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Here we will assume the symmetry breaking at high density, from

nuclear matter (locally color white) to quark gluon-plasma QGP

(globally color white), this possibly gives the second-order phase

transition. In accordance to the conditions used before we can add

two extra constrains based on the conditions given by the

phenomena of matter close to the critical density of a second-order

phase transition at zero temperature T=0. Then we will get five

constrains:

Solving these nonlinear equations we get A , B, C and D and

therefore we obtain the critical point ρc =2.9354 ρ0 at T=0

For interactions between five particles O(ρ5) we obtain the same

behavior like O(ρ3). If we take more variables on our expansion,

until the sixth order and if we assume that n=1,..,4

and solve again this nonlinear system of seven unknown variables

we get A, B, C, D, E, F . Here the critical point ρc =5.523 ρ0 at T=0

where and ρ0 is the normal nuclear density, the first

term of the eq. (1) refers to the kinetic energy of a free energy of a

free Fermi gas. The other terms are due to potential interactions

and correlations. Let us consider for simplicity a classical system

with an EOS in the form [2].

where K is the (isothermal) compressibility

It is important to emphasize that the nuclear EOS strongly

influences the phase transition and the phase diagram. The three

parameters A, B and σ are determinate using the conditions at

ρ=ρ0 eq. (1) has a minimum, the binding energy is E=-15MeV and

finally the compressibility is of order of 200MeV, as inferred from

the vibrational frequency of the giant monopole resonance. Using

these conditions, we get A=356 MeV, and σ=7/6 .

VIRIAL EXPANSION OF THE NUCLEAR EQUATION OF STATE*Ruslan I. Magana Vsevolodovnaab, Aldo Bonaseraa, Hua Zhenga

aCyclotron Institute, Texas A&M University , College Station, TX 77843,USA.bREU student from National Autonomous University of Mexico, A.P. 70-543, 04510,Mexico D.F. Mexico.

For a system interacting through two body forces having a short-

range repulsion and a longer-range attraction the equation of state

(EOS) resembles a Van Der Waals one. This is indeed the case for

nuclear matter [1]. A popular approach is to postulate an equation

of state which satisfies known properties of nuclei. The equation for

energy per particle

Now if our purpose is to obtain the dependence of our EOS with the

temperature we can consider our system like Fermi gas [4]. The

temperature of the system can be derived experimentally [4] from the

momentum fluctuations or particles in center of mass frame of the

fragmenting source.

the equation for energy per particle takes the form

The saturation pint corresponds to the equilibrium point (at zero

temperature) of nuclear matter hence characterized by vanishing

pressure then

4. Theoretical Results

For the case of fourth order O(ρ4).of our EOS, the critical temperature

and volume have the values Tc=18.05MeV and ρc =0.3724 ρ0

respectively. This could be the critical point for the second-order

phase transition In this approach we are assuming interactions

between pair of particles and between groups of three and four

particles. This could be fully justified by the fact that nucleus are made

of quarks and gluons. For higher terms, for the sixth order O(ρ6).the

critical temperature and density are Tc=17.932MeV and ρc =0.3717 ρ0.

If we take the free energy in terms of a power series and we consider

an external field Pc and we stop our expansion at the fourth order

O(ρ4) , then introduce the parameter η=(V– Vc) it is possible find the

minimum at the critical temperature . Thus we obtain

Therefore for O(ρ4) has the critical exponent δ=3 and for O(ρ6) we

obtain δ=5 .

.

1. IntroductionIn recent years the availability of new heavy-ion accelerators capable

to accelerate ions from few MeV/nucleon to GeV/nucleon has fueled a

new field of research loosely referred to as Nuclear Fragmentation.

The characteristics of these fragments depend on the beam energy

and the target-projectile combinations which can be externally

controlled to some extent. This kind of experiments provides

information about the nuclear matter. This is very useful to make the

best equation of state of nuclear matter. The conventional EOS

provide only limited information about the nuclear matter : the static

thermal equilibrium properties. In heavy ion collisions nonequilibrium

processes are very important. After an energetic nucleus-nucleus

collisions many light nuclear fragments, a few heavy fragments and a

few mesons (mainly pions) are observed in the 100MeV-

4GeV/nucleon beam energy region. Thus the initial kinetic energy of

the projectile leads to the destruction of the ground state nuclear

matter and converts it into dilute gas (ρ<<ρ0) of fragments which then

loses thermal contacting during the break-up or freeze-out stage:

(these frozen-out fragments and their momentum distributions can be

measured by detectors).

2.Theoretical Nuclear Overview

2

322.52 1

A BE

0/

2 EP T

3. Nuclear EOS Approach

Now if we modify this approach [3] the compressibility condition is

given by the mean field potential

where

which means that the compressibility at ρ=ρ0 is equal to K=225

MeV, then using the conditions E=-15MeV, P=0 and K=225MeV at

ρ = ρ0 we get A=-210 MeV,B=157.5 MeV and σ=4/3. For a nuclear

system we expect to see a liquid-gas phase transition at a

temperature of the order of 10MeV and at low density. Under these

conditions we can assume that nuclear matter behaves like a

classical ideal gas eq. (2) however this is just our ansatz. Actually

in this work we will compare the theoretical behavior of the EOS

assuming classical ideal gas and Fermi gas.

( ) ( ) ( )U A BA

0

( )0 at =

U

To calculate the critical point, we will impose the conditions that the

first and second derivative of eq. (2) respect to ρ are equal to zero

therefore we can obtain the critical temperature Tc =9MeV and

density ρc=0.35ρ0. Such values are in some agreement with

experimental results [1]. This is the critical point for the second

order liquid-gas phase transition. The region where K<0 is a first

order which terminates in a second order. Now we will propose a

new equation for the energy per particle

where the first term refers to the kinetic energy of a free Fermi gas,

the other terms are due to potential interactions and correlations,

the third term is obtaining by taking into account the interaction

between pair of particles, and the subsequent terms must involve

the interactions between groups of three, four, etc., particles. Let’s

consider three body forces O(ρ3), take the same conditions as in eq

(4). Unfortunately, the solution has no physical sense because the

energy diverges to minus infinity when densities approaches infinity.

That would mean that the core would collapse and that is not

possible. For the fourth order of our expansion O(ρ4), the eq. (5)

takes the form

2

3

1

22.51

knn

n

AE

n

(1)

(2)

2

2 3 4322.52 3 4 5

A B C DE

0

2

0 2

0

) ( ) 15MeV ) 0

) P( ) 0 ) 0

) ( ) 225MeV

c

c

Pa E d

Pb e

c K

Table 1.0 Values of the parameters of the EOS’s of nuclear matter.

Interactions A

[MeV]

B

[MeV]

C

[MeV]

D

[MeV]

E

[MeV]

F

[MeV]

σ Emin

[MeV]

K[MeV]

Conventional -356 303 7/6 -16 200

-210 157.5 4/3 -15 225

3 -135 112.5 -30 -15 225

4 -136.89 120.99 -41.32 4.72 -15 225

5 -137.59 124.41 -46.51 7.67 -0.47 -15 225

6 -137.96 126.25 -49.46 9.55 -0.92 0.0035 -15 225

The Conventional EOS corresponds to eq. (1) and the remaining EOSs correspond to eq. (5). Emin is the ground

state binding energy density and K is the compressibility at normal respectively.

0

c

n

n

P

32 2

0

*

0

a T

E

5 1( 1)23 3

0 0

1

215

( 1) 3

nkn

n

nAP a T

n

2 2

23 30

1

22.51

knn

n

AE a T

n

1

3

4

4

6 ( )cc

Vc

P V

EV

0, 0

9T

PK

Fig. 1.0 The energy per particle

of nuclear matter as a function of

density using different density-

dependent interactions in

comparison with the conventional

formulation.

Fig. 4. Some properties or our EOS at many temperatures. The first four plots are the comparison of the

compressibility and speed of sound between classical ideal gas and Fermi gas for the conventional EOS. The

remaining plots belong to our EOS where we are assuming that the nuclear matter behaves like Fermi gas.

Fig. 2. The pressure per particle of nuclear matter as a function of density at some temperatures. Here we

are assuming that nuclear matter behaves like classical ideal gas.

O(ρ6).O(ρ4).Conventional

K=225MeV

Conventional

K=225MeV O(ρ4). O(ρ6).

Fig. 3. The pressure per particle of nuclear matter as a function of volume at some temperatures. Here

we are assuming that nuclear matter behaves like Fermi gas.

(5)

(6)

O(ρ4).

Conventional

K=225MeV

Conventional

K=225MeV

Ideal gas

Fermi gas

Fermi gas

Fermi gas

O(ρ6).5.ConclusionThe virial expansion of the nuclear equation

of state reproduce some elementary

properties of our nuclear system. In this work

we have determined the critical density for

the second order phase transition at MeV. It

would of interest to see how our equation of

state affects the microscopic dynamical

evolution of the nuclear system. We found

that for odd orders this approach is not

suitable to describe the basic properties of

our core system however for even orders we

have more acceptable behavior of the energy

per particle as function of density

It could be possible that the collective

character in regions with high density of

particles are associated with even

interactions [5,6].

Fig. 12 Behavior near to the critical point. The pressure per particle of nuclear matter as a function of

volume at some temperatures

O(ρ4).

O(ρ6).

(4)

(6)

(7)

(8)

(9)

(10)

6.References

*Funded by DOE and NSF-REU Program

(3)

[1] G. Sauer, H. Chandra and Moselu, Nucl. Phys. A, 264

[2] A. Bonasera et al., Rivista del Nuovo Cimento, 23, (2000).

[3] A. Bonasera and Toro M. D. Lettere at Nuovo Cimento

Vol44 N. 3 (1985).

[4] Wuenschel S, et al., Nucl. Phys. A, 843 (2b) 1-13

[5] A. Arima and F. Iachello, “Collective nuclear states as

representaitons of a SU(6) group”,Phys. Rev. Lett. 35

1069(1975).

[6] A. Bohr and B.R. Mottelson, Mat. Phys. Dan. Vid. Selks.

27 No 16 (1953).