/A)=E_{sym}=25(I/A)^{2}$ MeV/A, i.e. the symmetry energy of a nucleus in
its ground state pres . We will use this relationship in order to infer an
approximate value of the temperature of the system. However, we stress that in
actuality, F(I/A) is a function of density, temperature and all other relevant
quantities near the critical point. According to the Fisher equation given
above, we can compare all systems on the same basis by normalizing the yields
and factoring out the power law term. For this purpose we have chosen to
normalize the yield data for each system to the 12C yield ($I=0$) in that
system, i.e. we define a ratio:

$R=\frac{YA^{\tau}}{Y(^{12}C)12^{\tau}}.$ (2)

The normalized ratios for the system 64Ni + 64Ni at 40 MeV/nucleon are plotted
as a function of the (ground state) symmetry energy in Fig.(3), bottom panel.
The data display an exponential decrease with increasing symmetry energy,
except for the isotopes for which $I=0$. The yields of these I = 0 isotopes
are of course not sensitive to the symmetry energy but rather to the Coulomb
and pairing energies and possibly to shell effects. A fit to the exponentially
decreasing portion of the data using the ground state symmetry energy gives an
‘apparent temperature’ T of 6.0 MeV. This value of T would be the real one if
only the symmetry energy is important, if entropy can be neglected, if
$a_{sym}=25$ MeV (the g.s. symmetry energy coefficient value) and if secondary
decay effects are negligible. In general we expect that the symmetry energy
coefficient is density and temperature dependent. Further, secondary decay
processes may modify the primary fragment distributions huang10_2 ; chen10 .
we will discuss these questions in the framework of the Landau free energy
approach below. We stress that the appearance of two branches in Fig.3
(bottom), indicates that the total free energy must contain an odd power term
in (I/A) at variance with the common expression for the ground state symmetry
energy. For reference in the top part of Fig.(3) we have plotted the ratio
versus the total ground state binding energy of the fragments. No clear
correlations are observed which might suggest that the symmetry energy
dominates the process.

Figure 3:  Ratio versus fragments ground state binding energy (top panel) and
symmetry energy (bottom panel) for the 64Ni + 64Ni case at 40 MeV/nucleon.
$a_{s}ym$=25 MeV is used. The $I<0$ and $I>0$ ($I=0$)isotopes are indicated by
the open and full circles respectively (full squares). The dashed lines
(bottom panel) are fits using a ground state symmetry energy, Eq.1, and a
‘temperature’ of 6 MeV. Notice that the given experimental 8He yield is the
upper limit.

It is surprising that such a scaling appears as a function of the symmetry
energy only. In fact we might wonder about the role of the Coulomb energy if
we accept that surface and volume terms give negligible contribution. In
figure 4 we have plotted the same normalized ratios as a function of the
quantity $\alpha E_{coul}+\beta E_{sym}$, $\alpha$ and $\beta$ are arbitrary
parameters given in the figure and $E_{coul}=0.7Z(Z-1)A^{-1/3}$ is the Coulomb
contribution to the ground state energy of the nucleus. We see from the figure
that by decreasing the relative contribution of the Coulomb energy compared to
the symmetry energy the scaling appears. This implies that the Coulomb energy
is much less important than the symmetry energy near the critical point, which
suggests that either the density dependence of those two terms is different or
that, at the time of formation the fragments are strongly deformed, reducing
the Coulomb effect. Such deformations have been seen in CMD calculations of
fragmentation bon00 .

Figure 4:  Ratio versus symmetry energy + Coulomb energy for the 64Ni + 124Sn
case at 40 MeV/nucleon. The panels from top to bottom are for different
combinations of the symmetry and Coulomb energy. The $I<0$ and $I>0$
($I=0$)isotopes are indicated by the open and full circles respectively (full
squares).

To further explore the role of the relative nucleon concentrations we plot in
Fig.(5) the quantity $\frac{F}{T}=-\frac{ln(R)}{A}$ versus $m=(I/A)$, the
difference in neutron and proton concentration of the fragment. As expected
the normalized yield ratios depend strongly on m.

Pursuing the question of phase transition we can perform a fit to these data
within the generalized Landau free energy description huang . In this approach
the ratio of the free energy to the temperature is written in terms of an
expansion:

$\frac{F}{T}=\frac{1}{2}am^{2}+\frac{1}{4}bm^{4}+\frac{1}{6}cm^{6}-m\frac{H}{T},$
(3)

where $m$ is an order parameter, $H$ is its conjugate variable and $a-c$ are
fitting parameters huang . We observe that the Free energy is even in the
exchange of $m\rightarrow-m$, reflecting the invariance of the nuclear forces
when exchanging N and Z. This symmetry is violated by the conjugate field $H$
which arises when the source is asymmetric in chemical composition. We stress
that m and H are related to each other through the relation $m=-\frac{\delta
F}{\delta H}$.

Figure 5:  Free energy versus m for the case 64Ni+232Th. The full line is a
free fit based on Landau O($m^{6}$) free energy. The dashed-dotted-dotted-
dotted line is obtained imposing in the fit $b=-\sqrt{16/3ac}$ and it is
located on a line of first order phase transitions. The short dashed line
corresponds to $b=-\sqrt{4ac}$, i.e. superheating. The $O(m^{2})$ case,
$F/T=a(m-m_{s})^{2},$i.e. $b=c=0$, $m_{s}=0.1$, is given by the long dashed
line.

The use of the Landau approach is for guidance only. While the approximation
to $O(m^{4})$ does not work prl , the $O(m^{6})$ case is in good agreement
with the data. This is not surprising since, if fluctuations are important, a
higher order approximation to the free energy is better, i.e. gives critical
exponents closer to those seen in the data and satisfies the Ginzburg
criterion huang . A free fit using Eq.3 is displayed in Fig.5 (full line).
Notice the change of curvature near $m=0.3$, which incidentally is close to
$m_{cn}$ of the compound nucleus. For comparison in the same figure we have
displayed the $O(m^{2})$ case, i.e., $F/T=a(m-m_{s})^{2}$ ($b=c=0$)
111$F/T=a(m-m_{s})^{2}=(a/2)m^{2}-H/Tm+(a/2)m_{s}^{2};H/T=am_{(}s)$. The last
term is dropped out when the yields are normalized by 12C. As seen in the plot
last assumption also produces a reasonable fit, although it does not reproduce
shoulders near m $\sim\pm$0.3. As we will discuss in more detail below the
appearance of two minima for $m\neq 0$ (when $H/T=0$) might be a signature for
the existence of a first order phase transition occurring in these reactions.

In general the coefficients entering the Landau free energy Eq.(3), depend on
temperature, pressure or density of the source. Usually one assumes $c>0$,
$a=a_{0}(\rho)(T-T_{0})$ and $b=b(T,\rho)$, where $T_{0}$ is some ‘critical’
temperature discussed below. The precise determination of these parameters
determines the nuclear equation of state (NEOS) near the critical point. The
data we have do not allow such a complete constraining of the NEOS but do
suggest some interesting possible scenarios which we discuss below.

We begin by noting that the conjugate variable H which appears in equation (3)
is determined by the chemical composition of the source. Since, in general,
the source has $N\neq Z$, the extreme of $F/T$ are displaced from the values
obtained when H=0. In fact if we take the first derivative of the free energy
we get:

$(\frac{F}{T})^{\prime}=am+bm^{3}+cm^{5}-\frac{H}{T}.$ (4)

When $H/T=0$ the first derivative is zero for the following values of m huang
:

$m_{0}=0;m^{2}_{\pm}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2c}.$ (5)

If we now assume $H\neq 0$ but small, we can expand the solutions above as
$m=m_{0\pm}+\eta$ with $\eta$ small. Equating the first derivative to zero,
Eq.(4), and neglecting terms $O(\eta^{2})$ we get:

$\eta=\frac{H/T}{a+3bm_{0\pm}^{2}+5cm_{0\pm}^{4}}.$ (6)

The shift of the minimum from $m_{0}=0$ should be given by the equation above
and should be proportional to $m$ of the emitting source. We can easily check
this feature in our data. In Fig.(6) we plot the values of $H/T$ obtained from
the fits to our data for all systems using Eq.(3) versus $m_{cn}=(I/A)_{cn}$.

Figure 6:  H/T versus (I/A) of the compound nucleus obtained from the data fit
to the Landau free energy, Eq.(3). The full circles are for 64Ni, the full
triangles are for 70Zn and the squares for 64Zn projectiles impinging on
various targets, see text.

The linear fit in Fig.6 is given by $H/T=0.47+1.6(I/A)_{cn}$ which agrees with
the linear dependence of Eq.(6). However, for this fit $H/T\neq 0$ for
$I_{cn}=0$ which could indicate the favoring of $N>Z$ fragments by the Coulomb
field. Another possibility is that $(I/A)_{source}\propto(I/A){cn}$ which then
gives $H=0$ when $I_{source}=0$. Finally we should consider that together with
H also the temperature may also be changing some since the collisions are
between different target-projectile combinations at the same beam energy. If
the temperature were the same then the coefficients of the free energy, Eq.3,
should be independent of the source size, only H/T should change. In Fig.(7)
we plot the parameters a, b and c as a function of the compound nucleus
$m_{cn}$. As we see there is some dependence which may reflect differences in
temperature. However, we note that the error bars and fluctuations are large
which may also indicate important secondary decay effects. Thus is not too so
easy to draw definite conclusions.

Figure 7:  The parameters a,b and c versus (I/A) of the compound nucleus
obtained from the data fit to the Landau free energy, Eq.(3). The symbols are
like in figure 6.

Given the information on the parameters of the Landau free energy contained in
figures 6 and 7 we can discuss some features regarding the NEOS. In particular
for each reaction system we can estimate F/T when $H/T=0$. In figure 8 we plot
this quantity versus m of the fragments for various reactions.

Figure 8:  F/T (H/T=0) versus m of the fragments obtained from the a, b and c
parameters fit to the Landau free energy, Eq.(3). Results for all
experimentally investigated reactions are displayed.

The curves do not differ much suggesting that temperatures are quite similar.
The fits exhibit curvature near $m=\pm 0.4$ which may suggest the presence of
additional minima at larger absolute values of m. This could indicate either a
first order phase transition or superheating (see below). The lack of data at
very large $m$ makes it difficult to constrain the fit. However, we can study
other situations of particular physical interest which arise when the
relationships among the parameters a, b and c are constrained. huang ; land ;
prl :

We have considered four such cases as follows:

1) superheating. This case corresponds to $b=-\sqrt{4ac}$ and gives two minima
at $m\neq 0$ and is plotted in figure 5 for the 64Ni+232Th system with a short
dashed line. These are not absolute minima, which occur only at $m=0$, and
they correspond to metastable states. They might be observed in high quality
data for collisions of more neutron rich or proton rich systems making a hot
source with $m_{source}\approx\pm 0.4$. In fact if the system could be gently
brought to the right temperature $T_{s}$, with the correct isotopic
composition, it might stay in the minimum, i.e. more fragments of that $m$
should appear;
2) line of first order phase transition. This corresponds to the condition
$b=-\sqrt{16ac/3}$ at a temperature $T_{3}$, which, if imposed on the fit of
the free energy, results in the dashed-dotted-dotted-dotted line of Fig.5.
This fit is of similar quality to the previous cases. Now the minima are at
$m\approx 0.6$, i.e. for more neutron rich fragments due to the fact that
$H/T\neq 0$. This suggests that in this situation we might produce a large
number of neutron rich fragments. However, most of those fragments are
probably unstable, thus coincidence measurements may be required to determine
their yields. Of course this feature should become important in neutron rich
stars.
3) first order phase transition. Corresponds to the case $a=0$ and determines
the ‘critical temperature’ $T_{0}$ where the minimum at $m=0$ disappears and
only the ones at $m\neq 0$ survive. This case is excluded by our present data.
However, the fit in Fig.(5) suggests an intermediate situation between this
and case (2) above.
4) line of second order phase transition, tri-critical point. Corresponds to
$a=0$ and $b>0$ ($T=T_{c}$). When $b=0$ as well we have a tricritical point
($T=T_{3c}$), i.e. the point where the line of first order phase transition
terminates into a second order phase transition. This case is also excluded by
our data.

Figure 9:  F/T (H/T=0) versus m of the fragments obtained from the a, b and c
parameter fit to the Landau free energy, Eq.(3) for 64Ni+232Th. The four
curves correspond to vapor(full line), superheating(short dashed), line of
first order transition(3 critical dashed-dotted-dotted-dotted line), first
order phase transition(long dashed line), see text.

We can extrapolate the cases discussed above to $H/T=0$ as was done for Fig.8.
In Fig.9 we plot F/T (H/T=0) (extrapolated from the data) vs. $m$. Purists
will not call this the EOS but reserve that for the pressure vs. $m$ case
(that we discuss below). Since H/T is zero, the curves are symmetric with
respect to $m$. We see in the plot: vapor (dashed-dotted-dotted line)
$T>T_{s}$, superheating $(T=T_{s})$(dotted line), a point in the line of a
first order phase transition $(T=T_{3})$(dashed-dotted line) which displays
three equal minima at $m_{0}$ and $m_{\pm}$, see Eq.5, the experimental data
fit line(full line)$T<T_{3}$. We have also added the case $a=0$ which should
be obtained at $T=T_{0}$, where the minimum at $m=0$ becomes a maximum. There
is a series of cases not displayed in the figure, corresponding to the
temperatures between $T_{0}$ and $T_{3}$ where $m=0$ is still a minimum but
$\it{not}$ an absolute minimum. This corresponds to supercooling and might be
observed in gentle collisions of $N=Z$ nuclei similarly to the superheating
case.

The features in Fig.9 are reminiscent of the superfluid $\lambda$ transition
observed as some 3He is added to 4He huang . Pure 4He has a critical
temperature of 2.18 K. The critical temperature for the second-order
transitions decreases with increasing 3He concentration until at temperature,
$T=0.867$K, a first-order transition appears. This point is known as the tri-
critical point for this system. In a similar fashion, a nucleus, which can
undergo a liquid-gas phase transition, should be influenced by the different
neutron to proton concentrations. Thus the discontinuity observed in Fig.5
($m=0$) could be a signature for a tri-critical point as in the 4He-3He case.
We believe that our data, analyzed in terms of the the Landau O($m^{6}$) free
energy, suggest such a feature but are not sufficient to clearly demonstrate
this. Some other work campi ; gulm , also suggests that a line of critical
points might be found away from its ‘canonical’ position, i.e. at the end of a
first-order phase transition and, for small systems, even extending into the
coexistence region.

##  III Critical exponents

In the fits discussed above the parameters a, b and c were left free since we
do not have any particular values to fix the scale. Nevertheless, we saw in
figure 8 that the free energy (H/T=0) looks very similar for the different
systems. Thus the values of the fitting parameters are similar apart from a
scaling factor. We can avoid unnecessary factors by defining suitable
dimensionless quantities. This can be accomplished by looking at the solutions
of the minima of the free energy, cf. Eq.5. In particular from the value at
the minimum, $m_{+}$ we can define the following quantities ($b\neq 0$):

$x=\frac{4ac}{b^{2}}.$ (7)

Recalling that $a$ is related to the distance from the critical temperature
while b and c should only depend on density huang , we deduce that x is a
measure of the distance $T-T_{3}$ from the critical temperature in a suitable
dimensionless fashion. Similarly we can define a reduced order parameter from
Eq.5:

$y=\frac{2cm^{2}}{|b|}.$ (8)

Thus Eq.5 can be rewritten as:

$y=1+\sqrt{1-x}.$ (9)

Near the critical point we know that the order parameter has a singular part
that behaves in a power law fashion, thus we can define the singular part as:

$M=\pm\sqrt{y-1}=\pm{(1-x)}^{1/4},$ (10)

defining the temperature ‘distance’ from the critical point, $|t|=|1-x|$,
immediately gives the value of a critical exponent: $\beta=\frac{1}{4}$. This
exponent is very close to the accepted experimental value as is well known in
the O($m^{6}$) Landau theory huang .

Figure 10:  Order parameter versus reduced temperature for all studied
systems. The dashed line is given by Eq.(10), the vertical line indicates the
critical temperature $T_{3}$. To the right of this line the system is in a
superheated state. Supercooling occurs on the left of the vertical line and
M=0 huang .

In Fig.(10) the experimental values of M and x obtained within the Landau
theory are plotted together with the equilibrium condition given by Eq.(10).
Supercooling and Superheating regions, as discussed in the previous section,
can be identified as well huang .

As is the case for macroscopic systems we can now ‘turn ’ the external field H
on and off. In our case this is done with a suitable choice of the colliding
systems. In this way we can study the ‘EOS’ at the critical point by turning
on H:

$M=H^{1/\delta},$ (11)

which defines the critical exponent $\delta$. In the Landau theory this
exponent can be determined at the critical point where $a=b=0$. From equation
(4) we easily get $\delta=5$, which is the accepted value for such a critical
exponent huang . In order to exactly determine this exponent we need to bring
the system to the critical point. This does not appear to be the case for our
data as we saw in figure 10. Nevertheless a plot of the order parameter versus
H should display a power law behavior as it is well known in macroscopic
systems huang . A precise determination of the critical exponent requires the
knowledge of the temperature T both above and below the critical point. This
is feasible but requires precise experimental data. ¿From Eq.(6) assuming the
only minimum is at m=0 we get

$\eta=\frac{H/T}{a}.$ (12)

The temperature (a) dependence of the order parameter shows that we are away
from the critical point. Nevertheless we can study the behavior close to the
critical point by suitably defining scaling forms huang :
$\frac{M}{|t|^{\beta}}=\frac{\eta}{|t|^{\beta}}$ vs.
$\frac{H/T}{|t|^{\beta\delta}}$. These quantities are plotted in figure 11 and
compared to magnetization data for nickel metal. The scaled magnetization is
plotted versus the scaled external magnetic field huang . The nuclear data
have been shifted in the region near the crossing of data above and below the
critical temperature where we expect our data to be, see Fig.10. Of course it
is not possible at this stage to directly compare to the macroscopic data
since we have no information for the absolute values of the temperatures.
Furthermore the role of the density (or pressure) is not clear since we expect
that the parameter $a$ (or equivalently x) depends on the ‘distance’ from the
critical temperature and critical pressure. These quantities could however be
obtained in $4\pi$ experiments where charges, masses and their velocities are
carefully determined.

Figure 11:  Scaling form for magnetization M vs. external field for nickel
huang , open symbols. The corresponding quantities for nuclei normalized to
the metal case are given by the full symbols.

Once we have derived the ‘reduced’ parameters of the Landau $O(m^{6})$ theory,
we can write a ‘reduced’ free energy as ($b\neq 0$):

$\frac{f}{T}=\frac{1}{2}xm^{2}-|z|m^{4}+\frac{2}{3}z^{2}m^{6}-\frac{{\it
h}}{T}m,$ (13)

where: $\frac{f}{T}=\frac{4c}{b^{2}}\frac{F}{T}$, $z=\frac{c}{|b|}$,
$\frac{h}{T}=\frac{4c}{b^{2}}\frac{H}{T}$. These quantities together with the
temperature, Eq.(7), and the reduced order parameter y,of Eq.(8), constitute
the Landau $O(m^{6})$ theory in dimensionless form. It is instructive to study
how these quantities change with the reaction system as we did in figures (7)
and (8).

Figure 12:  The parameters x, z and h/T versus (I/A) of the compound nucleus
obtained from the data fit to the Landau free energy, Eq.(3). The symbols are
like in figure 6.

In Fig.(12) we plot these normalized quantities vs. difference in neutron
proton concentration of the compound nucleus. Compare to figure 7. A feature
worth noticing is the following, while the parameter $a$ is decreasing with
increasing (I/A) of the compound nucleus, the opposite holds for the parameter
$x$ which gives the ‘distance’ from the critical temperature, see Fig.(12).
This is very important since only normalized quantities should be used when
inferring the properties of the EOS (i.e. temperature, density etc.) near the
critical point.

##  IV Symmetry and Pairing compared to the Coulomb Energy.

In the previous sections we have seen that the Coulomb energy might become
important especially for large values of the charges. We can now try to derive
some qualitative understanding of when and why Coulomb corrections might
become important and might even hinder a possible phase transition. From the
mass formula we can write the Coulomb energy for large Z as pres :

$\frac{E_{c}}{A}=0.77\frac{Z^{2}}{A^{2}}A^{2/3}=\frac{0.77}{4}(1-m)^{2}A^{2/3},$
(14)

which explicitly introduces the order parameter $m$ in the Coulomb energy. We
can define an ‘effective’ symmetry energy (per particle) as:

$\frac{E_{eff}}{A}=(a_{sym}+\frac{0.77}{4}A^{2/3})m^{2}-\frac{0.77}{2}A^{2/3}m+\frac{0.77}{4}A^{2/3},$
(15)

where the symmetry energy coefficient $a_{sym}=25MeV$. Ignoring for a moment
density corrections we see that $O(m^{2})$ term should be affected by Coulomb
corrections for large fragment mass numbers. Furthermore, a linear term in $m$
is introduced which will then modify the ‘external’ field even in collisions
where the source $m_{s}=0$ as we discussed in Fig.(6). Finally there is a term
not dependent on $m$ that will destroy the scaling for large mass (charge)
numbers. We should also notice that assuming a spherical expansion, at low
densities the Coulomb energy will decrease as $\rho^{1/3}$ while contributions
to the symmetry energy should depend both on $\rho^{2/3}$ reflecting the Fermi
energy of the nuclei and on $\rho$, the latter coming from different n-p
interactions. At low densities we would expect Coulomb to be stronger than it
appears to be in the data. This may be indicative that the fragments must be
highly deformed, reducing the Coulomb energy. Coulomb corrections should
become more important when $m=0$ for the detected fragment. We have plotted
the yields of $m=0$ nuclei in Fig.(2) and pointed out that pairing appears to
be playing a role. From Eq.(15) above we should expect that, if Coulomb is
dominant for such fragments, the free energy should depend on $A^{2/3}$. In
Fig.(13) (top panel) we plot F/T versus $A$ for $m=0$ fragments. The expected
dependence with mass number in the free energy suggested from ‘effective’
symmetry energy Eq.(15) is not seen in the figure.

Figure 13:  Free energy versus mass for $m=0$ isotopes for the 70Zn+124Sn
system (top panel), the dashed line is a fit using Coulomb and pairing
contributions. Free energy times $\delta$ (see text-bottom panel) versus mass
for $m=0$ isotopes. The lines are separate fits suggested by the
Coulomb(dashed line) and pairing(full line) energy mass number dependence.

Rather, a staggering between odd-odd and even-even nuclei is clearly visible.

To better clarify these arguments we can write the pairing energy from the
mass formula as pres :

$\frac{E_{p}}{A}=12\frac{\delta}{A^{3/2}},$ (16)

where $12MeV$ is the ground state pairing energy coefficient and $\delta$:

$\delta=\frac{(-1)^{N}+(-1)^{Z}}{2}.$ (17)

The suggested mass dependence from pairing, Eq.16, is completely different
from the Coulomb one when $m=0$, see Eq.14.

Notice that it is the $\delta$ factor from pairing that changes the sign of
the contribution for odd-odd to even-even nuclei. A combined fit to the data
using Coulomb plus pairing contributions results in the dashed line in
Fig.(13), top panel. The agreement with data is very good. If we multiply the
pairing energy by the factor $\delta$ we should get no discontinuities when
plotting this quantity versus mass number. Similarly if the properties of the
free energy depend on the pairing term, as for the ground state case, then it
should be a monotonic function of $A$ after multiplying it by $\delta$. In
Fig.(13) (bottom panel), we plot the quantity $\frac{F}{T}\delta$ versus mass
number for the same system of Fig.(13) (top panel). The fit using the pairing
mass dependence is also good. The Coulomb mass dependence fails especially for
small mass number. From the values of the fit, using the ground state
coefficients we can derive a temperature for the Coulomb case of
$T=9.2(\frac{\rho}{\rho_{0}})^{1/3}MeV$ where we have explicitly indicated a
possible density correction. For the pairing case we get $T=6.45MeV$. Notice
that in this case we have not suggested any density correction since the fate
of the pairing energy at low density and finite temperature is ‘terra
incognita’. When making a combined fit using pairing and Coulomb energy we get
a good reproduction of the data (dashed line in Fig.(13),top panel). While the
fitting value for pairing results in a ‘temperature’ T=5.13 MeV, we get an
increase of the Coulomb contribution to T=12.1 MeV. Assuming that pairing is
independent on density, we could derive a density from the Coulomb result. A
simple calculation give $\frac{\rho}{\rho_{0}}=(6.45/9)^{3}=0.34$ which could
be a reasonable indication of the density of the system when it breaks into
fragments.

In summary in this section we have shown that the role of the Coulomb energy
appears to be rather reduced in the reactions analyzed in this paper. We
expect it to become more important for large nuclei. On the other hand large
nuclei have smaller symmetry and pairing energies per nucleon, thus a precise
determination of the EOS can be obtained from measurements of isotopes having
relatively small masses.

##  V dynamics of the phase transition

As we have seen we have been able to discuss some observables in the
fragmentation of nuclei using a language common to macroscopic systems
undergoing a phase transition. In the nuclear case we have a finite system
composed at most of hundreds of particles which evolves in time under the
influence of a long range Coulomb force. This poses many questions on why
techniques of statistical mechanics should apply in such evolving nuclear
systems. This also offers the possibility of dealing with statistical
mechanics of open systems and the problem of extending the description of a
phase transition to such a system.

We start by observing that even though we are dealing with a dynamical system,
the order parameter defined in this work, $m$, is confined between -1 and +1.
In this sense we have a somewhat ‘closed’ system. Also the density at which
the transition occurs should be smaller then normal density and thus Coulomb
effects are reduced. However, if we deal with larger sources, such as in U+U
collisions the phase transition might be washed out by the strong Coulomb
field. We expect our current considerations to be valid for small sources
only.

From statistical mechanics we know that in a first order phase transition
huang  a small seed increases in size depending upon the surface tension at a
given T and density $\rho$. If the pressure of the surrounding matter is
smaller than the internal pressure of the drop, the drop will grow by
capturing surrounding matter. On the other hand if the opposite is true then
the drop will decrease in size to balance the external pressure. The entire
process is driven by surface tension. Drops of a given size will survive only
when their internal pressure balances the external pressure. If the system is
at a very low density the interaction between different parts might take a
relatively long time. In these conditions a big nuclear drop whose internal
pressure is larger than that of the surroundings could be considered to be a
nucleus which is evaporating particles in order to balance the external (zero
in the case of an isolated nucleus) pressure. If we accept this picture than
the evaporation step is part of the dynamics of the phase transition. Thus a
very low density system might be thought of as many isolated drops evaporating
particles and reaching their equilibrium conditions before they collide with
other parts of the system or as small fragments being evaporated by other
drops. In a finite system this does not happen, but we might think of a
process where at some point the finite system becomes unconfined and an
infinite system is approximated by an infinite number of repetitions or
‘events’. Of course in a statistically equilibrated system we know that time
averages and event averages are the same. Here we are extending this concept
to finite systems where only event averages can be used. A major question here
is whether the properties of the phase transition are decided very early, i.e.
when the system ‘enters’ the instability region. As we said above if we have
an infinite system at a very low density undergoing a first order phase
transition, then the drops can explode, evaporate, and fuse with other
particles over a very long time. Our finite system might behave similarly but
without the fusion at later times. If this were the case than the detected
fragments carry all the information of the phase transition, if not then we
need to reconstruct the primary fragment distributions coming out of the
instability region.

Figure 14:  Free energy vs. time in AMD calculations (see text) for 64Zn+112Sn
system at 40MeV/A and central collisions, i.e. impact parameter less than 3
fm. Different picture correspond to T=200, 300, 500, 1000, 1500 and 2000 fm/c
respectively.

We can try to clarify some of these questions by means of microscopic models
such as Antisymmetrized Molecular Dynamics (AMD) or similar approaches where
the time evolution of the system is followed ono . However, we have to stress
that in such microscopic models some assumptions are made in order to
recognize the fragments at particular times during the time evolution. In
simpler approaches, fragments are recognized if particles are close in
coordinate space (of the order of the range of the attractive nuclear forces)
bon00 . In such a case the recognized fragments are ‘excited’ and they evolve
in time until a final state is reached after a long time of the order of
thousands of fm/c. A more refined approach for fragment recognitions is given
by defining clusters when its components are within a given distance in phase
space. The naive expectation would be that in this case we should recognize
fragments earlier than the previous case and this is the method that we will
adopt here for simplicity following ref. ono . In an ambitious approach dorso
the claim is that fragments are recognized very early during the time
evolution, of the order of tens of fm/c, if one searches for particles
connected in phase space to form fragments and minimize the energy. This case
probably corresponds to minimizing the entropy of the fragmenting system. If
this last picture will hold true then a picture of an infinite system at low
density will be equivalent to an ‘infinite’ repetition of events. Finally in
all the considerations above we have to add the necessary and interesting
complication that we have a mixture and not a single fluid, thus we can have
more situations to explore than discussed in the previous sections and we can
‘turn on and off’ an external field as well.

Figure 15:  Fit parameters a, b and c vs. time (see text) for the same system
of Fig.14. Solid circles refer to AMD calculations while the open symbol in
the M vs. x plot is the experimental value for this system.

We have performed AMD calculations for the same systems investigated
experimentally. After some time t, fragments are separated enough in phase
space that we can recognize within a simple phase space coalescence approach
as discussed in ono . In this way we can define a yield at a given time and
from this derive the free energy exactly as we did with the experimental data.
Characteristic results for the free energy versus time is given in Fig.(14)
together with a Landau $O(m^{6})$ fit. Some time evolution is observed. Using
a more sophisticated fragment recognition approach dorso  might even decrease
the time over which this evolution occurs. We can study the time evolution in
more detail by plotting the variables a, b and c. M defined in the previous
sections versus time. The results of the fits to the free energy at different
times is given in Fig.(15). While the quantities a, b and H/T change somewhat
during the time evolution, smaller changes are observed in the time evolution
of normalized quantities, x, z and h/T. Nevertheless the time evolution of the
fitting parameters influences the time evolution of the order parameter M
versus reduced temperature x as seen in the bottom right of Fig.(15). It is
very interesting to see that in these units the system is initially very hot
(superheated) and cools down when coming to equilibrium below the critical
temperature. The final result is very close to the observed values given by
the open points.

Thus in this model most qualitative features of the phase transition are
decided very early during the time evolution. This might correspond to an
entropy saturation early during the evolution. However, different models and
fragment recognition approaches might change the picture somewhat.

##  VI Equation of state

Once we know the free energy (at least in some cases) we can calculate the
NEOS by means of the Fisher model fisher . Since we do not have at present
experimental information on the density $\rho$, temperature T and pressure P
of the system we can only estimate the ‘reduced pressure’ paolo :

$\frac{P}{\rho T}(m)=\frac{M_{0}}{M_{1}},$ (18)

where $M_{i}$ are moments of the mass distribution given by:

$\displaystyle
M_{k}=\sum_{A}A^{k}Y(A,m)=Y_{0}\sum_{A}A^{k}A^{-\tau}e^{-F/T(m)A};$
$\displaystyle k=0,...n.$ (19)

Notice that the quantities above are now dependent on the order parameter m.
From the knowledge of $F/T$ $(H/T=0)$ from the previous section we can easily
calculate the reduced pressure near the critical point. In particular given
the simple expression for the moments we can also derive some analytical
formulas following paolo :

$\frac{P}{\rho
T}(m)=\frac{3.072|F/T|^{4/3}+1.417-3.631|F/T|+...}{-4.086|F/T|^{1/3}+3.631+0.966|F/T|+...},$
(20)

which gives at the critical point a critical compressibility factor ($F/T=0$):
$\frac{P}{\rho T}|_{c}=\frac{1.417}{3.631}=0.39$.
This value is essentially that derived from the Van-der-Waals gas equation but
is well above the values observed for real gases. Using the relations above we
can calculate the NEOS for the situations illustrated in Fig.9. The results
are displayed in Fig.16 where the reduced pressure is plotted versus m for
vaporization, superheating and first order phase transitions on the tri-
critical line. Notice that there is not a large difference between the first
two cases, while the last case displays two critical points (a third one is on
the negative m axis).
We have seen in Fig.2 that $N=Z$ nuclei display a power law. We can also
estimate the critical reduced pressure for this case noticing that the sums in
Eq.(VI) above must be restricted to $A=2Z$ nuclei. This leads to a critical
compressibility factor $\frac{P}{\rho T}_{c}=0.20$ which is a value closer to
that estimated from other multi-fragmentation studies before mor .

Figure 16:  Reduced pressure versus m of the fragments obtained from the a, b
and c parameters fit to the Landau free energy, Eq.(3) for the 64Ni+232Th. The
curves correspond to vapor(open circles), superheating(open squares), first
order (3 critical line-solid stars), see text. The solid circle is for $N=Z$
nuclei at the critical point.

We can compare our analytical result given in Eq.(20) with the numerical
values obtained above. This is displayed in Fig.(17) and we see that the
numerical approximation is especially good near the critical point(s) as
expected. If, from detailed comparison to experimental data, we are able to
extract the temperature and pressure dependence of the parameters entering the
Landau free energy, then Eq.(18) would be the Nuclear equation of state near a
critical point. From the actual data at our disposal we can only estimate the
behavior of the reduced pressure as function of the order parameter m.

Figure 17:  Comparison to the analytical result, Eq.(20)(solid circles) for a
first order phase transition.

On similar ground we can define a reduced compressibility as

$\chi\rho T(m)=\frac{M_{2}}{M_{1}}.$ (21)

Figure 18:  Reduced compressibility versus m of the fragments as in Fig.(16)

Its behavior is displayed in Fig.(18) for the cases outlined above.
Divergences near the critical point(s) are obtained.

##  VII conclusions

In conclusion, in this paper we have presented and discussed experimental
evidence for the observation of a quantum phase transition in nuclei, driven
by the neutron/proton asymmetry. Using the Landau approach, we have derived
the free energies for our systems and found that they are consistent with the
existence of a line of first-order phase transitions terminating at a point
where the system undergoes a second-order transition. The properties of the
critical point depend on the symmetry. This is analogous to the well known
superfluid $\lambda$ transition in 3He-4He mixtures. We suggest that a
tricritical point, observed in 3He-4He systems may also be observable in
fragmenting nuclei. These features call for further vigorous experimental
investigation using high performance detector systems with excellent isotopic
identification capabilities. Extension of these investigations to much larger
asymmetries should be feasible as more exotic radioactive beams become
available in the appropriate energy range.

It is important to stress that the observables discussed here represents only
$necessary$ conditions for a critical behavior. A definite proof of a phase
transition and a tricritical point could be given by a precise determination
of yields of fragments whose $m\approx\pm 0.5$, i.e. very unstable nuclei
which, most probably, decay before reaching the detectors. Thus fragment-
particle correlation measurements for exotic primary fragments such as
${}^{4}Li$, ${}^{5}Be$ (proton rich) or extremely neutron rich ${}^{10}He$ are
needed. More generally, such correlation experiments can also shed light on
the effects of secondary decay on the fragment observables. This remains a key
question in many equation of state studies and model calculations differ in
their assessment of these effects wci ; bon00 . Higher quality data over a
wider range of beam energies and colliding systems should also help in
clarifying the role of other energy terms, such as surface, Coulomb etc.,
which are important at lower excitation energies. In particular the role of
pairing and the possibility of Bose-Einstein condensation, should be more
deeply investigated. Our data for $I=0$ fragments already show that pairing is
important. This might be due to its importance during the phase transition or
to its role during secondary decay of the excited primary fragments.
Exploration of quantum phase transitions in nuclei is important to our
understanding of the nuclear equation of state and can have a significant
impact in nuclear astrophysics, helping to clarify the evolution of massive
stars, supernovae explosions and neutron star formation.

###### Acknowledgements.

We thank the staff of the Texas A$\&$M Cyclotron facility for their support
during the experiment. We thank L. Sobotka for letting us to use his spherical
scattering chamber. This work is supported by the U.S. Department of Energy
under Grant No. DE-FG03-93ER40773 and the Robert A. Welch Foundation under
Grant A0330. One of us(Z. Chen) also thanks the “100 Persons Project” of the
Chinese Academy of Sciences for the support.

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