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

**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. Comment: 14 pages, 18 figures

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**ABSTRACT:**Thermal and isospin composition effects on the heat capacity of infinite nuclear matter are studied within the binodal coexistence region of the nuclear phase diagram. Assuming the independent conservation of both proton and neutron densities, a second-order phase transition is expected, leading to a discontinuous behavior of the heat capacity. This discontinuity is analyzed for the full range of the thermodynamical variables consistent with the equilibrium coexistence of phases. Two different effective models of the nuclear interaction are examined in the mean-field approximation: the nonrelativistic Skyrme force and the covariant QHD formulation. We found qualitative agreement between both descriptions. The discontinuity in the specific heat per particle is finite and decreases with both the density of particles and the isospin asymmetry. As a byproduct, the latent heat for isospin-symmetric matter is considered.Physical Review C 01/2012; 85(1). · 3.72 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Projectile fragmentation cross sections are calculated for reactions of 20Ne on the C, Al, Cu, Sn and Pb targets at 400 A MeV by using the improved quantum molecular dynamics model together with the statistical model code GEMINI. The improved quantum molecular dynamics model is applied to describe the dynamical process, and the GEMINI model is used to simulate the de-excitation process of excited primary fragments. It is found that the total cross sections increase as a function of the target mass, which is in good agreement with the experimental results and other theoretical predictions. The odd–even effect of the partial cross sections observed in experiments is well reproduced, which appears in the de-excitation process of the excited primary fragments as a result of pairing effect and is mainly formed in the grazing collisions. The results of isospin distributions demonstrate that the odd–even effect of partial cross sections mainly comes from the fragments with TZ = 0, ±0.5.Journal of Physics G Nuclear and Particle Physics 05/2012; 39(5). · 5.33 Impact Factor - J. Mabiala, A. Bonasera, H. Zheng, A. B. McIntosh, Z. Kohley, P. Cammarata, K. Hagel, L. Heilborn, L. W. May, A. Raphelt, G. A. Souliotis, A. Zarrella, S. J. Yennello[Show abstract] [Hide abstract]

**ABSTRACT:**The critical phenomenon of the nuclear liquid-gas phase transition has been investigated in the reactions 64Zn+64Zn, 64Ni+64Ni and 70Zn+70Zn at beam energy of 35 MeV/nucleon. Yields of fragments arising from fragmenting quasi-projectiles (QPs) with different neutron-proton asymmetries were analyzed within the framework of the Landau free energy approach. Fits to the free energy of fragments as a function of fragment asymmetry showed three minima, indicating the system to be in a regime of a first-order phase transition. The QP temperature estimates were extracted from the analysis of N=Z fragment data. Additionally, we make use of a recent method based on quantum fluctuations of fermions to derive temperatures and densities of selected QPs. Critical scaling of these observables is found for systems which differ in neutron to proton asymmetry. The derived critical exponent β = 0.35 ± 0.01, belongs to the liquid-gas universality class.Journal of Physics Conference Series 03/2013; 420(1):2110-.

Page 1

arXiv:1002.1738v1 [nucl-ex] 9 Feb 2010

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

M. Huang,1,2,3A. Bonasera,1,4, ∗Z. Chen,1,2R. Wada,1, †K. Hagel,1J.B. Natowitz,1P.K. Sahu,1L. Qin,1

T. Keutgen,5S. Kowalski,6T. Materna,1J. Wang,2M.Barbui,1C.Bottosso,1and M.R.D.Rodrigues1

1Cyclotron Institute, Texas A&M University, College Station, Texas 77843

2Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000,China.

3Graduate University of Chinese Academy of Sciences, Beijing, 100049, China.

4Laboratori Nazionali del Sud, INFN,via Santa Sofia, 62, 95123 Catania, Italy

5FNRS and IPN, Universit´ e Catholique de Louvain, B-1348 Louvain-Neuve, Belgium.

6Institute of Physics, Silesia University, Katowice, Poland.

We discuss experimental evidence for a nuclear phase transition driven by the different concentra-

tion 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 in4He-

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.

PACS numbers: 25.70Pq,21.65.Ef,24.10.-i

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

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.

have a Fermi motion [1] 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 [1]. Analogous to the properties of mirror nuclei,

we could expect that if we study nuclei at finite tem-

peratures, T, and low densities, ρ, 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).

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[2, 3]. On the other hand, there could be a

symmetry breaking favoring N ?= Z at lower T. In this

they

That

∗E-mail at:bonasera@lns.infn.it

†E-mail at:wada@comp.tamu.edu

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

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 ?= Z. Thus a phase transition is driven by

the difference in isospin concentration of the fragments

m = (N − Z)/A.

which clearly demonstrate that m is an order parameter

of the phase transition.Its conjugate field [2] 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 [4, 5]. We also note that

the phase transition has a strong resemblance to that

observed in superfluid mixtures of liquid4He-3He near

the λ point. In both systems, changing the concentration

of one of the components of the mixture, changes the

characteristics of the Equation Of State (EOS) [2, 3].

[2] which terminates

In this paper we will discuss data

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, ρ, less

than half of the normal ground state nuclear den-

sity [6].Even though strong signals for a first and a

second-order phase transition have been found [6, 7],

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 [8]. At the last stage of the expansion the role

Page 2

2

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 Dynam-

ics (CMD) simulations of expanding drops with and

without a Coulomb field [9, 10]. The expansion with the

Coulomb force included is very slow in the later stage

and nearly isothermal.

Even though at high T and small ρ the nucleus behaves

as a classical fluid, the analogy to classical systems should

not be overemphasized as, in the (T,ρ) region of interest,

the nucleus is still a strongly interacting quantum sys-

tem. 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 [6, 11–16].

The paper is organized as follows: in the next sec-

tion we discuss the experimental setup in detail. This is

followed by a description of the data analysis and a dis-

cussion in terms of the Landau O(m6) free energy. We,

then derive some critical exponents and the EOS corre-

sponding 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 super-

conducting cyclotron facility at Texas A&M Univer-

sity.

112,124Sn,197Au and232Th targets at 40 A MeV. Interme-

diate mass fragments (IMF) were detected by a detector

telescope placed at 20o. The telescope consisted of four

Si detectors. Each Si detector had 5cm × 5cm area. The

thicknesses were 129, 300, 1000 and 1000 µm. All Si de-

tectors were segmented into four sections and each quad-

rant had a 5oacceptance in polar and azimuthal angles.

The fragments were detected at average angles of 17.5o

± 2.5oand 22.5o± 2.5o. Typically 6-8 isotopes were

clearly identified for a given Z up to Z=18 with an en-

ergy threshold of 4-10 A MeV, using the ∆E-E technique

for any two consecutive detectors. The ∆E-E spectrum

was linearized by an empirical code based on a range-

energy table. In the code, isotopes are identified by a

parameter ZReal. For the isotopes with A=2Z, ZReal=

Z is assigned and other isotopes are identified by inter-

polating between them. The energy spectrum of each

isotope was extracted by gating on lines corresponding

to the individual identified isotopes. In order to com-

pensate for the imperfectness of the linearization, actual

gates for isotopes were made on the 2D plot of ZReal

versus energy. The multiplicity of each isotope was eval-

64,70Zn and64Ni beams were incident on58,64Ni,

uated from the extracted energy spectra, using a mov-

ing source fit at the two given angles. Since the energy

spectra of some isotopes have very low statistics, the fol-

lowing 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 multi-

plicity 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 iso-

tope 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 ta-

ble 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 ≤ 2) in coin-

cidence 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 discrim-

ination method was used to identify p, d, t, h and α

particles. The energy calibration for these particles were

performed using Si detectors (50 -300 µm) in front of the

CsI detectors in separate runs. The yield of each iso-

tope 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 ve-

locity. The parameters were searched globally for all 16

angles. Detailed procedures of the data analysis are also

given in refs. [17, 18].

Special care has been taken for8He identification. All

He isotopes are identified in the Si telescope, using the

∆E-E technique, in a narrow energy range. When a pro-

ton and an α hit the same quadrant and when both of

them stop in the E detector, their ∆E-E points overlap

with those of8He. Since the multiplicities of protons

and alphas are about three orders of magnitude larger

than that of8He, the contribution of accidental events

becomes significant, especially for the reaction systems

with lower numbers of neutrons, in which8He production

is suppressed. Since Z=1 ∆E-E spectra are not available

in this experiment, ∆E-E for Z ≤ 3 were measured in a

separate run. Using the light charged particle multiplic-

ity extracted from the 16 CsI detectors, the accidental

events were simulated for each reaction for the observed

Page 3

3

α yield in the ∆E-E spectra in this experiment, the solid

angle of the quadrant and the multiplicity of Z=1 parti-

cles. In order to minimize the accidental events, the runs

with a low beam intensity were selected in each reaction.

Typical linearized ZReal spectra with these accidentals

are shown in Fig.(1) for70Zn +232Th (N/Z ∼ 1.5) and

64Ni +112Sn ( N/Z ∼ 1.25). As one can see, the Zreal

values for the accidental events of proton and α pileup

is nearly identical to that of8He, while6He is clearly

identified. The contributions from d + α and t + α are

also reasonably consistent with the observed background

yields. A significant excess of8He yield beyond the ac-

cidentals is only observed for the reaction systems with

the124Sn,197Au and232Th targets. After the correction

of the accidental contributions, the multiplicities of6He

and8He were calculated using the source fit parameters

obtained for Li isotopes.

1

5

10

2

10

3

10

4

10

5

10

He

3

He

4

He

6

He

8

Li

6

Li

7

Li

8

Li

9

Th

232

Zn +

70

1.52 2.5

Z

3 3.5

1

10

2

10

3

10

4

10

10

α

d+

p+

α

t+

α

Sn

112

Ni +

64

Real

Counts

FIG. 1: Typical ZReal spectra for He and Li isotopes. Ac-

cidental events are generated only for p, d, t +α 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−τ

Fig.(2) for the64Ni+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 τ appear which suggests that pairing

is playing a role in the dynamics [1], leading to higher

yields for even-even nuclei.

The observation of the power law behavior suggests

that the mass distributions may be discussed in terms of

a modified Fisher model [7, 19]:

[4, 19]. This is shown in

Y = y0A−τe−β∆µA, (1)

where y0 is a normalization constant, τ = 2.3 is a crit-

ical exponent[7], β is the inverse temperature and

060605,Ni64Sn124,exp

10

2010-02-02 14:41:07

A

1

2 3 4 5

10

20 30

Yield

-5

10

-4

10

-3

10

-2

10

-1

10

1

FIG. 2: Mass distribution for the64Ni+124Sn system at 40

MeV/nucleon for I=0. The lines are power law fits with ex-

ponents 2.3±0.02 (odd-odd nuclei, dashed line), and 3.4±0.06

(even-even nuclei, full line) respectively.

∆µ = 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), A2/3(surface) and

the chemical composition m of the fragments and pos-

sibly pairing. The region we are studying in this paper

seems near the critical point for a liquid gas phase tran-

sition (volume and surface equal to zero) but modified

by m =

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 tran-

sition [21]. Our data indicate that those terms are not im-

portant in the present case [7] as we will show more in de-

tail 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. Ex-

periments 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) = Esym = 25(I/A)2

MeV/A, i.e. the symmetry energy of a nucleus in its

ground state [1]. 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

I

Page 4

4

normalize the yield data for each system to the12C yield

(I = 0) in that system, i.e. we define a ratio:

R =

Y Aτ

Y (12C)12τ.(2)

The normalized ratios for the system64Ni +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 sensi-

tive 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 us-

ing the ground state symmetry energy gives an ‘appar-

ent temperature’ T of 6.0 MeV. This value of T would

be the real one if only the symmetry energy is impor-

tant, if entropy can be neglected, if asym= 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 tempera-

ture dependent. Further, secondary decay processes may

modify the primary fragment distributions [17, 18]. we

will discuss these questions in the framework of the Lan-

dau free energy approach below. We stress that the ap-

pearance 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.

It is surprising that such a scaling appears as a function

of the symmetry energy only.

der about the role of the Coulomb energy if we accept

that surface and volume terms give negligible contribu-

tion. In figure 4 we have plotted the same normalized

ratios as a function of the quantity αEcoul+ βEsym, α

and β are arbitrary parameters given in the figure and

Ecoul= 0.7Z(Z−1)A−1/3is 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 [7].

To further explore the role of the relative nucleon con-

centrations we plot in Fig.(5) the quantity

versus m = (I/A), the difference in neutron and proton

concentration of the fragment. As expected the normal-

ized yield ratios depend strongly on m.

Pursuing the question of phase transition we can per-

In fact we might won-

F

T= −ln(R)

A

Ground state Binding Energy (MeV)

-400 -350 -300 -250 -200 -150 -100-500 50

Ratio

-4

10

-3

10

-2

10

-1

10

1

10

symmetry Energy (MeV)

01020304050

Ratio

-4

10

-3

10

-2

10

-1

10

1

10

FIG. 3: Ratio versus fragments ground state binding energy

(top panel) and symmetry energy (bottom panel) for the64Ni

+64Ni case at 40 MeV/nucleon. asym=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

experimental8He yield is the upper limit.

form a fit to these data within the generalized Landau

free energy description [2]. In this approach the ratio of

the free energy to the temperature is written in terms of

an expansion:

F

T=1

2am2+1

4bm4+1

6cm6− mH

T,

(3)

where m is an order parameter, H is its conjugate vari-

able and a−c are fitting parameters [2]. We observe that

the Free energy is even in the exchange of m → −m,

reflecting the invariance of the nuclear forces when ex-

changing N and Z. This symmetry is violated by the

conjugate field H which arises when the source is asym-

metric in chemical composition. We stress that m and H

are related to each other through the relation m = −δF

The use of the Landau approach is for guidance only.

While the approximation to O(m4) does not work [4],

the O(m6) 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 [2]. 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 mcnof the compound nucleus. For comparison

in the same figure we have displayed the O(m2) case, i.e.,

F/T = a(m − ms)2(b = c = 0) [26] As seen in the plot

δH.

Page 5

5

=2.3 NNsource,40 A MeV,z>=0

α

τ

021705,exp.

2010-02-02 15:45:11

-100 10203040

=1.0

β

5060 7080

Ratio

-4

10

-3

10

-2

10

-1

10

1

10

=0.0

β

=1.0,

-10010203040

=1.0

β

50607080

Ratio

-4

10

-3

10

-2

10

-1

10

1

10

=1.0,

α

-10

1

0102030 4050607080

Ratio

-4

10

10

-3

10

-2

10

-1

10

1

10

=0.5,

α

sym

E

β

+

c

E

α

-1001020304050607080

Ratio

-4

10

-3

10

-2

10

-1

10

=1.0

β

=0.1,

α

FIG. 4: Ratio versus symmetry energy + Coulomb energy for

the64Ni +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).

last assumption also produces a reasonable fit, although

it does not reproduce shoulders near m ∼ ±0.3. As we

will discuss in more detail below the appearance of two

minima for m ?= 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 en-

ergy Eq.(3), depend on temperature, pressure or den-

sity of the source.Usually one assumes c > 0, a =

a0(ρ)(T −T0) and b = b(T,ρ), where T0is 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 chem-

ical composition of the source.

source has N ?= 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:

Since, in general, the

(F

T)′= am + bm3+ cm5−H

T.

(4)

When H/T = 0 the first derivative is zero for the follow-

m

-1-0.500.51

F/T

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

b=c=0

(m

Ο

b= -

b= -

)

6

16/3ac

4ac

FIG. 5: Free energy versus m for the case64Ni+232Th. The

full line is a free fit based on Landau O(m6) free energy. The

dashed-dotted-dotted-dotted line is obtained imposing in the

fit b = −?

phase transitions. The short dashed line corresponds to b =

−√4ac, i.e. superheating. The O(m2) case, F/T = a(m −

ms)2,i.e. b = c = 0, ms = 0.1, is given by the long dashed

line.

16/3ac and it is located on a line of first order

ing values of m [2]:

m0= 0;m2

±=−b ±√b2− 4ac

2c

.(5)

If we now assume H ?= 0 but small, we can expand the

solutions above as m = m0±+ η with η small. Equating

the first derivative to zero, Eq.(4), and neglecting terms

O(η2) we get:

η =

H/T

0±+ 5cm4

a + 3bm2

0±

.(6)

The shift of the minimum from m0= 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) ver-

sus mcn= (I/A)cn.

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 ?= 0 for Icn= 0 which

could indicate the favoring of N > Z fragments by the

Coulomb field. Another possibility is that (I/A)source∝

(I/A)cn which then gives H = 0 when Isource = 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 combi-

nations 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

Page 6

6

cn

(I/A)

00.050.1 0.150.20.25

H/T

0

0.5

1

1.5

2

FIG. 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

squares for64Zn projectiles impinging on various targets, see

text.

64Ni, the full triangles are for

70Zn and the

and c as a function of the compound nucleus mcn. As

we see there is some dependence which may reflect dif-

ferences 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.

a

15

20

25

b

-400

-300

-200

-100

0

cn

(I/A)

0 0.05 0.1 0.15 0.2 0.25

c

0

100

200

300

400

FIG. 7: The parameters a,b and c versus (I/A) of the com-

pound 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. The curves do

m

-1-0.500.51

F/T

-1

0

1

2

3

FIG. 8:

from the a, b and c parameters fit to the Landau free energy,

Eq.(3). Results for all experimentally investigated reactions

are displayed.

F/T (H/T=0) versus m of the fragments obtained

not differ much suggesting that temperatures are quite

similar.The fits exhibit curvature near m = ±0.4

which may suggest the presence of additional minima

at larger absolute values of m.

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. [2–4]:

This could indicate

We have considered four such cases as follows:

1) superheating. This case corresponds to b = −√4ac

and gives two minima at m ?= 0 and is plotted in fig-

ure 5 for the64Ni+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 msource≈ ±0.4. In fact if the system could

be gently brought to the right temperature Ts, with the

correct isotopic composition, it might stay in the mini-

mum, i.e. more fragments of that m should appear;

2) line of first order phase transition. This corresponds

to the condition b = −?16ac/3 at a temperature T3,

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 ≈ 0.6, i.e. for more neutron rich fragments due

to the fact that H/T ?= 0. This suggests that in this

situation we might produce a large number of neutron

rich fragments. However, most of those fragments are

Page 7

7

probably unstable, thus coincidence measurements may

be required to determine their yields. Of course this fea-

ture should become important in neutron rich stars.

3) first order phase transition. Corresponds to the case

a = 0 and determines the ‘critical temperature’ T0where

the minimum at m = 0 disappears and only the ones at

m ?= 0 survive. This case is excluded by our present data.

However, the fit in Fig.(5) suggests an intermediate sit-

uation between this and case (2) above.

4) line of second order phase transition, tri-critical point.

Corresponds to a = 0 and b > 0 (T = Tc). When b = 0 as

well we have a tricritical point (T = T3c), 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.

We can extrapolate the cases discussed above to H/T =

m

-1-0.500.51

F/T

-4

-2

0

2

4

FIG. 9:

from the a, b and c parameter fit to the Landau free energy,

Eq.(3) for

por(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.

F/T (H/T=0) versus m of the fragments obtained

64Ni+232Th. The four curves correspond to va-

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 > Ts, superheating

(T = Ts)(dotted line), a point in the line of a first or-

der phase transition (T = T3)(dashed-dotted line) which

displays three equal minima at m0 and m±, see Eq.5,

the experimental data fit line(full line)T < T3. We have

also added the case a = 0 which should be obtained at

T = T0, where the minimum at m = 0 becomes a maxi-

mum. There is a series of cases not displayed in the fig-

ure, corresponding to the temperatures between T0and

T3where m = 0 is still a minimum but 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 λ

transition observed as some3He is added to4He [2]. Pure

4He has a critical temperature of 2.18 K. The critical

temperature for the second-order transitions decreases

with increasing3He concentration until at temperature,

T = 0.867K, 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 the4He-3He case. We believe that

our data, analyzed in terms of the the Landau O(m6)

free energy, suggest such a feature but are not sufficient

to clearly demonstrate this. Some other work [23, 24],

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 unnec-

essary factors by defining suitable dimensionless quanti-

ties. 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 ?= 0):

x =4ac

b2.(7)

Recalling that a is related to the distance from the crit-

ical temperature while b and c should only depend on

density [2], we deduce that x is a measure of the dis-

tance T − T3from the critical temperature in a suitable

dimensionless fashion. Similarly we can define a reduced

order parameter from Eq.5:

y =2cm2

|b|

.(8)

Thus Eq.5 can be rewritten as:

y = 1 +√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 = ±?y − 1 = ±(1 − x)1/4,

defining the temperature ‘distance’ from the critical

point, |t| = |1 − x|, immediately gives the value of a

critical exponent: β =1

4. This exponent is very close to

(10)

Page 8

8

x

0 0.20.4 0.60.811.2

M

0

0.2

0.4

0.6

0.8

1

1.2

Equilibrium

Equilibrium

EquilibriumSupercool

Superheat

FIG. 10:

all studied systems. The dashed line is given by Eq.(10), the

vertical line indicates the critical temperature T3. 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 [2].

Order parameter versus reduced temperature for

the accepted experimental value as is well known in the

O(m6) Landau theory [2]. 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 dis-

cussed in the previous section, can be identified as well

[2].

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 = H1/δ,(11)

which defines the critical exponent δ.

theory this exponent can be determined at the critical

point where a = b = 0. From equation (4) we easily get

δ = 5, which is the accepted value for such a critical ex-

ponent [2]. 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 [2]. A precise determina-

tion 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

In the Landau

η =H/T

a

. (12)

The temperature (a) dependence of the order parameter

shows that we are away from the critical point. Never-

theless we can study the behavior close to the critical

point by suitably defining scaling forms [2]:

M

|t|β =

η

|t|β

vs.

compared to magnetization data for nickel metal. The

scaled magnetization is plotted versus the scaled exter-

nal magnetic field [2]. 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 di-

rectly 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 equiva-

lently x) depends on the ‘distance’ from the critical tem-

perature and critical pressure. These quantities could

however be obtained in 4π experiments where charges,

masses and their velocities are carefully determined.

H/T

|t|βδ. These quantities are plotted in figure 11 and

δβ

H/|t|

3

10

4

10

5

10

6

10

β

M/|t|

1

10

2

10

3

10

c

T < T

c

T > T

FIG. 11: Scaling form for magnetization M vs. external field

for nickel [2], 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(m6) theory, we can write a ‘reduced’ free en-

ergy as (b ?= 0):

f

T=1

2xm2− |z|m4+2

3z2m6−h

Tm,

(13)

where:

ties together with the temperature, Eq.(7), and the re-

duced order parameter y,of Eq.(8), constitute the Landau

O(m6) theory in dimensionless form. It is instructive to

study how these quantities change with the reaction sys-

tem as we did in figures (7) and (8). In Fig.(12) we plot

these normalized quantities vs. difference in neutron pro-

ton 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

f

T=

4c

b2F

T, z =

c

|b|,

h

T=

4c

b2H

T. These quanti-

Page 9

9

x

0.2

0.4

0.6

0.8

z

1

1.5

cn

(I/A)

0 0.05 0.1 0.15 0.2 0.25

h/T

0

0.02

0.04

FIG. 12:

compound nucleus obtained from the data fit to the Landau

free energy, Eq.(3). The symbols are like in figure 6.

The parameters x, z and h/T versus (I/A) of the

x which gives the ‘distance’ from the critical temperature,

see Fig.(12). This is very important since only normal-

ized quantities should be used when inferring the prop-

erties 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 val-

ues of the charges. We can now try to derive some qual-

itative understanding of when and why Coulomb correc-

tions 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 [1]:

Ec

A

= 0.77Z2

A2A2/3=0.77

4

(1 − m)2A2/3,(14)

which explicitly introduces the order parameter m in the

Coulomb energy. We can define an ‘effective’ symmetry

energy (per particle) as:

Eeff

A

= (asym+0.77

4

A2/3)m2−0.77

2

A2/3m+0.77

4

A2/3,

(15)

where the symmetry energy coefficient asym= 25MeV.

Ignoring for a moment density corrections we see that

O(m2) 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 ‘ex-

ternal’ field even in collisions where the source ms= 0 as

we discussed in Fig.(6). Finally there is a term not de-

pendent 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 ρ1/3while contributions to the symmetry

energy should depend both on ρ2/3reflecting the Fermi

energy of the nuclei and on ρ, the latter coming from

different n-p interactions. At low densities we would ex-

pect 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 ap-

pears 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 A2/3. In Fig.(13) (top

panel) we plot F/T versus A for m = 0 fragments. The

expected dependence with mass number in the free en-

ergy suggested from ‘effective’ symmetry energy Eq.(15)

is not seen in the figure. Rather, a staggering between

F/T

-0.2

0

0.2

0.4

0.6

0.8

A

05 10 15 20 25 30 35 40

δ

F/T

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

FIG. 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 δ (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.

odd-odd and even-even nuclei is clearly visible.

To better clarify these arguments we can write the pair-

ing energy from the mass formula as [1]:

Ep

A

= 12

δ

A3/2, (16)

where 12MeV is the ground state pairing energy coeffi-

Page 10

10

cient and δ:

δ =(−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 δ factor from pairing that changes

the sign of the contribution for odd-odd to even-even nu-

clei. A combined fit to the data using Coulomb plus pair-

ing 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 δ we should

get no discontinuities when plotting this quantity versus

mass number. Similarly if the properties of the free en-

ergy depend on the pairing term, as for the ground state

case, then it should be a monotonic function of A after

multiplying it by δ. In Fig.(13) (bottom panel), we plot

the quantity

Tδ versus mass number for the same sys-

tem of Fig.(13) (top panel). The fit using the pairing

mass dependence is also good. The Coulomb mass de-

pendence 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(ρ

a possible density correction. For the pairing case we get

T = 6.45MeV. Notice that in this case we have not sug-

gested any density correction since the fate of the pair-

ing 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 fit-

ting 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 re-

sult. A simple calculation give

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 be-

come 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 rela-

tively small masses.

F

ρ0)1/3MeV where we have explicitly indicated

ρ

ρ0= (6.45/9)3= 0.34

V. DYNAMICS OF THE PHASE TRANSITION

As we have seen we have been able to discuss some ob-

servables 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 statisti-

cal mechanics should apply in such evolving nuclear sys-

tems. This also offers the possibility of dealing with sta-

tistical 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 deal-

ing with a dynamical system, the order parameter de-

fined 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 re-

duced. 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 [2] a small seed increases in size depend-

ing upon the surface tension at a given T and density ρ.

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 bal-

ance 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 in-

teraction 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 evaporat-

ing 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 evaporat-

ing 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 re-

gion. As we said above if we have an infinite system at

a very low density undergoing a first order phase tran-

sition, 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 frag-

ments 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.

Page 11

11

zn64sn112 primary g0as DSTCL5 c>100,0<b<3fm Z>=0

2

2010-02-02 22:47:04

-2

0

-ln(Ratio)/A

-2

2

0

2

(N-Z)/A

-1 -0.5 0 0.51

-2

0

-2

0

2

-2

2

0

2

(N-Z)/A

-1 -0.500.51

-2

0

FIG. 14: Free energy vs. time in AMD calculations (see text)

for64Zn+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 Molec-

ular Dynamics (AMD) or similar approaches where the

time evolution of the system is followed [22]. However, we

have to stress that in such microscopic models some as-

sumptions are made in order to recognize the fragments

at particular times during the time evolution. In sim-

pler approaches, fragments are recognized if particles are

close in coordinate space (of the order of the range of the

attractive nuclear forces) [7]. In such a case the recog-

nized 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 com-

ponents 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 simplic-

ity following ref. [22]. In an ambitious approach [10] 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 fragment-

ing system.If this last picture will hold true then a

picture of an infinite system at low density will be equiv-

alent 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.

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 ap-

proach as discussed in [22]. In this way we can define a

zn64sn112 primary g0as DSTCL5 c>=100, 0<b<3fm Z>=0

20

2010-02-02 23:13:58

a

10

15

25

b

-200

-100

0

H/T

0.5

0.6

0.7

0.8

time

10002000

c

0

100

200

300

x

0.5

1

1.5

z

1.15

1.2

1.25

1.3

time

10002000

h/T

0

0.02

0.04

0.06

0.08

x

00.51

M

0

0.5

1

Equilibrium

Equilibrium

Equilibrium

Supercool

Superheat

FIG. 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.

yield at a given time and from this derive the free energy

exactly as we did with the experimental data. Charac-

teristic results for the free energy versus time is given in

Fig.(14) together with a Landau O(m6) fit. Some time

evolution is observed. Using a more sophisticated frag-

ment recognition approach [10] 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 dif-

ferent times is given in Fig.(15). While the quantities

a, b and H/T change somewhat during the time evolu-

tion, 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 sys-

tem is initially very hot (superheated) and cools down

when coming to equilibrium below the critical tempera-

ture. 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 satura-

tion early during the evolution. However, different mod-

els and fragment recognition approaches might change

the picture somewhat.

Page 12

12

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 [21]. Since we do not have at present experimental

information on the density ρ, temperature T and pres-

sure P of the system we can only estimate the ‘reduced

pressure’ [25]:

P

ρT(m) =M0

M1,(18)

where Miare moments of the mass distribution given by:

Mk=

?

A

AkY (A,m) = Y0

?

A

AkA−τe−F/T(m)A;

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 cal-

culate the reduced pressure near the critical point. In

particular given the simple expression for the moments

we can also derive some analytical formulas following [25]:

P

ρT(m) =

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):

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.(19) above must

be restricted to A = 2Z nuclei. This leads to a critical

compressibility factor

ρTc= 0.20 which is a value closer

to that estimated from other multi-fragmentation studies

before [11].

P

ρT|c=1.417

P

We can compare our analytical result given in Eq.(20)

with the numerical values obtained above. This is dis-

played in Fig.(17) and we see that the numerical approx-

imation 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 pres-

sure 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 re-

duced pressure as function of the order parameter m. On

m

-0.2 0 0.2 0.4 0.6 0.81

T

ρ

P/

0

0.2

0.4

0.6

0.8

1

1.2

FIG. 16:

tained from the a, b and c parameters fit to the Landau free

energy, Eq.(3) for the

to vapor(open circles), superheating(open squares), first or-

der (3 critical line-solid stars), see text. The solid circle is for

N = Z nuclei at the critical point.

Reduced pressure versus m of the fragments ob-

64Ni+232Th. The curves correspond

m

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

T

ρ

P/

-1

-0.5

0

0.5

1

1.5

2

FIG. 17:

circles) for a first order phase transition.

Comparison to the analytical result, Eq.(20)(solid

similar ground we can define a reduced compressibility as

χρT(m) =M2

M1.(21)

Its behavior is displayed in Fig.(18) for the cases out-

lined 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 neu-

tron/proton asymmetry. Using the Landau approach, we

Page 13

13

m

-0.2 0 0.2 0.4 0.6 0.81

T

ρ

χ

2

4

6

8

FIG. 18: Reduced compressibility versus m of the fragments

as in Fig.(16)

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.

properties of the critical point depend on the symmetry.

This is analogous to the well known superfluid λ transi-

tion in3He-4He mixtures. We suggest that a tricritical

point, observed in3He-4He systems may also be observ-

able in fragmenting nuclei. These features call for fur-

ther vigorous experimental investigation using high per-

formance detector systems with excellent isotopic iden-

tification capabilities. Extension of these investigations

to much larger asymmetries should be feasible as more

exotic radioactive beams become available in the appro-

priate energy range.

It is important to stress that the observables dis-

cussed 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 ≈ ±0.5,

The

i.e. very unstable nuclei which, most probably, decay

before reaching the detectors.

correlation measurements for exotic primary fragments

such as

rich10He are needed. More generally, such correlation

experiments can also shed light on the effects of sec-

ondary 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 [6, 7]. 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.

Thus fragment-particle

4Li,

5Be (proton rich) or extremely neutron

Acknowledgments

We thank the staff of the Texas A&M Cyclotron facil-

ity 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. Depart-

ment 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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[26] F/T=a(m − ms)2

(a/2)m2

when the yields are normalized by12C.

=(a/2)m2− H/Tm +

s;H/T = am(s). The last term is dropped out

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