November 15, 2007 11:35 WSPC/143-IJMPE 00900

International Journal of Modern Physics EVol. 16, No. 9 (2007) 3041–3044c© World Scientific Publishing Company

HOT NUCLEAR MATTER PROPERTIES

TOMAZ PASSAMANI∗ and MARIA LUIZA CESCATO†

Departamento de Fısica, Universidade Federal da Paraıba

CEP 58059-970 Joao Pessoa, PB, Brazil∗tomaz@fisica.ufpb.br

†mcescato@fisica.ufpb.br

Received 25 May 2007Revised 3 June 2007

Accepted 14 June 2007

The nuclear matter at finite temperature is described in the relativistic mean field theoryusing linear and nonlinear interactions. The behavior of effective nucleon mass with tem-perature was numerically calculated. For the nonlinear NL3 interaction we also observedthe striking decrease at temperatures well below the nucleon mass. The calculation ofNL3 nuclear matter equation of state at finite temperature is still on progress.

1. Introduction

Field theory descriptions of the nuclear system, based on relativistic mean field

approximation (RMF), have been extremely successful in reproducing the ground-

state properties of spherical as well as deformed nuclei by the whole table of

nuclides.1–3 It is important to extend these studies to finite temperatures, regard-

ing the relevance of this degree of freedom in nuclear systems at extreme conditions

of pressure and temperature, as in the case of relativistic heavy-ion reactions, and

the study of neutron star properties.

In this work the RMF framework at finite temperature was used, in σ-ω-ρ model,

with the Walecka linear parametrization and the NL3 nonlinear parametrization.1,3

The thermodynamic averaging of the mesonic fields could be calculated through the

construction of the grand-partition function of the system.1 In a previous work we

have made analytical calculations with the same approach in a low temperature

limit.4 Here we present numerical calculations for any temperature. Solving the

effective mass equation at finite temperature we could analyze the behavior of

effective mass as a function of temperature.

In Sec. 2, a brief resume of mean field theory for the σ-ω-ρ model at finite

temperature is presented, as well as its application to the nuclear matter system.

In Sec. 3, our results are presented and discussed.

3041

Int.

J. M

od. P

hys.

E 2

007.

16:3

041-

3044

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

VA

ND

ER

BIL

T U

NIV

ER

SIT

Y o

n 05

/01/

13. F

or p

erso

nal u

se o

nly.

November 15, 2007 11:35 WSPC/143-IJMPE 00900

3042 T. Passamani & M. L. Cescato

2. The Relativistic Mean Field Model at Finite Temperature

The model considers a system of nucleons interacting through the meson fields

σ(J = 0, T = 0), ωµ(J = 1, T = 0) and ~ρµ(J = 1, T = 1). Considering these

degrees of freedom, the Lagrangian density is

L = Ψiγµ∂µ − MΨ +1

2∂µσ∂µσ − U(σ) − gσΨσΨ

−1

4ΩµνΩµν +

1

2m2

ωωµωµ − gωΨγµωµΨ

−1

4~Rµν ~Rµν +

1

2m2

ρ~ρµ~ρµ − gρΨγµ~τ~ρµΨ , (1)

where Ψ is the spinor of the nucleon and γµ are the usual Dirac matrices. The

parameters M , mσ , mω and mρ are the nucleon mass and the σ, ω and ρ meson

masses, respectively, while gσ, gω and gρ the corresponding coupling constants. The

potential U(σ) is given by

U(σ) =1

2m2

σσ2 +1

3g2σ

3 +1

4g3σ

4 . (2)

The nonlinear terms in the sigma field are included in this potential. The vector

meson tensor fields are

Ωµν = ∂µων − ∂νωµ , (3)

~Rµν = ∂µ~ρν − ∂ν~ρµ − gρ(~ρµ × ~ρν) . (4)

In the mean field approximation, the fields are replaced by their mean values and

for rotation invariant systems only the time component of the vector fields survive.

The thermodynamic potential and grand-partition function are then

Φ(µ, V, T ; σ, ω, ρ) = −kBT ln Ξ , (5)

Ξ = Tr e−(H−µN)/kBT , (6)

where µ is the chemical potential, V is the volume, T is the temperature, σ, ω and ρ

are the thermodynamic averages of the scalar and vector mean fields. H and N are

the RMF Hamiltonian and baryon number operators, and kB is the Boltzmann’s

constant. Under these conditions, for stationary systems, we have

M∗ = M + gσσ , (7)

defining the effective mass for the nucleon. Nuclear matter is an uniform system, in

which the σ, ω and ρ fields are position independent. Here we describe only nuclear

symmetric matter (γ = 4) and neutron matter (γ = 2). From the equations of

motion, the solutions for the mesonic fields are then

σ = −gσ

m2σ

ρs , (8)

ω =gω

m2ω

ρv . (9)

Int.

J. M

od. P

hys.

E 2

007.

16:3

041-

3044

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

VA

ND

ER

BIL

T U

NIV

ER

SIT

Y o

n 05

/01/

13. F

or p

erso

nal u

se o

nly.

November 15, 2007 11:35 WSPC/143-IJMPE 00900

Hot Nuclear Matter Properties 3043

The calculation of the energy-momentum tensor provides

E =1

2

g2ω

m2ω

ρ2v + U(σ)

+γ

(2π)3

∫d3k(k2 + M∗

2

)1

2 [nk(T ) + nk(T )] , (10)

with

ρv =γ

(2π)3

∫d3k[nk(T ) − nk(T )] , (11)

where nk(T ) and nk(T ) are the familiar baryon and antibaryon thermal distribution

functions

nk(T ) =1

e(E∗−ν)/kBT + 1

, (12)

nk(T ) =1

e(E∗+ν)/kBT + 1, (13)

with ν = µ−gωω, E∗ = (k2+M∗2

)1

2 and k is the nucleon wave number.1 Minimizing

the energy density with respect to σ we have

M∗ = M −γ

(2π)3g2

σ

m2σ

∫d3k

M∗

(k2 + M∗2)

1

2

[nk(T ) + nk(T )]

− gσg2σ

2

m2σ

− gσg3σ

3

m2σ

. (14)

3. Results and Discussion

The behavior with temperature of the effective nucleon mass was numerically calcu-

lated, using the model described above with the linear Walecka parametrization and

the nonlinear NL3 parametrization.1,3 Figure 1 shows the results for nuclear matter

0 50 100 150 200 250 300 350 400KT (MeV)

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

M*/

M

WaleckaNL3

NUCLEAR MATTER

ν = 0

Fig. 1. Behavior of the M∗/M with temperature for ν = 0 for nuclear matter, using Waleckaand NL3 parametrizations.

Int.

J. M

od. P

hys.

E 2

007.

16:3

041-

3044

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

VA

ND

ER

BIL

T U

NIV

ER

SIT

Y o

n 05

/01/

13. F

or p

erso

nal u

se o

nly.

November 15, 2007 11:35 WSPC/143-IJMPE 00900

3044 T. Passamani & M. L. Cescato

0 50 100 150 200 250 300 350 400KT (MeV)

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

M*/

M

WaleckaNL3

NEUTRON MATTER

ν = 0

Fig. 2. Behavior of the M∗/M with temperature for ν = 0 for neutron matter, using Waleckaand NL3 parametrizations.

and Fig. 2 for neutron matter, for ν = 0. The striking decrease of the effective mass

at temperatures well below the nucleon mass, already observed by Walecka, is also

observed for NL3 parametrization.1 The difference is that the decrease occurs at a

little bit lower temperature for NL3 parametrization in both nuclear and neutron

matter. For both interactions, in nuclear matter this decrease occurs at a lower tem-

perature than observed in neutron matter. The next step in this investigation is the

construction of the NL3 finite temperature neutron and nuclear matter equation of

state. Work in this direction is on progress.

Acknowledgments

This work was partially supported by the Brazilian agency Conselho Nacional de

Desenvolvimento Cientıfico e Tecnologico (CNPq).

References

1. B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16 (1986) 1.2. P. Ring, Prog. Part. Nucl. Phys. 37 (1996) 193.3. G. A. Lalazissis, J. Konig and P. Ring, Phys. Rev. C 55 (1997) 540.4. T. Passamani and M. L. Cescato, Int. J. Mod. Phys. D 16 (2007) 297.

Int.

J. M

od. P

hys.

E 2

007.

16:3

041-

3044

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

VA

ND

ER

BIL

T U

NIV

ER

SIT

Y o

n 05

/01/

13. F

or p

erso

nal u

se o

nly.