^{1,2}

^{3,2}

^{1,2}

^{3,2,4}

^{3}.

The noncongruent liquid-gas phase transition (LGPT) in asymmetric nuclear matter is studied using the recently developed quantum van der Waals model in the grand canonical ensemble. Different values of the electric-to-baryon charge ratio,

An infinite hypothetical system of interacting neutrons and protons in equal proportions is called symmetric nuclear matter. The known phenomenology of the nucleon-nucleon interaction suggests short range repulsion and intermediate range attraction. This yields a first-order liquid-gas phase transition (LGPT) from a diluted (gaseous) to a dense (liquid) phase in symmetric nuclear matter, and, correspondingly, a discontinuity of the particle number density as a function of pressure.

Experimentally, evidence for a LGPT in nuclear matter was first reported in Refs.

The present paper treats the more complex situation when the densities of neutrons and protons are not equal, i.e., the ratio of the electric-to-baryon charge,

Asymmetric nuclear matter is of interest for both heavy ion collisions and nuclear astrophysics: neutron-rich matter is present in compact stars and binary neutron star mergers

For asymmetric nuclear matter, the order parameter therefore is not longer given by the difference between the net baryon densities of the liquid and the gaseous phases,

The properties of nuclear matter can be described by a variety of different models. Here we employ an extension of the classical van der Waals (vdW) model which was recently generalized to include the effects of quantum statistics, special relativity, grand canonical ensemble, and mixtures of different sized constituents. This quantum vdW (QvdW) model has been further developed and applied to the description of symmetric nuclear matter in Refs.

The present paper studies the LGPT in asymmetric nuclear matter using the QvdW model. Section

We consider an infinite system of interacting nucleons consisting of neutrons and protons which differ only by the electric charge they carry. The total baryonic,

Here

As the values of the charges

The thermodynamical functions at given

The LGPT in symmetric nuclear matter was studied within the QvdW model in Ref.

The QvdW interactions are taken to be equal for all pairs of nucleons:

The position of the critical point (CP) of symmetric nuclear matter within QvdW model is

The LGPT line in the

(a) Congruent LGPT lines in the

Another limiting case is the completely asymmetric nuclear matter consisting of neutrons only.^{1}

Note that neutron matter as discussed here does not match neutron star matter: the latter must include light and heavy nuclei, beta equilibrium, leptons, strange hadrons, and eventually also a quark matter contribution. So the term neutron matter is used here only for a hypothetical state of neutrons only.

This corresponds to a zero asymmetry parameter,The values of the thermodynamic quantities in the nuclear GS of symmetric nuclear matter are fixed to known empirical values; see Eq.

One sees that the neutrons' binding energy in the GS is positive,

Mixed phase boundaries at given

The LGPT lines in the

In the Boltzmann approximation, Eqs.

Hence, the noncongruence of the LGPT vanishes^{2}

Note that this is true only if the isospin-independent interaction parameters are considered as in the present section. Within a more general QvdW formalism, which is described in Sec.

In Sec.

A QvdW system of nucleons with a large number of internal degrees of freedom,

The nuclear GS is at

The LGPT regions in

The position of the CP in the QvdW model with large degeneracy is close to the position of the CP in the Boltzmann approximation even for smaller values of

The so-called noncongruent LGPTs occur for

Dependence on the asymmetry parameter of (a) the ground state binding energy and (b) of the baryon density.

Dependence of the binding energy on baryon density at zero temperature for the constant values of the asymmetry parameter. The ground states are represented by the squares.

Figure

Noncongruent LGPT regions (a) in the

While points

The infinitesimal fractions

The phase transformation starts from a gas in point

The mixed phase boundaries are found as the sets of point

The standard equations

Another interesting feature of noncongruent PTs is that the locations of temperature and pressure endpoints (the point with, respectively, maximum

The zoomed in picture of the CP region for the phase diagrams shown in Fig.

Figure

Figure

(a) Temperature of coexistence as a function of local charge fractions in coexistent phases,

(a)

The asymmetry in the gaseous fraction of the mixture is always larger then the asymmetry in the liquid fraction; this is the isospin distillation phenomenon, which is just the equivalent of the strangeness distillation considered in Refs.

The system can be described in the mixed phase for an arbitrary proportion of fractions by introducing an additional parameter

Figure

Figure

Local baryon densities in the gaseous and in the liquid fractions of the mixed phase as functions of the total baryon density for

Figure

Fraction of volume occupied in the mixed phase by liquid,

Figure

Congruent and noncongruent LGPT regions in the

The scaled variances of baryonic and electric charges fluctuations can be calculated as

From Eq.

The scaled variances given by Eq.

The correlation between the baryonic and the electric charges is calculated as follows:

Figure

The scaled variance of the baryonic (a), (b) and of the electric (c), (d) charge in the

Throughout this work we have assumed that the QvdW interaction parameters are the same for proton-proton, proton-neutron, and neutron-neutron interactions; i.e., the isospin dependence of the

In Ref.

The symmetric nuclear matter,

The blue line in Fig.

Dependence of the binding energy on baryon density at zero temperature. The black line is the same as in Fig.

Figures

(a)

The situation is different in the case of a large asymmetry. In the case of isospin-independent interactions, the CP is present for all values of

The comparison between isospin-independent and isospin-dependent parametrizations of the QvdW interactions in Fig.

The noncongruent liquid-gas phase transition in asymmetric nuclear matter with two globally conserved charges is studied within the quantum van der Waals model. The features of those noncongruent phase transitions, such as the continuous phase transformation, a change in the location of the critical point, the separation of the critical point and of the endpoints, and the retrograde condensation, have been analyzed. The magnitudes of these phenomena tend to zero if the composition of the nuclear matter approaches the composition of either the limit of symmetric nuclear matter

The quantum van der Waals model with isospin-dependent interaction parameters, constrained to empirical values of the symmetry energy and its slope, has also been considered. It yields results which are similar to the case of isospin-independent interaction parameters for sufficiently symmetric systems,

The authors thank D. V. Anchishkin, B. I. Lev, A. G. Magner, M. Gazdzicki, A. Motornenko, K. Taradiy, A. I. Sanzhur, L. M. Satarov, and G. M. Zinovjev for fruitful discussions and useful comments. This research was supported by a theme grant of the Department of Physics and Astronomy of NAS of Ukraine, “Dynamics of formation of spatially non-uniform structures in many-body systems,” PK 0118U003535. H.S. appreciates the support through the Judah M. Eisenberg Laureatus Chair at Goethe University, and the Walter Greiner Gesellschaft, Frankfurt.