Consistent relativistic mean-field models: critical parameter values

Consistent relativistic mean-field models: critical parameter values

M. Dutra Departamento de Ciências da Natureza - IHS, Universidade Federal Fluminense, 28895-532 Rio das Ostras, RJ, Brazil    O. Lourenço 111Present address: Departamento de Física, Instituto Tecnológico de Aeronáutica, CTA, São José dos Campos, 12228-900. SP, Brazil Universidade Federal do Rio de Janeiro, 27930-560, Macaé, RJ, Brazil    D. P. Menezes Depto de Física - CFM - Universidade Federal de Santa Catarina, Florianópolis - SC - CP. 476 - CEP 88.040 - 900 - Brazil
July 20, 2019

We revisit the study published in prc-critical (), related to the behavior of relativistic mean-field models, previously selected because they satisfy bulk nuclear matter properties, here used to compute the critical parameters of the symmetric nuclear matter. We evaluate their critical temperature, pressure, and density and compare with some values encountered in the literature. We also show that these parameters are correlated with the incompressibility calculated at the zero temperature regime.

I Introduction

Nuclear matter is an idealized medium, and all its properties, are derived from experiments indirectly in a model-dependent way. However, the understanding of its properties is of fundamental importance as a guide towards more specific subjects, such as nuclear and hadron spectroscopy, heavy-ion collisions, nuclear multifragmentation, caloric curves, and others. It is well known that theoretical hadronic models predict phase transitions at moderate temperatures. All these models share the prediction that a liquid-gas phase transition occurs for symmetric and asymmetric nuclear matter at finite temperature and density. Qualitatively, the isotherms of these hadronic mean-field models typically show a van der Waals-like behavior, where liquid and gaseous phases can coexist vdw5 ().

It is important to note that the critical temperature is defined in such a way that it always takes place in symmetric matter. In Ref.avancini2006 () the authors have shown that the instability region decreases with the increase of the temperature up to a certain value, which is related to a critical pressure and critical density. The values of these critical parameters are model dependent and there are many nonrelativistic and relativistic models in the literature, which can be used to calculate them. In this work we use the Relativistic Mean-Field (RMF) Approximation.

Ii Choice of models

Our study is based on the study presented in Ref. rmf () where RMF models were analyzed. These parametrizations had their volumetric and thermodynamical quantities compared with theoretical and experimental data available in the literature. These data were divided into three groups: symmetric nuclear matter (SNM), pure neutron matter (PNM) and a third group named MIX (the mixture of PNM + SNM). This last one encompasses the symmetry energy and its slope at the saturation density as well as reduction of the symmetry energy at half of the saturation density. In Table 1 we present a summary of these constraints. For more details see Ref. rmf ().

The analysis has shown that only 35 parametrizations were approved. They are named consistent relativistic mean field (CRMF) parametrizations. We consider in the present study only of them because the point-coupling parametrization does not generate a mass-radius curve, according to Ref. stars (), so, it was excluded. The remaining of them are part of two groups out of the seven presented in Ref. rmf (). We have shown them in Subsection  II.1 and II.2 of that reference”.

ii.1 Nonlinear RMF models

The group of the nonlinear RMF parametrizations with and terms and cross terms involving these fields encompasses thirty parametrizations. The Lagrangian density that describes this model is:


with and . The nucleon mass is and the meson masses are ,, and .

Constraint Quantity Density Region Range of constraint
SM1 at 190 270 MeV
SM3a Band Region
SM4 Band Region
PNM1 Band Region
MIX1a at 25 35 MeV
MIX2a at 25 115 MeV
MIX4 at and 0.57 0.86
Table 1: Set of updated constraints (SET2a) used in Ref. rmf (). For more details concerning each constraint see the reference.

We can derive from Eq.(1) the equation of state for symmetric nuclear matter (). The pressure is given by



are the Fermi-Dirac distributions for particles and antiparticles, respectively. The effective energy, nucleon mass, and chemical potential are , , and , respectively. Furthermore, the (classical) mean-field values of and are found by solving the following system of equations,




ii.2 Density-dependent models

The four remaining parametrizations belong to the density-dependent group. Two of them include the meson. Their Lagrangian density is given by




for , and .

The expression for the pressure for these models can be obtained from Eq. (8) and reads:

with the rearrangement term defined as


The mean-fields and are given by


with the functional forms of and given as in the nonlinear model, Eqs. (6)-(7), with the same distributions functions of Eq. (3), and the same form for the effective energy . The effective nucleon mass and chemical potential are now given, respectively, by , and .

Iii Results

The necessary conditions used in the calculation of the critical point are given by the following expressions:


where , and are, respectively, the critical pressure, density and temperature.

The critical parameters , , and are then obtained for each of the CRMF parametrizations. The results can be seen in Table 2.

Model Ref. (MeV) (fm)  (MeV/fm)
BKA20 PRC81-034323 ()
BKA22 PRC81-034323 ()
BKA24 PRC81-034323 ()
BSR8 PRC76-045801 ()
BSR9 PRC76-045801 ()
BSR10 PRC76-045801 ()
BSR11 PRC76-045801 ()
BSR12 PRC76-045801 ()
BSR15 PRC76-045801 ()
BSR16 PRC76-045801 ()
BSR17 PRC76-045801 ()
BSR18 PRC76-045801 ()
BSR19 PRC76-045801 ()
BSR20 PRC76-045801 ()
FSU-III PRC85-024302 ()
FSU-IV PRC85-024302 ()
FSUGold PRL95-122501 ()
FSUGold4 NPA778-10 ()
FSUGZ03 PRC74-034323 ()
FSUGZ06 PRC74-034323 ()
IU-FSU PRC82-055803 ()
G2* PRC74-045806 ()
Z271s2 PRC82-055803 ()
Z271s3 PRC82-055803 ()
Z271s4 PRC82-055803 ()
Z271s5 PRC82-055803 ()
Z271s6 PRC82-055803 ()
Z271v4 PRC82-055803 ()
Z271v5 PRC82-055803 ()
Z271v6 PRC82-055803 ()
DD-F PRC74-035802 ()
TW99 NPA656-331 ()
DDH NPA732-24 ()
DD-ME PRC84-054309 ()
Table 2: Critical values for Consistent RMF models

The results we have computed can be compared with the ones obtained from eight experimental data Refs. karn1 (); natowitz (); karn2 (); karn3 (); karn4 (); karn5 (); elliott (). In Table 3 we show a brief compilation of these results. In elliott (), the authors estimate not only the value for  MeV, but also for  MeV/fm, and  fm, all of them related to symmetric nuclear matter.

By first analyzing the critical temperature, we can see that only the family Z271 (that encompasses all 8 related parametrizations), presents compatible with five of the eight experimental points, including the more recent one elliott (). The density-dependent models present the critical temperature inside the range of  MeV proposed by karn3 (). The other critical parameters of Ref. elliott (), namely, pressure and density, are also compatible with the ones computed for the Z271 family. The density dependent family also agrees with this experiment.

Reference (MeV) (fm)  (MeV/fm)
karn1 () - -
natowitz () - -
karn2 () - -
karn3 () - -
karn4 () - -
karn5 () - -
elliott ()
Table 3: Summary of experimental data of Refs. karn1 (); natowitz (); karn2 (); karn3 (); karn4 (); karn5 (); elliott ()

If we look at the structure of Eq. (LABEL:pnl), we can understand this agreement with the experimental data based on the only term that distinguish such model from the one, which is those containing the constant. In this case .

In the case of the density-dependent model, we can think of a similar structure, since the nonlinear behavior of the field can be represented somehow in the thermodynamical quantities, by the density-dependent constant . The same occurs with the field, i.e., the strength of the repulsive interaction is also a density-dependent quantity, .

We have also tried to verify if there are correlations between the critical parameters and the observables of nuclear matter at zero temperature and at the saturation density. We investigate possible correlations between , and with the symmetry energy, its slope and incompressibility. The results are shown in Figs. 1, 2, and 3, respectively.

Figure 1: Critical (a) temperature, (b) pressure, and (c) density of CRMF parametrizations versus symmetry energy at saturation density. Circles: nonlinear model. Squares: density dependent model.
Figure 2: Critical (a) temperature, (b) pressure, and (c) density of CRMF parametrizations versus the slope of symmetry energy at saturation density. Circles: nonlinear model. Squares: density dependent model.
Figure 3: Critical (a) temperature, (b) pressure, and (c) density of CRMF parametrizations versus incompressibility at saturation density. Circles: nonlinear model. Squares: density dependent model.

Note that for the symmetry energy and its slope, Figs. 1 and 2, there are no indications of possible correlations. A unique pattern for the nonlinear and density-dependent models are not seen. However, the picture changes when we look at the incompressibility (. From Fig. 3, one can observe an increasing behavior of , and as increases.

Iv Summary

In this work, we present the results obtained in the calculation of the critical parameters: temperature, pressure, and density in symmetric nuclear matter. In our analysis, we verified that the nonlinear models, whose parameterizations were grouped in the family Z271, show a good agreement with the experimental data elliott () for all critical parameters analyzed. The density-dependent family also shows an agreement with the data given in  elliott () for the pressure and density. Concerning , the agreement is only found with data presented in Ref. karn3 ().

In the search for possible correlations, we can see that the incompressibility at zero temperature and at saturation density show a clear increasing behavior with the critical parameters analyzed. The same does not occur with the symmetry energy and its slope.


This work was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil under grants 301155/2017-8 and 310242/2017-7. This work is also a part of the project CNPq-INCT-FNA Proc. No. 464898/2014-5.


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