Nuclear matter equation of state including few-nucleon correlations (A\leq 4)

Nuclear matter equation of state including few-nucleon correlations

G. Röpke gerd.roepke@uni-rostock.de Institut für Physik, Universität Rostock, D-18051 Rostock, Germany
July 21, 2019
Abstract

Light clusters (mass number ) in nuclear matter at subsaturation densities are described using a quantum statistical approach. In addition to self-energy and Pauli-blocking, effects of continuum correlations are taken into account to calculate the quasiparticle properties and abundances of light elements. Medium-modified quasiparticle properties are important ingredients to derive a nuclear matter equation of state applicable in the entire region of warm dense matter below saturation density. Moreover, the contribution of continuum states to the equation of state is considered. The effect of correlations within the nuclear medium on the quasiparticle energies is estimated. The properties of light clusters and continuum correlations in dense matter are of interest for nuclear structure calculations, heavy ion collisions, and for astrophysical applications such as the formation of neutron stars in core-collapse supernovae.

pacs:
21.65.-f, 21.60.Jz, 25.70.Pq, 26.60.Kp

I Introduction

We investigate nuclear matter in thermodynamic equilibrium, confined in the volume at temperature . It consists of neutrons (total neutron density ) and protons (total proton density ). We are interested in the subsaturation region where the baryon density with the saturation density fm, the temperature MeV, and proton fraction between 0 and 1. This region of warm dense matter is of interest not only for nuclear structure calculations and heavy ion collisions explored in laboratory experiments Natowitz (), but also in astrophysical applications. For instance, core-collapse supernovae at post-bounce stage are evolving within this region of the phase space Tobias (), and different processes such as neutrino emission and absorption, which strongly depend on the composition of warm dense matter, influence the mechanism of core-collapse supernovae.

In particular, the standard versions LS (); Shen () of the nuclear matter equation of state (EOS) for astrophysical simulations have been improved recently, see SR (); Arcones2008 (); Armen2009 (); Typel (); Gulminelli2010 (); NSE (); Hempel (); Furusawa (); shenTeige (); Providencia (); Avancini (); Jaqaman (); Vergleich (); Hempel2013 (); Gulminelli2013 (); Gulminelli2014 (); NSETabellen (); Hempel2014 (). Here, we will not discuss different approaches but rather contribute to a special question, the treatment of light clusters which is a long-standing problem RMS (). A simple chemical equilibrium of free nuclei is not applicable up to saturation density because medium modifications by self-energy shifts and Pauli blocking become relevant. Concepts such as the heuristic excluded volume approach or in-medium nuclear cluster energies within the extended Thomas-Fermi approach may be applied to heavier clusters but are not satisfactory to describe light clusters that require a more fundamental quantum statistical (QS) approach.

We consider the total proton number density , the total neutron number density , and the temperature as independent thermodynamic variables. Weak interaction processes leading to equilibrium are not considered. The chemical potentials are an alternative to and in characterizing thermodynamic equilibrium of warm dense matter. The relations

(1)

are equations of state that relate the set of thermodynamic quantities to . We give solutions for these EOS for warm dense matter. Further thermodynamic variables are consistently derived after a thermodynamic potential is found by integration, see App. A.

To treat the many-nucleon system (nuclei and nuclear matter) at densities up to saturation, semi-empirical mean-field approaches have been worked out. Based on the Hartree-Fock-Bogoliubov approximation and related quasiparticle concepts such as the Dirac-Brueckner-Hartree-Fock (DBHF) approach for the nuclear matter EOS, see Klahn:2006ir (), semi-empirical approaches such as the Skyrme parametrization Skyrme () or relativistic mean-field (RMF) approaches give an adequate description of the properties of nuclear matter near the saturation density. For a discussion of different versions of these models see, for instance, Ref.  Providencia (); Avancini (); Hempel2014 (). The mean-field potentials may be considered as density functionals that include various correlations, beyond a microscopic Hartree-Fock-Bogoliubov approximation. In this work we use the DD-RMF parametrization according to Typel, Refs. Typel2005 (); Typel1999 (). Other parametrizations of the nucleon quasiparticle energies can alternatively be used to optimize the description of nuclear matter near saturation density.

For strongly interacting nuclear matter considered here, in particular warm dense matter in the low-density region, correlations are important so that a simple mean-field description is not satisfactory. A quantum statistical (QS) approach FW (); AGD (), see Sec. II, can treat the many-particle aspects in a systematic way, using the methods of thermodynamic Green functions, diagram techniques, or path integral methods. A signature of strong correlations is the formation of bound states. In the low-density limit, we can consider the many-nucleon system as an ideal mixture of clusters (nuclei) where the interaction is reduced to accidental (reacting) collisions, leading to chemical equilibrium as given by the mass-action law. This so-called nuclear statistical equilibrium (NSE), see NSE (), has several shortcomings, such as the exclusion of excited states, in particular continuum correlations, and the failure to account for the interaction between the different components (single nucleons as well as nuclei) that is indispensable when approaching saturation density. Both problems are discussed in the present work.

To describe correlations in warm dense matter we restrict our treatment to light elements: deuteron (H), triton (H), helion (He), and (He), in addition to free neutrons () and protons (). The QS approach can be extended to describe further clusters with , see Debrecen (), but is not well worked out for this regime until now. An alternative approach to include heavy nuclei is the concept of the excluded volume (EV), see Hempel (). In a simple semi-empirical approach, the effect of Pauli blocking is replaced by the strong repulsion determined by the excluded volume. The comparison between the EV model and the QS approach HempelRoepke () gives qualitatively similar results, although in the EV model the center of mass motion of clusters is not systematically treated (for instance, effective mass and quantum condensation effects), the light clusters such as the deuteron are not well described by a hard core potential, not depending on the energy, and correlations in the continuum are not considered. Here, we present a theory for nuclear systems under conditions where clusters with are irrelevant, (see Refs. Shen (); Furusawa (); shenTeige (); Vergleich () for an illustration of the parameter space where such regions are shown). Heavier clusters can be included as e.g. done in Ref. SR (), and the combination with the EV model to treat heavier nuclei will be discussed in future work.

Our aim is to describe nuclear matter in the entire region of subsaturation densities, connecting the single-nucleon quasiparticle approach that reproduces the properties near to the low-density limit where a cluster-virial expansion clustervirial () is possible. This is achieved by considering the constituents as quasiparticles with energy which depends not only of the center of mass momentum , but also the set of thermodynamic parameters or, according to the EOS (II.1), . These medium-dependent quasiparticle cluster energies are obtained from an in-medium few-body wave equation derived within a Green-function approach, see Sec. III. More details including fit formulae for the cluster quasiparticle energies are given in R (); R2011 (). We account for the contributions of self-energy and Pauli blocking to the quasi-particle energies that describe the light elements moving in warm dense matter. The Coulomb energy that is screened in dense matter can be omitted for .

The treatment of excited states and continuum correlations leads to a virial expansion Huang () that describes rigorously the low-density limit. The generalized Beth-Uhlenbeck approach SRS () allows to implement both limits, the low-density NSE with the virial expansion as benchmark HS (), and the behavior near saturation density, where the quasi-particle concept for the nucleons is applicable. The low density region, where the contribution of the continuum correlations to the virial expansion has to be taken into account, was investigated recently in context with the generalized RMF approach VT (). The bound states (light elements) gradually disappear due to Pauli blocking before saturation density is reached. Introducing the quasiparticle concept, we have to be careful to avoid double counting because part of the continuum correlations is already implemented in the quasiparticle energy shift, see clustervirial (). We give expressions for the remaining residual continuum correlations that are extrapolated to higher densities in Sec. IV.

Another problem is the treatment of correlations in the medium. Although the formalism has been worked out RMS2 (); cmf (); schuckduk (), the resulting cluster mean-field equations have not yet been solved in a self-consistent way. Comparing with the ideal Fermi distribution of free nucleons, the occupation of the phase space is changed if correlations in the medium are taken into account. We propose a simple parametrization to improve on the Fermi distribution of the ideal nucleon gas. Results are given in Sec. V, and some general issues that need to be resolved to devise an improved EOS are discussed in the final Sec. VI.

Ii Green functions approach and quasiparticle concept

ii.1 Cluster decomposition of the equation of state

The nuclear matter EOS ( is the system volume, )

(2)

is obtained from the spectral function which is related to the self-energy, see FW (); AGD ():

(3)

The single-nucleon quantum state can be chosen as which denotes wave number, spin, and isospin, respectively. The EOS (2) relates the total nucleon numbers or the particle densities to the chemical potentials of neutrons or protons so that we can switch from the densities to the chemical potentials. On the other hand, if this EOS is known in some approximation, all other thermodynamic quantities are obtained consistently after changing over to a thermodynamic potential, see App. A.

The spectral function and the corresponding correlation functions are quantities, well-defined in the grand canonical ensemble characterized by . The self-energy depends, besides the single-nucleon quantum state , on the complex frequency and is calculated at the Matsubara frequencies. Within a perturbative approach it can be represented by Feynman diagrams. A cluster decomposition with respect to different few-body channels () is possible, characterized, for instance, by the nucleon number , as well as spin and isospin variables.

Using the cluster decomposition of the self-energy which takes into account, in particular, cluster formation, we obtain

(4)

where denotes the center of mass (c.o.m.) momentum of the cluster (or, for , the momentum of the nucleon). The internal quantum state contains the proton number and neutron number of the cluster,

(5)

is the Bose or Fermi distribution function for even or odd , respectively, that is depending on . The integral over is performed within the quasiparticle approach, the quasiparticle energies are depending on the medium characterized by . These in-medium modifications will be detailed in the following sections II.2, II.3, and III.

We analyze the contributions of the clusters (), suppressing the thermodynamic variables . We have to perform the integral over the c.o.m. momentum what, in general, must be done numerically since the dependence of the in-medium quasiparticle energies on is complex. The summation over concerns the bound states as far as they exist, as well as the continuum of scattering states. Solving the few-body problem what is behind the -nucleon T matrices in the Green function approach, we can introduce different channels () characterized, e.g., by spin and isospin quantum numbers. This intrinsic quantum number will be denoted by , and we have in the non-degenerate case

(6)

with the degeneration factor in the channel . The partial density of the channel at

(7)

contains the intrinsic partition function. It can be decomposed in the bound state contribution and the contribution of scattering states .

The summations of (7) over and remain to be done for the EOS (II.1), and may be included in . The region in the parameter space, in particular , where bound states exist, may be restricted what is expressed by the step function else. The continuum edge of scattering states is denoted by , see Eq. (21) below.

For instance, in the case the deuteron is found in the spin-triplet, isospin-singlet channel () as bound state. In addition, in the same channel we have also contributions from scattering states, i.e. continuum contributions, characterized by the relative momentum as internal quantum number. Thus, in the isospin-singlet (spin-triplet) channel of the two-particle case () we have in the zero density limit the deuteron as bound state [, , MeV], and according to the Beth-Uhlenbeck formula Huang (), see RMS (),

(8)

with being the baryon thermal wavelength (the neutron and proton mass are approximated by MeV), and the isospin-singlet () scattering phase shifts with angular momentum as function of the energy of relative motion. A similar expression can also be derived for the isospin-triplet channel (e.g. two neutrons) where, however, no bound state occurs, see also HS () where detailed numbers are given.

At this point we mention that the NSE is recovered if the summation over is restricted to only the bound states (nuclei), neglecting the contribution of correlations in the continuum. Furthermore, for the bound state energies of the isolated nuclei are taken, neglecting the effects of the medium such as mean-field terms or contributions due to correlations in the medium. Both aspects will be investigated in the present work. In contrast to the semi-empirical treatment of the medium contribution to the EOS within the excluded volume concept, see Hempel (), we use the Pauli blocking terms within a systematic quantum statistical approach.

Note that the subdivision (7), (8) into a bound state contribution and a contribution of continuum states is not unique. We will use another decomposition which follows after performing an integration by parts, see Eq. (36) below.

ii.2 Cluster mean-field approximation

To go to finite densities, the main problem is the medium modification of few-body properties which defines also in the contribution (6) of the different components. Quasiparticles are introduced considering the propagation of few nucleon cluster (including ) in warm dense matter. The Green function approach describes the propagation of a single nucleon by a Dyson equation governed by the self-energy, and the few-particle states are obtained from a Bethe-Salpeter equation containing the effective interaction kernel. Both quantities, the effective interaction kernel and the single-particle self-energy, should be approximated consistently. Approximations which take cluster formation into account have been worked out RMS2 (); cmf (); clustervirial (), the cluster mean-field approximation is outlined in App. B.

For the -nucleon cluster, the in-medium Schrödinger equation

(9)

is derived from the Green function approach. This equation contains the effects of the medium in the single-nucleon quasiparticle shift

(10)

(nonrelativistic case), as well as in the Pauli blocking terms given by the occupation numbers in the phase space of single-nucleon states . Thus, two effects have to be considered, the quasiparticle energy shift and the Pauli blocking.

In the lowest order of perturbation theory with respect to the nucleon-nucleon interaction , the influence of the medium on the few-particle states () is given by the Hartree-Fock shift

(11)

and, consistently, the Pauli blocking terms

(12)

for , see (B), (56) neglecting the contributions with .

Both terms have a similar structure, besides the nucleon-nucleon interaction the single-nucleon Fermi distribution occurs. In this simplest approximation, only the free nucleons contribute to the self-energy shift and the Pauli blocking. The distribution function is the Fermi distribution with the parameter set .

It is obvious that also the nucleons found in clusters contribute to the mean field leading to the self-energy, but occupy also phase space and contribute to the Pauli blocking. The cluster mean-field approximation which considers also the few-body T matrices in the self-energy and in the kernel of the Bethe-Salpeter equation leads to similar expressions (11), (12) but the free-nucleon Fermi distribution replaced by the effective occupation number (57)

(13)

with contains also the distribution function for the abundance of the different cluster states and the respective wave functions . (The variable has not been given explicitly.) For the quantum statistical derivation see the references given in App. B.

Because the self-consistent determination of for given is very cumbersome, we consider appropriate approximations. In particular, we use the Fermi distribution with new parameters (effective temperature and chemical potentials),

(14)

These effective parameters allow to reproduce some moments of the occupation number distribution. For instance, besides the normalization (total neutron/proton number)

(15)

also

(16)

can be used to fix the values of as functions of .

This ansatz contains the special case where the medium is described by non-interacting single-nucleons states. Then, the effective parameter values coincide with . We come back to the parametrization, Eq. (14), where are functions of according to (15) and (16), below in Sec. IV.6.

ii.3 Medium modification of few-body properties in warm dense matter

For the -nucleon cluster, the in-medium Schrödinger equation (9) is derived, depending on the occupation numbers of the single-nucleon states . As a consequence, the solutions (the energy eigenvalues and the wave functions) will also depend on the parameters which characterize the occupation numbers.

We change the parameter introducing new variables according (14) to characterize the occupation number distribution. We calculate the eigenvalues as function of these new variables that are related to the original variables . We go a step further and switch from the chemical potentials to densities so that we use as variables to characterize the occupation number distribution,

(17)

The tilde denotes a Fermi distribution as a function of densities instead the chemical potentials. Implicitly, depends on the effective parameter values so that the normalization holds, i.e. are the solutions of the normalization conditions (15).

The use of and realizes that all nucleon participate in the phase space occupation [see (13)], and is a further parameter the takes into account the formation of correlations in the medium, see Eq. (42) given below. As a consequence, the solutions (the energy eigenvalues and the wave functions) will also depend on the parameters which now characterize the occupation numbers, but are, in principle, functions of .

To evaluate the dependence of the cluster energy eigenvalues on we solve the in-medium Schrödinger equation

(18)

obtained from (9) replacing the occupation numbers by a Fermi distribution . This equation contains the effects of the medium in the single-nucleon quasiparticle shift , Eq. (10), as well as in the Pauli blocking terms given by the occupation numbers in the phase space of single-nucleon states .

Obviously the bound state wave functions and energy eigenvalues as well as the scattering phase shifts become dependent on the effective temperature and the densities . In particular, we obtain the cluster quasiparticle shifts

(19)

with the free contribution . Expressions for the in-medium self-energy shift and Pauli blocking are given in Sec. III.2 and App. D below. We added the Coulomb shift due to screening effects which can be approximated by the Wigner-Seitz expression. For the light elements with considered here, the Coulomb corrections are small compared with the other contributions and are omitted.

Of special interest are the binding energies

(20)

with

(21)

that indicate the energy difference between the bound state and the continuum of free (scattering) states at the same total momentum . This binding energy determines the yield of the different nuclei according to Eq. (4), where the summation over is restricted to that region where bound states exist, i.e. .

In addition, the continuum states solving Eq. (18) are also influenced by the medium effects, but the results are less obvious. We give some estimations in Sec. IV.

Iii Quasiparticle contributions to the EOS

We analyze the contributions of different mass numbers to the EOS (II.1). The single-nucleon contribution is extensively discussed, the quasiparticle picture is well elaborated and broadly applied. An exhaustive discussion of the two-nucleon contribution () has been given in Ref. SRS (). Besides the bound state part, also the scattering states have been treated. A generalized Beth-Uhlenbeck equation has been considered where not only the low-density limit (second virial coefficient) is correctly reproduced, but also the mean-field terms are consistently included avoiding double counting. The results can be applied to finite densities of warm dense matter up to saturation density. Correlations in the medium have been neglected so that the effective occupation numbers (14) are approximated by .

We are interested in including all light clusters (). Besides the cluster-virial expansion in the low-density limit, the medium modifications raising up with increasing density are of interest, and the effect of correlations in the medium is discussed. The behavior of bound states has been investigated in previous work R (); R2011 (), and some usable results are available, but the continuum contributions remain until now very difficult to treat. We give some estimations and simple interpolation formulae.

In the present work, the sum over will be restricted to (light elements). The contribution of heavier clusters to the EOS is not considered in calculating the EOS. This confines the region of thermodynamic parameters to that region in the phase space where heavier clusters are not of relevance, see, e.g., Shen (); Furusawa (); shenTeige (); Vergleich (). Heavier clusters may be included, however, different considerations (such as the excluded volume concept) in addition to the NSE have to be performed to treat higher densities. Besides some QS calculations Debrecen (), mostly the excluded volume concept, see LS (); Shen (); Hempel (), is used to give a semi-empirical treatment of the medium-modified contribution of heavier clusters to the EOS.

iii.1 Single-nucleon quasiparticle approximation

Before improving the low-density limit of the EOS considering NSE and the cluster-virial expansion, we discuss the influence of the medium what is unavoidable to describe warm dense matter up to saturation density. We consider the approximation of the EOS (II.1) where only the single-nucleon contributions are taken, i.e. the sum over is reduced to which contains the neutron () and proton () quasiparticle contribution to the EOS.

In the quasi-particle approximation, the imaginary part of is neglected in (3). The spectral function is like, and the densities are calculated from Fermi distributions with the single-nucleon quasiparticle energies so that (spin factor 2)

(22)

The quasiparticle approximation is well elaborated in nuclear physics, see Talmi (); RS (). Starting from a microscopic approach with suitable nucleon-nucleon interaction potentials, standard approximations for the single-nucleon self-energy shift are the Hartree-Fock-Bogoliubov or the Dirac-Brueckner-Hartree-Fock approximation, see Sec. I. In the spirit of the density-functional approach, semi-empirical expressions such as the Skyrme forces or relativistic mean-field approaches have been worked out. The relativistic quasiparticle energy

(23)

gives in the non-relativistic limit and . Explicit expressions for and in form of Padé approximations which are suitable for numerical applications, are given in Appendix C. They are obtained from the DD-RMF parametrization of Typel Typel2005 () and can be replaced by alternative parametrizations Providencia (); Avancini (); Hempel2014 ().

Fitted to properties near the saturation density, the description of warm dense matter at densities near is adequate. No cluster formation can be described in the single-nucleon quasiparticle (mean field) approach. We have to go beyond this approximation and have to treat the imaginary part of in (3) to include cluster formation and to reproduce the correct low-density limit.

iii.2 Shifts of light cluster binding energies in dense matter

Now we come back to the EOS (II.1) and add the contributions of clusters with . We consider the bound state parts. In the low-density limit we use the empirical binding energies , see below Tab. 1, as also used in the NSE. In the case of there is no excited bound state above the ground state. In the case of , binding energy = 28.3 MeV, there exists an excited state with excitation energy 20.2 MeV to be included into the partial density (7) which also contains the contribution of scattering states.

Going to finite densities, the eigenvalues (quasiparticle energies of the light clusters) depend, in the last consequence, on the temperature and chemical potentials of the nuclear matter, as derived from the in-medium wave equation (9). The light clusters are considered as quasiparticles with dispersion relation depending on the single-nucleon occupation number which is parametrized by a Fermi distribution (17) with effective parameter values .

We discuss the contribution to the dispersion relation of the cluster quasiparticles according to (19). The most significant medium effect is the Pauli blocking which is also strongly dependent on temperature. The Pauli blocking shift of the binding energies , see Eq. (63), has been evaluated within a variational approach. The results are presented in R2011 (), and a parametrization has been given which allows to calculate the medium modification of the bound state energies with simple expressions in good approximation, see Eq. (14) of Ref. R2011 (). We collect the results for the Pauli-blocking medium shifts of the bound state energies in App. D.

The contribution of the single nucleon energy shift to the cluster self-energy shift is easily calculated in the effective mass approximation, where the single-nucleon quasiparticle energy shift

(24)

can be represented by the energy shift and the effective mass . We use the empirical value

(25)

In the rigid shift approximation where , the self-energy shift cancels in the binding energy because the continuum is shifted by the same value. It can be absorbed in the chemical potential of the EOS (4).

In general, in the kinetic part of the the wave equation (18) which consists of the single-nucleon quasiparticle energies , we can introduce the c.o.m. momentum and the intrinsic motion described by Jacobi coordinates. In the effective mass approximation, the separation of the c.o.m motion is simple because the single-particle dispersion relations are quadratic. The self-energy shift consists of the c.o.m. part which coincides with the edge of the continuum (21) for the intrinsic motion, and the intrinsic part

(26)

The intrinsic part of the cluster self-energy shift is easily calculated for given wave functions SR (); R (), see also Typel (), within perturbation theory. Values for for the light elements are given below in Tab. 5. It results as the averages of for , for , and for , where denote the respective Jacobian momenta R ().

We introduce the intrinsic part of the bound state energies as

(27)

With Eq. (20), the intrinsic parts of the bound state energies are the negative values of the binding energies, .

iii.3 Mott points

A consequence of the medium modification is the disappearance of bound states with increasing density what is of significance for the physical properties. To calculate the composition one has to check for given parameter values whether the binding energy of the cluster with quantum numbers is positive. We denote the density as Mott density where the binding energy of a cluster with c.o.m. momentum vanishes, with (21), (27)

(28)

(Note that is determined by , see Eq. (42) below.) For baryon densities we can introduce the Mott momentum , where the bound state disappears,

(29)

At , the summation over the momentum to calculate the bound state contribution to the composition is restricted to the region .

Crossing the Mott point by increasing the baryon density, part of correlations survive as continuum correlations so that the properties change smoothly. Therefore, the inclusion of correlations in the continuum is of interest.

Iv Virial expansion and correlated medium

In the low-density limit, rigorous expressions for the EOS are obtained for the virial expansion. The second virial coefficient is related to experimental data such as the bound state energies and scattering phase shifts, according to the Beth-Uhlenbeck formula Huang (). The application to nuclear matter RMS (); HS () as well as the generalized Beth-Uhlenbeck formula SRS () and the cluster-virial expansion Arcones2008 (); clustervirial () allow for the account of continuum correlations for the EOS.

The virial coefficients are determined also by continuum correlations and are neglected in the simple NSE. However, in particular for the deuteron contribution where the binding energy is small, the account for the correct second virial coefficient is of relevance, see the comparison of quantum statistical with generalized RMF calculations in Typel (). A detailed description of the virial expansion in the context of a RMF treatment has been given by Voskresenskaya and Typel VT (). We are interested in the extension of the virial expansion to higher densities up to . For the two-nucleon case rigorous results can be given, whereas for the treatment of higher order correlations only some estimations can be made.

iv.1 Two-nucleon contribution

The virial expansion of the EOS (II.1) reads RMS (); SRS (); HS (); VT (); Huang ()

(30)

Already the noninteracting, i.e. ideal Fermi gas of nucleons contains two effects in contrast to the standard low-density, classical limit:
i) The relativistic dispersion relation results in a first virial coefficient where the value follows from the dispersion relation . For a more detailed investigation see VT ().
ii) The degeneration of the fermionic nucleon gas leads to the contribution to Huang ().

The remaining part of the second virial coefficient is determined by the two-nucleon interaction. We can introduce different channels, in particular the isospin triplet (, neutron matter) and isospin singlet (, deuteron) channels which are connected with the spin singlet and spin triplet state, respectively, if even angular momentum is considered, e.g. S-wave scattering. The second virial coefficient in both channels can be derived from and . Empirical values are given as function of in HS () (isospin symmetry is assumed).

iv.2 Generalized Beth-Uhlenbeck formula

The second virial coefficients and cannot directly used within a quasiparticle approach. Because part of the interaction is already taken into account when introducing the quasi-particle energy, we have to subtract this contribution from the second virial coefficient to avoid double counting, see clustervirial (); SRS (); VT (). We expand the density in the quasiparticle approximation picture (22), (23) with respect to the fugacities. We identify the residual isospin-triplet contribution from the neutron matter case as

(31)

and the residual isospin-singlet contribution from the symmetric matter case according to

(32)

dots indicate higher orders in densities. The residual second virial coefficients are given by SRS ()

(33)

Compared with the ordinary Beth-Uhlenbeck formula (8) there are two differences:
i) After integration by parts, the derivative of the scattering phase shift is replaced by the phase shift, and according to the Levinson theorem for each bound state the contribution appears.
ii) The contribution