Momentum and density dependence of the isospin part of nuclear mean field and equation of state of asymmetric nuclear matter

Nuclear Physics A 753 (2005) 367–386

Momentum and density dependence of the isospin part of nuclear mean field and equation of state of asymmetric nuclear matter B. Behera a , T.R. Routray a,∗ , A. Pradhan a , S.K. Patra b , P.K. Sahu b a Department of Physics, Sambalpur University, Jyoti Vihar, Burla, Sambalpur, Orissa PIN-768019, India b Institute of Physics, Sachivalaya Marg, Bhubaneswar, Orissa PIN-751005, India

Received 12 July 2004; received in revised form 2 March 2005; accepted 3 March 2005 Available online 23 March 2005

Abstract Momentum and density dependence of the isospin part of nuclear mean field uτ (k, ρ) which is still, in part, the open problem of the old Lane potential is analysed using density dependent finite range effective interactions. The behaviour of uτ (k = kf , ρ) around the Fermi momentum kf is found to be related to the density dependence of nuclear symmetry energy Jτ (ρ) and nucleon effective mass M ∗ (k = kf , ρ)/M in symmetric nuclear matter. The momentum dependence of uτ (k, ρ) is separated out in terms of a simple functional uex τ (k, ρ) which vanishes at k = kf and involves only the finite range parts of the exchange interactions between pairs of like and unlike nucleons. Depending on the choice of the parameters of these exchange interactions two conflicting trends of momentum dependence are noticed which lead to two opposite types of splitting of neutron and proton effective masses. The equation of state of asymmetric nuclear matter and the high density behaviour of nuclear symmetry energy Jτ (ρ) are studied by constraining the additional parameters involved on the basis that pure neutron matter should not be predicted to be bound by any reasonable nuclear interaction. Emphasis is also given on the need of experimental data sensitive to the differences between neutron and proton transport properties in highly asymmetric dense nuclear matter and its analysis to constrain the high density behaviour of nuclear symmetry energy as well as to resolve the controversy on the two opposite types of splitting of neutron and proton effective masses.  2005 Published by Elsevier B.V.

* Corresponding author.

E-mail addresses: [email protected], [email protected] (T.R. Routray). 0375-9474/$ – see front matter  2005 Published by Elsevier B.V. doi:10.1016/j.nuclphysa.2005.03.002

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PACS: 21.65.+f Keywords: Isospin; Asymmetric nuclear matter; Mean field; Lane potential; Neutron and proton effective masses; Symmetry energy; Finite range effective interactions; Beta-stable matter

1. Introduction The equation of state (EOS) of nuclear matter under extreme conditions has become a major area of research in nuclear physics in the past decade. Currently an increasing attention is being given to the study of EOS of highly isospin asymmetric dense nuclear matter because of its implications and connections beyond standard nuclear physics, such as, astrophysical phenomena like supernova explosion and neutron star structure. The most important quantity in the calculation of EOS of asymmetric nuclear matter (ANM) is the isospin part of the nuclear mean field uτ (k, ρ) as function of momentum k and total nucleon density ρ = ρn + ρp . It is defined as the difference between zerotemperature neutron and proton mean fields, un (k, ρ, Yp ) and up (k, ρ, Yp ), as functions of k, ρ and the proton fraction Yp = ρp /ρ normalized to the asymmetry (1 − 2Yp ), uτ (k, ρ) =

un−p (k, ρ, Yp ) un (k, ρ, Yp ) − up (k, ρ, Yp ) = . 2(1 − 2Yp ) 2(1 − 2Yp )

(1)

In other words it is just the famous Lane potential; υ1 = 4uτ (k, ρ) [1]. A study of momentum and density dependence of uτ (k, ρ) is a very important topic for fundamental reasons, better knowledge of the isospin part of the in medium interaction as well as for the new experimental possibilities offered by the radioactive beam facilities. The growing experimental facilities for the study of nuclear reactions induced by high energy intense radioactive ion beams and their analysis in terms of transport model calculations have provided an opportunity for the first time to explore the EOS of highly asymmetric dense nuclear matter experimentally [2–12]. However, the progress in this direction seems to be rather low and the momentum and density dependence of uτ (k, ρ) is still largely unknown [6,7,9–12] particularly at high density and at high momentum. The various theoretical approaches used in the study of EOS of ANM include Dirac– Brueckner–Hartree–Fock (DBHF) calculations [13–20], Brueckner–Hartee–Fock (BHF) approximation to Brueckner–Bethe–Goldstone (BBG) calculations [21–24] and variational methods [25–27]. Besides these microscopic approaches, effective theories such as relativistic mean field (RMF) approximations [28–37] and non-relativistic effective interactions [38–44] have also been used extensively to study the EOS and mean field properties of asymmetric nuclear matter. In the simulations of dynamical evolution of high energy heavy ion collisions using isospin dependent transport models, usually very simple parametrizations of the energy density and mean fields are used [2–4,9–12]. The momentum and density dependence of the isospin part of nuclear mean field uτ (k, ρ) predicted by these different theoretical model calculations are rather extremely divergent and even contradicting. All theoretical results on the behaviour of uτ (k, ρ) around the Fermi surface k = kf which is directly connected with the density dependence of nuclear symmetry energy Jτ (ρ) can be roughly classified into two groups, i.e., a group

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where Jτ (ρ) increases monotonically and another where it decreases after attaining a maximum value and then ultimately becomes negative with increasing density. The high density behaviour of Jτ (ρ) is very important for the understanding of many astrophysical phenomena, such as, chemical composition and related mechanisms in the process of formation of neutron stars [45–53]. On the other hand, all theoretical predictions on the momentum dependence of uτ (k, ρ) can also be classified into two distinct opposite groups depending on whether it decreases or increases with increase in momentum k. An increasing trend of uτ (k, ρ) with momentum implies proton effective mass going above the neutron one and the other way around for a decreasing trend. The conflicting predictions regarding high density behaviour of Jτ (ρ) and momentum dependence of uτ (k, ρ) mentioned above are not interlinked with each other, rather the two problems are clearly distinct. The purpose of the present work is to analyse the momentum and density dependence of the isospin part of nuclear mean field uτ (k, ρ) and to focus on the existing controversies in the high density behaviour of nuclear symmetry energy as well as the two opposite types of splitting of neutron and proton effective masses using a simple parametrization of the energy density in ANM based on density dependent finite range effective interactions which were used to investigate the mean field properties and EOS of symmetric nuclear matter (SNM) in our earlier work [54]. The simplicity of the energy density allows analytical calculations of the isospin part of nuclear mean field and other properties of ANM with a minimum number of adjustable parameters and can, therefore, provide a physical insight as to how these different properties are connected with each other. In Section 2, we give a brief account of mean field properties and EOS of SNM investigated in our earlier work. The extension of the formalism to the case of asymmetric nuclear matter and the analysis of momentum and density dependence of the isospin part of nuclear mean field is given in Section 3. In Section 4 we focus on the EOS of pure neutron matter (PNM) to constrain the additional adjustable parameters involved. In the absence of any experimental/empirical constraints, we look for simple physical considerations to constrain the EOS of PNM for this purpose. The high density behaviour of the nuclear symmetry energy is also given in this section. Section 5 contains a brief summary and conclusion on the work presented in this paper.

2. Equation of state of symmetric nuclear matter Our starting point is a simple density dependent finite range effective interaction
γ ρ(R) t3 δ(r) veff (r) = t0 (1 + x0 Pσ )δ(r) + (1 + x3 Pσ ) 6 1 + bρ(R) + (W + BPσ − H Pτ − MPσ Pτ )f (r)

(2)

which was used in an earlier work [54] to study the EOS as well as mean field properties of SNM. Here f (r) represents a short range interaction of conventional form, such as, Yukawa, Gaussian or exponential and is specified by a single parameter α, the range of the interaction. The other symbols in Eq. (2) have their usual meanings. The energy density HT (ρ) and the single particle energy T (k, ρ) in SNM derived from this effective interaction can be written as

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 γ 1/2 3  εγ ρ ε0 ρ 2 fT (k) c2 h¯ 2 k 2 + M 2 c4 d k+ + γ +1 ρ2 2ρ0 1 + bρ 2ρ0    εex + fT (k)fT (k )gex |k − k | d3 k d3 k  , 2ρ0

 HT (ρ) =

and

 γ +1   2 2 2 εγ ρ ε0 ρ 2 4 1/2 T (k, ρ) = c h¯ k + M c + + γ +1 (1 + bρ + γ /2) ρ0 1 + bρ ρ0      εex + fT k gex |k − k | d3 k  , ρ0

(3)

(4)

where, ρ0 is the normal nuclear matter density and gex (k) is the normalized Fourier transform of the finite range interaction f (r)  ik·r e f (r) d3 r . (5) gex (k) =  f (r) d3 r The connection of new parameters ε0 , εγ , and εex with the interaction parameters are given in Ref. [54]. In Eqs. (3) and (4) we have used the relativistic relation between kinetic energy and momentum to take account of possible relativistic effects that may arise at high temperature T and/or high density ρ. The denominator (1 + bρ) in the density dependent term is necessary to prevent the supraluminous behaviour in nuclear matter at very high density [55]. This modification has also the advantage of softening the EOS at high density [56] in better agreement with the flow data. The single particle momentum distribution function fT (k) is subject to the constraint  fT (k) d3 k = ρ. (6) At non-zero temperature (T = 0), fT (k) depends on the single particle energy T (k, ρ) and Eq. (4) implies a self-consistent calculation of fT (k). However, at zero-temperature the single particle momentum distribution function takes the form of a step function and the exchange integrals involving fT (k) in Eqs. (3) and (4) can be calculated analytically for all the three types of conventional interactions, namely, Yukawa, Gaussian and exponential. The EOS of SNM which is completely described in terms of the energy density and the single particle energy contains all together six adjustable parameters, namely, ε0 , εγ , γ , b, εex and α. As discussed in our earlier work [54,57], the range α and the strength parameter εex must be determined by requiring to give a reasonable account of momentum dependence of the zero-temperature mean field in SNM at normal density ρ0 over a wide range of momentum as demanded by optical model fits to nucleon-nucleus scattering data [58]. The parameter b appearing in the εγ term in Eqs. (3) and (4) is constrained by requiring to prevent the supraluminous behaviour of zero-temperature SNM at high densities [54,55]. In the present work we have assumed standard values; Mc2 = 939 MeV, energy per particle in zero-temperature SNM, e(ρ0 ) = 923 MeV and (c2 h¯ 2 kf20 + M 2 c4 )1/2 = 976 MeV (corresponding to ρ0 = 0.1658 fm−3 ). The values of the two parameters εex and α obtained through an optimization procedure for the Yukawa form of the exchange interaction are εex = −121.8 MeV and α = 0.4044 fm which gives an effective nucleon mass

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Fig. 1. The energy per particle e(ρ) in SNM as a function of density ρ for γ = 1/12, 1/6, 1/3 and 1/2. The empirical region of saturation is depicted by the Coester rectangle.

M ∗ (k = kf0 , ρ0 )/M = 0.67 at the Fermi surface corresponding to normal nuclear matter density ρ0 . It may be mentioned here that in this work we will be restricting only to the Yukawa form of the exchange interaction. The rest three parameters ε0 , εγ and γ are determined from the saturation conditions and the value of nuclear matter incompressibility K(ρ0 ). Unlike the earlier work [54] where γ was constrained to give K(ρ0 ) = 210 MeV, we now restrict it between 1/12 and 1/2 to give an incompressibility K(ρ0 ) in the range 190-240 MeV to study its influence on the EOS of PNM in Section 4. The energy per particle e(ρ) in SNM is shown in Fig. 1 as a function of density ρ for several values of the parameter γ between 1/12 and 1/2. The saturation point is depicted by the Coester rectangle [59] in the same figure for comparison. The saturation curves for all the γ values mentioned above are quite the same inside the depicted rectangle and pass through the middle of the allowed empirical region. Recently Danielewicz et al. [4] have assessed the pressure–density relationship in zerotemperature SNM in the density range 2
ρ/ρ0
4.6 from an analysis of experimental data on flow of matter in high energy heavy-ion collisions. This experimental determination of nuclear EOS has provided for the first time an important test for all theoretical models that extrapolate the EOS from the properties of finite nuclei near normal density and from nucleon–nucleon scattering, in terms of laboratory measurements of high density matter. The pressure–density relationship in zero-temperature SNM assessed from analysis of flow of matter in high energy heavy-ion collisions is depicted in Fig. 2 by the bounded region in the density range 2
ρ/ρ0
4.6. In the same figure, the calculated pressure–density relationships for different values of the parameter γ in the range 1/12 and 1/2 are also shown for comparison. All the calculated EOS agree quite well with the experimental EOS. In particular the EOS corresponding to γ = 1/6 and 1/3 pass almost along the middle of the experimentally allowed region. However, it also appears that an increase in γ beyond the value 1/2 will raise the pressure–density curves more and more above the bounded region around the density ρ = 2ρ0 and the incompressibility in the present parametrization of the EOS of SNM is restricted to K(ρ0 ) < 240 MeV.

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Fig. 2. Pressure–density relationships for four different EOS of SNM with γ = 1/12, 1/6, 1/3 and 1/2 compared with the experimental EOS of Danielewicz et al. [4] depicted by the shaded region.

3. Momentum and density dependence of the isospin part of nuclear mean field in asymmetric nuclear matter In this section, we extend the simple parametrizations of energy density and single particle energy in SNM given in the previous section to analyse the EOS of asymmetric nuclear matter. In ANM we have two different densities ρn , ρp and correspondingly two differp ent single particle momentum distribution functions fTn (k) and fT (k). As a result there will be separate contributions to the kinetic energy density from neutrons as well as protons. Similarly, the potential energy density will now split into three parts corresponding to contributions arising out of proton–proton, neutron–neutron and neutron–proton interactions. In this connection we note that the parameters α, γ and b are fixed from the zero-temperature EOS of SNM and therefore the strength parameters ε0 , εγ and εex will l (ε ul ) for interactions between now split into two channels namely, ε0l (ε0ul ), εγl (εγul ) and εex ex like (unlike) nucleons, subject to the conditions ε0l + ε0ul = 2ε0 ,

εγl + εγul = 2εγ ,

l ul εex + εex = 2εex .

(7)

The different strength parameters for interactions between like and unlike nucleons appearing in this equation can be easily connected with the parameters of the effective interaction given in Eq. (2). With these parameters the energy density in asymmetric nuclear matter HT (ρn , ρp ) can now be expressed in terms of neutron and proton densities and their p respective single particle momentum distribution functions fTn (k) and fT (k) as HT (ρn , ρp )  n 1/2 3

 p = fT (k) + fT (k) c2 h¯ 2 k 2 + M 2 c4 d k +


γ  γ   
ul εγl εγul  2  ε ρ ρ 1 ε0l ρn + ρp2 + 0 + γ +1 ρn ρp + γ +1 2 ρ0 ρ 1 + bρ ρ0 1 + bρ ρ0 0

B. Behera et al. / Nuclear Physics A 753 (2005) 367–386 l  

  εex p p fTn (k)fTn (k ) + fT (k)fT (k ) gex |k − k | d3 k d3 k  2ρ0 ul  

  εex p p + fTn (k)fT (k ) + fT (k)fTn (k ) gex |k − k | d3 k d3 k  . 2ρ0

373

+

(8)

At zero temperature (T = 0), the neutron and proton single particle momentum distribution functions take the form of step functions and the corresponding energy density H (ρn , ρp ) can be calculated analytically. n,p The neutron and proton single particle energies T (k, ρn , ρp ) can be obtained as the respective functional derivatives of the energy density HT (ρn , ρp ) in Eq. (8) and are given by 1/2  n,p n,p T (k, ρn , ρp ) = c2 h¯ 2 k 2 + M 2 c4 + uT (k, ρn , ρp ). (9) In this equation the first term is the kinetic energy of the neutron or proton under considern,p ation and uT (k, ρn , ρp ) are the respective mean fields. The more important quantity for our present purpose is the difference in the mean fields between a neutron and a proton having the same kinetic energy

n−p p (10) uT (k, ρ, Yp ) = unT (k, ρ, Yp ) − uT (k, ρ, Yp ) , expressed as a function of momentum k, total nucleon density ρ and proton fraction Yp at n−p given temperature T . The functional uT (k, ρ, Yp ) obtained from the energy density in Eq. (8) can be written as   l n−p ul GT (k, ρ, Yp ), − εex (11) uT (k, ρ, Yp ) = (1 − 2Yp )ρA(ρ) + εex where

 γ (ε0l − ε0ul ) (εγl − εγul ) ρ A(ρ) = + γ +1 ρ0 1 + bρ ρ0

and GT (k, ρ, Yp ) =

1 ρ0



  n  p fT (k ) − fT (k ) gex |k − k | d3 k  .

(12)

(13)

n−p

The first part of uT (k, ρ, Yp ) is independent of temperature and momentum and is directly proportional to (1 − 2Yp ), the proportionality factor depending only on the total nucleon density ρ. On the other hand, the dimensionless functional GT (k, ρ, Yp ) appearing in the second part has a very complicated dependence on temperature T , momentum k, total nucleon density ρ and proton fraction Yp . However, the functional GT (k, ρ, Yp ) at zero-temperature considerably simplifies and Eq. (11) can be expressed as   l ul G(k, ρ, Yp ), − εex (14) un−p (k, ρ, Yp ) = (1 − 2Yp )ρ A(ρ) + εex where G(k, ρ, Yp ) =

ρ0



1 f (r) d3 r


ρn

 3j1 (kp r) 3j1 (kn r) − ρp j0 (kr)f (r) d3 r. kn r kp r

(15)

Here j0 and j1 are spherical Bessel functions of order 0 and 1, and kn , kp are neutron, proton Fermi momenta corresponding to densities ρn and ρp , respectively.

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Fig. 3. Momentum dependence of the functional G(k, ρ, Yp ) given in Eq. (15) is shown at three different densities ρ = ρ0 , 3ρ0 and 5ρ0 for Yp = 0.1 and compared with its asymptotic and Taylor expansion versions in Eqs. (16) and (17), respectively.

The functional GT (k, ρ, Yp ) in Eq. (13) as well as its zero-temperature version G(k, ρ, Yp ) in Eq. (15) have a complicated dependence on the proton fraction Yp which can be simplified in two situations. In the limit of large momenta k, so that gex (|k − k |) appearing inside the integral in Eq. (13) can be approximated by gex (k), the functional GT (k, ρ, Yp ) simplifies to ρ GT (k, ρ, Yp ) large gex (k). (16)
k (1 − 2Yp ) ρ0 This result is quite important in the sense that the asymptotic behaviour of GT (k, ρ, Yp ) for large k is independent of temperature and is directly proportional to the asymmetry parameter (1 − 2Yp ), the proportionality factor being a simple function of momentum k and total nucleon density ρ. The asymptotic behaviour of G(k, ρ, Yp ) at zero-temperature for large momenta k is also given by the same result in Eq. (16) and is valid for k  kn , kp . However, the actual values of k beyond which the asymptotic result in Eq. (16) will be valid depend on the total nucleon density ρ and temperature T . The second simplification arises from the fact that G(k, ρ, Yp ) at zero-temperature given in Eq. (15) is approximately proportional to the asymmetry parameter (1 − 2Yp ). The proportionality factor can be n r) approximated analytically by making a Taylor expansion of the functional [ρn 3j1k(k − nr ρp

3j1 (kp r) kp r ]

inside the integral in Eq. (15) about Yp = 1/2,  (1 − 2Yp )ρ j0 (kf r)j0 (kr)f (r) d3 r. G(k, ρ, Yp ) ≈  ρ0 f (r) d3 r

(17)

The validity of Eqs. (16) and (17) are examined in Fig. 3 for the Yukawa form of exchange interaction, f (r), where the dimensionless functional G(k, ρ, Yp ) in Eq. (15) which can be calculated analytically is shown as a function of momentum k at three dif-

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ferent densities, ρ0 , 3ρ0 and 5ρ0 for Yp = 0.1 and compared with these two approximate versions. The value of momentum k beyond which the asymptotic behaviour agrees with the exact one gradually increases with increase in density. This critical value of k also increases with temperature T as evident from the temperature evolution of the mean field in symmetric nuclear matter in our earlier work [54]. It may be noted here that such limiting values of k could be hardly reached in the non-equilibrium case of a reaction dynamics in high energy heavy-ion collisions. The approximate version of G(k, ρ, Yp ) in Eq. (17) although obtained from a Taylor expansion about Yp = 1/2 compares quite well with the exact one even at Yp = 0.1 over the entire range of momentum. However, small discrepancies arise at very low momenta which slowly increases as Yp → 0 at higher densities. This fact also justifies the linear isospin dependence of the famous Lane potential [1]. The nucleonic effective mass is an important property that is related to the momentum dependence of the in medium interaction of a nucleon and is defined as [54] 
∗ M (k, ρ, Yp ) M n,p  
 n,p 2 h¯ k 2 −1/2 M ∂uT (k, ρ, Yp ) −2 h¯ 2 k 2 1/2 = 1+ 2 2 + 2 − 2 2 . (18) ∂k M c M c h¯ k ∂u

n−p

(k,ρ,Y )

p T The asymptotic behaviour of the dimensionless quantity M2 calculated from ∂k h¯ k Eqs. (11) and (16) for Yukawa form of exchange interaction is given by

n−p

M ∂uT h¯ 2 k

ul )α 2 (k, ρ, Yp ) 1 Mρ (ε l − εex → −2(1 − 2Yp ) 2 ex . ∂k ρ0 (1 + α 2 k 2 )2 h¯

(19)

In obtaining this result we have used gex (k) = (1 + α 2 k 2 )−1 for the normalized Fourier transform of Yukawa form of interaction, f (r). The asymptotic behaviour of n−p

M ∂uT h¯ 2 k

(k,ρ,Yp ) ∂k

l − ε ul ). for large k given in Eq. (19) is positive for negative values of (εex ex ∂un (k,ρ,Y )

p

∂u (k,ρ,Y )

p p T T > M2 and asymptotically the neutron effective This means that M2 ∂k ∂k h¯ k h¯ k ∗ mass [M (k, ρ, Yp )/M]n is less than the proton effective mass [M ∗ (k, ρ, Yp )/M]p . On n−p

the other hand, the asymptotic behaviour of l − ε ul ) (εex ex

M ∂uT h¯ 2 k

(k,ρ,Yp ) ∂k

is negative for positive values

and asymptotically the proton effective mass will be less than the neutron efof fective mass. This has been shown in Fig. 4, where the exact zero temperature neutron and proton effective masses which can be calculated analytically using Eqs. (14), (15) and (18), and would involve only the exchange strength and range parameters are plotted as functions of momentum k at ρ = ρ0 and Yp = 0.1 for two different sets of exchange strength paral = ε /3, ε ul = 5ε /3 (set A, in Fig. 4(a)) and ε l = 5ε /3, ε ul = ε /3 (set meters; εex ex ex ex ex ex ex ex B in Fig. 4(b)). The asymptotic behaviour of neutron and proton effective masses are also shown in the same figures for comparison. The asymptotic results agree quite well with the exact results for k  3.5 fm−1 . The density dependence of these two opposite types of splitting of neutron and proton effective masses could also be quite important for the differences between neutron and proton transport properties in highly asymmetric dense nuclear matter. This has been shown for neutron and proton effective masses at k = 0 for

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Fig. 4. Neutron and proton effective masses shown as functions of momentum k for ρ = ρ0 and Yp = 0.1 for the two different exchange parameter sets A panel and B panel (see text for details). The corresponding asymptotic behaviours are also shown for comparison.

the two sets of exchange strength parameters A and B in Figs. 5(a) and (b), respectively. The effective masses decrease with increase in density. However, the rate of decrease is considerably slowed down at higher densities. It may be worth noticing here that the abl − ε ul ) does not change for the two different sets A and B considered solute value of (εex ex here and as a result the curves for the neutron effective mass in Figs. 4(a) and 5(a) are converted to the curves for the proton effective mass in Figs. 4(b) and 5(b) and conversely. In connection with the splitting of neutron and proton effective masses in asymmetric nuclear matter, it is important to note that all theoretical results can be roughly classified into two groups; one in which the neutron effective mass goes above the proton effective mass as in Figs. 4(a) and 5(a) and the other showing a splitting in the opposite direction, similar to the trend shown in Figs. 4(b) and 5(b). The results obtained from BHF calculations with realistic nucleon–nucleon interactions [21,22] exhibit a splitting where the neutron effective mass goes above the proton effective mass as in Figs. 4(a) and 5(a). Recently Li et al. [8–11] have studied effects of momentum dependent symmetry potentials on heavy-ion collisions induced by neutron-rich nuclei in terms of simple parametrizations of neutron and proton single particle potentials guided by a Hartree–Fock calculation using the Gogny effective interaction. This simple parametrization of neutron and proton single particle potentials also gives a splitting similar to the one shown in Figs. 4(a) and 5(a). On the other hand, the recent parametrization, SLy-type [38,44] of Skyrme forces give a splitting of neutron and proton effective masses in the opposite direction similar to the trend shown in Figs. 4(b) and 5(b). The results obtained in the microscopic relativistic Dirac–Brueckner calculations [20] as well as relativistic mean field approximation using quantum hadrodynamics (QHD) [29–32] also exhibit this type of splitting of neutron and

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Fig. 5. Neutron and proton effective masses shown as functions of density for k = 0 and Yp = 0.1 for the two different exchange parameter sets A panel and B panel.

proton effective masses. In a recent work Rizzo et al. [12] have analyzed the influence of the two opposite types of splitting of neutron and proton effective masses on flow data in heavy-ion collisions using two different simple GBD type parametrizations [60] of the energy density. ∂u

n−p

(k,ρ,Y )

p T is proportional to the total density ρ Since the asymptotic behaviour of M2 ∂k h¯ k and the neutron–proton asymmetry parameter (1−2Yp ), the splitting of neutron and proton effective masses mentioned above may be small around normal nuclear matter density ρ0 and therefore, can be neglected in finite nuclei where the density ρ and neutron–proton asymmetry (1 − 2Yp ) are rather small [31]. However, these two opposite types of splitting of neutron and proton effective masses may be quite relevant for the differences between their transport properties in highly asymmetric dense nuclear matter. As we have seen above, the sign of this splitting depends on the choice of interaction parameters connected with the exchange part of the effective interaction and the puzzle can only be resolved by a detailed analysis of experimental observables sensitive to the differences between neutron and proton flow data in high energy heavy-ion collisions. The approximate version of G(k, ρ, Yp ) at zero-temperature given in Eq. (17) is a good approximation to the exact one and can be used in conjunction with Eq. (14) to obtain a simple result of the isospin part of nuclear mean field uτ (k, ρ) defined in Eq. (1) l − ε ul )ρ  (εex ρA(ρ) ex  + j0 (kr) j0 (kf r) f (r) d3 r. (20) uτ (k, ρ) = 2 2ρ0 f (r) d3 r

A common problem which frequently arises in comparing the momentum dependence of the isospin part of nuclear mean field uτ (k, ρ) derived from different effective interactions is the differences in the density dependence of the nuclear symmetry energy Jτ (ρ) for these interactions. In view of this it is desirable to separate out the density dependence of

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Jτ (ρ) from the isospin part of nuclear mean field uτ (k, ρ). The nuclear symmetry energy Jτ (ρ) is usually defined through the relation 
1 ∂ 2 H (ρ, Yp ) Jτ (ρ) = . (21) 8ρ ∂Yp2 Yp =1/2 Here, H (ρ, Yp ) is the energy density in asymmetric nuclear matter at zero temperature as a function of total nucleon density ρ and proton fraction Yp . For the simple form of energy density H (ρ, Yp ) used in this work, the nuclear symmetry energy defined in Eq. (21) can be calculated analytically and the result can be put in the form 
 ∗  h¯ 2 kf2 M (k, ρ) 2 h¯ 2 k 2 −1/2 ρA(ρ) Jτ (ρ) = + 2 2 + 6M M 4 M c k=kf  l − ε ul )ρ (εex ex  + j02 (kf r) f (r) d3 r, (22) 4ρ0 f (r) d3 r where M ∗ (k, ρ)/M is the zero-temperature nucleon effective mass in symmetric nuclear matter as function of momentum and density. The isospin part of the nuclear mean field uτ (k, ρ) in Eq. (20) can be connected to the nuclear symmetry energy Jτ (ρ) in Eq. (22) through the relation 
 ∗  h¯ 2 kf2 M (k, ρ) 2 h¯ 2 k 2 −1/2 uτ (k, ρ) = 2Jτ (ρ) − + 2 2 + uex (23) τ (k, ρ), 3M M M c k=kf where uex τ (k, ρ) =

l − ε ul )ρ (εex ex  2ρ0 f (r) d3 r





j0 (kr) − j0 (kf r) j0 (kf r) f (r) d3 r.

(24)

Although this result has been obtained for the simple form of energy density in ANM used in this work, it can be generalized to any other effective interactions. In such cases only the functional uex τ (k, ρ) involving the exchange integrals would change to 

l

ρ ul (k, ρ) = (r) − vex (r) j0 (kf r) d3 r, (25) uex j0 (kr) − j0 (kf r) vex τ 2 l (r) and v ul (r) are the exchange interactions between two like and unlike nuclewhere vex ex ons, respectively. It is important to note that only finite range parts of the exchange interactions between two like and unlike nucleons can contribute to the functional uex τ (k, ρ). (k = k , ρ) vanishes at the Fermi momentum k . Moreover, uex f f τ It should be remarked here that in Eqs. (23)–(25) we have a connection to very old open problems; momentum and density dependence of the Lane potential [1] defined in Eq. (1), which now we see in a much more general framework. The value of the Lane potential, v1f0 = 4uτ (k = kf0 , ρ0 ), at Fermi momentum corresponding to normal nuclear matter density ρ0 as assessed earlier from nucleon–nucleus scattering and reactions as well as BBG calculations using realistic nucleon–nucleon interactions [1,61,62] exhibit a large uncertainty. This is due to the existing uncertainty in the values of nuclear symmetry energy and the nucleon effective mass even at normal density ρ0 . Substitution of the result

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Fig. 6. Isospin part of nuclear mean field at normal density uτ (k, ρ0 ) shown as function of momentum k for exchange parameter sets A and B with Jτ (ρ0 ) = 30 MeV and M ∗ (k = kf0 , ρ0 )/M = 0.67.

M ∗ (k = kf0 , ρ0 )/M = 0.67 obtained in the previous section and a standard value Jτ (ρ0 ) = 30 MeV in Eq. (23) leads to a result v1f0 = 101.7 MeV which can be compared with the range v1f0 = 100 ± 50 MeV quoted in Ref. [61]. It is also evident from Eqs. (23)–(25) that different uτ (k, ρ) sharing the same behaviour around k = kf over a wide range of density but having quite different momentum dependence for k away from kf will lead to almost same results for properties like energy per particle, neutron and proton chemical potentials, pressure and incompressibility in ANM. However, these different uτ (k, ρ) can give rise to significantly different predictions on experimental observables sensitive to the differences between neutron and proton transport properties in highly asymmetric dense nuclear matter. A complete description of momentum and density dependence of the isospin part of nuclear mean field uτ (k, ρ) given in Eq. (23) will require the knowledge of density dependence of nuclear symmetry energy Jτ (ρ) in addition to finite range exchange interactions between two like and unlike nucleons. However, at normal nuclear matter density, ρ = ρ0 , one can make use of standard values of Jτ (ρ0 ) and M ∗ (k = kf0 , ρ0 )/M and examine the momentum dependence of uτ (k, ρ0 ) for different representative values of the paral − ε ul ). This is shown for the Yukawa form of exchange interaction in Fig. 6 meter (εex ex for the two different sets of exchange strength parameters A and B by using the result, M ∗ (k = kf0 , ρ0 )/M = 0.67 obtained in the previous section and assuming a standard value, Jτ (ρ0 )= 30 MeV. The two different sets of exchange strength parameters A and B exhibit quite contradicting momentum dependence of the isospin part of nuclear mean field. While uτ (k, ρ0 ) decreases with increase in momentum for set-A, it increases for set-B and the two curves intersect at k = kf0 . This contradiction in momentum dependence of the isospin part of nuclear mean field will however become less and less pronounced with decrease in l − ε ul ). It is interesting to note that a zero-crossing point occurs the absolute value of (εex ex in case of uτ (k, ρ0 ) for set-A at k = 3.3 fm−1 where it vanishes and then becomes negative with further increase in k, and the isospin effect on the difference between neutron and proton mean fields are inverted. Such inversion of the isospin effect at a crossing point

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is observed in BHF calculations using realistic nucleon-nucleon interactions [21–23] and also in the work of Li et al. [8–11] mentioned earlier in this section. However, it is worth noticing that such a crossing point can appear only if uτ (k, ρ) given in Eq. (23) vanishes at some value of momentum k and ultimately approaches a negative asymptotic value in the limit of k → ∞. On the other hand, such a zero-crossing point does not occur in case of set-B and consequently the difference between neutron and proton mean fields becomes more and more repulsive with increase in k and approaches higher and higher asymptotic values in the limit k → ∞ with decrease in the proton fraction Yp . To examine the momentum dependence of the isospin part of nuclear mean field at densities other than normal nuclear matter density ρ0 , we require the knowledge of additional strength parameters for interactions between like and unlike nucleons. The relationships among the strength parameters given in Eq. (7) and the energy density in Eq. (8) clearly show that a complete description of EOS of ANM is exactly equivalent to separate descriptions of EOS of symmetric nuclear matter and pure neutron matter [25,26]. In view of this, we now focus on the EOS of PNM to constrain the strength parameters for interactions between like and unlike nucleons in the next section.

4. Equation of state of pure neutron matter and high density behaviour of nuclear symmetry energy Pure neutron matter and symmetric nuclear matter at same total nucleon density and temperature constitute the two extremes of asymmetric nuclear matter with Yp = 0 and Yp = 1/2, respectively. The nuclear symmetry energy Jτ (ρ) which is a very important quantity in the description of EOS of ANM may also be defined as the difference between energy per particle in PNM and SNM Jτ (ρ) = en (ρ) − e(ρ),

(26)

which leads to the simple parametrization of energy per particle in zero-temperature ANM e(ρ, Yp ) = e(ρ) + (1 − 2Yp )2 Jτ (ρ).

(27)

The energy per particle and the single particle energy in zero-temperature PNM as well as SNM can be calculated analytically for the effective interaction in Eq. (1). Since the parameters α, γ and b are fixed by the EOS of SNM, a complete description of EOS of l . Unlike PNM requires the knowledge of three additional parameters namely, ε0l , εγl and εex the case of zero-temperature SNM there are no experimental/empirical constraints on the behaviour of EOS of PNM [4]. The only constraint available is the value of the nuclear symmetry energy Jτ (ρ0 ) at normal nuclear matter density ρ0 . The high density behaviour of Jτ (ρ) predicted by different model calculations [5,6,63,64] are extremely divergent and very often contradictory. Our experimental/empirical knowledge on the momentum dependence of the mean field in PNM is still worse and different calculations have predicted quite different and even contradictory results as evident from the discussion in the previous section. In the absence of any experimental/empirical constraints our objective is to analyse the EOS of PNM on the basis of simple physical considerations and to study the high density behaviour of nuclear symmetry energy Jτ (ρ).

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Fig. 7. Influence of parameter γ on the high density behaviour of nuclear symmetry energy for the stiffest EOS of l with J (ρ ) = 30 MeV. The left (ε l = 4ε /3) and right (ε l = 0) PNM is shown for two typical values of εex τ 0 ex ex ex panels exhibit the minimum and maximum influence of the parameter γ .

All reasonable nuclear interactions predict that the equilibrium density ρ0 (Yp ) at which the energy per particle of ANM is minimum, decreases with increase in the asymmetry (1 − 2Yp ) from zero onwards. As a result binding energy per particle and incompressibility in ANM at this equilibrium density also decrease with increase in (1 − 2Yp ). With further increase in asymmetry (1−2Yp ) the minimum of energy per particle disappears completely around a value (1 − 2Yp ) > 0.8 which is in accordance with the well-known fact that PNM is not bound by nuclear forces. In view of this PNM should not be predicted to be bound by any reasonable nuclear interaction and we impose the condition P n (ρ) > 0

and

dP n (ρ) > 0. dρ

(28)

While the first condition rules out the existence of any equilibrium density where the energy per particle in PNM en (ρ) can have a minimum, both the conditions are necessary to avoid any non physical behaviour leading to a collapse of PNM. It can be remarked here that many existing parametrizations of effective interactions which are very successful in many nuclear calculations violate these two conditions [44]. l and examine the EOS of PNM We now vary the three additional parameters ε0l , εγl , εex subject to this constraint. To start with we take a standard value of Jτ (ρ0 ) at normal nuclear l in the range 0−2ε and vary the remaining parameter matter density, assume a value of εex ex to obtain the stiffest EOS of PNM that does not violate the constraint given in Eq. (28). We have examined the density dependence of nuclear symmetry energy Jτ (ρ) obtained l and J (ρ ). It is observed from these critical EOS of PNM with different values of γ , εex τ 0 that the curves of Jτ (ρ) are least influenced by different values of the exponent γ when l is around 4ε /3, whereas, these curves are most influenced by different values of γ εex ex l = 0. This is shown in Fig. 7(a) and (b) for γ = 1/12, 1/6, 1/3 and 1/2 with the when εex

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Fig. 8. High density behaviour of nuclear symmetry energy for the stiffest EOS of PNM is shown for two different sets of exchange strength parameters A and B (see text) with γ = 1/6 and different values of Jτ (ρ0 ). l = 4ε /3 shown same value of Jτ (ρ0 ) = 30 MeV. All the four curves of Jτ (ρ) with εex ex in Fig. 7(a) are nearly same over the entire range of density involved. On the other hand these curves become more and more stiff at higher densities with increase in γ for the case l = 0 shown in Fig. 7(b). εex l over its range 0–2ε has a very It is also observed that variation of the parameter εex ex little influence on the density dependence of Jτ (ρ) when γ is around 1/6. This is shown in Fig. 8 for the two different sets of exchange strength parameters A and B with three standard values of Jτ (ρ0 ) = 28, 30 and 32 MeV. The curves of Jτ (ρ) are almost same over the entire range of density for the two sets of exchange strength parameters A and B corresponding to a given value of Jτ (ρ0 ). The stiffness of the curves increase steadily with increase in the standard values of Jτ (ρ0 ). Here we have an example of different EOS in ANM which share the same density dependence of nuclear symmetry energy Jτ (ρ) over a wide range of density but having quite contradicting momentum dependence of the isospin part of nuclear mean field leading to two opposite types of splitting of neutron and proton effective masses. The density dependence of Jτ (ρ) obtained from several other admissible EOS of PNM under the constraint in Eq. (28) but softer than the stiffest one discussed above are shown in l = 4ε /3, Fig. 9(a). The six different curves shown in this figure correspond to γ = 1/6, εex ex Jτ (ρ0 ) = 30 MeV and are distinguished from each other by the value of the gradient τ (ρ) ]ρ=ρ0 at normal density. These curves clearly exhibit the extremely diJτ (ρ0 ) = [ρ dJdρ vergent and even contradicting behaviour of Jτ (ρ) at high densities predicted by different theoretical calculations [5,6,63,64]. The density dependence of equilibrium proton fraction in beta-stable n + p + e + µ matter calculated with these six different EOS in ANM are also shown in Fig. 9(b). In particular, an increasing Jτ (ρ) leads to a relatively more proton rich beta-stable matter whereas, a decreasing Jτ (ρ) makes it almost a pure neutron matter at high densities. As a result of this chemical composition and cooling mechanisms [45– 47] in the process of formation of neutron stars will all be different for the two different types of high density behaviour of Jτ (ρ).

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Fig. 9. High density behaviour of nuclear symmetry energy Jτ (ρ) and equilibrium proton fraction Yp (ρ) in beta-stable n + p + e + µ matter for six different EOS of PNM including the stiffest one are shown in increasing order of stiffness from bottom to top. All the curves shown correspond to Jτ (ρ0 ) = 30 MeV, γ = 1/6 and l = 4ε /3 but differ in the values of the gradient J  (ρ ). εex ex τ 0

In connection with the extremely divergent and contradicting high density behaviour of nuclear symmetry energy shown in Fig. 9, it may be noted here that recently Stone et al. [44] and references therein have examined 87 Skyrme interactions on the basis of their predictions on equilibrium proton fraction and EOS of beta-stable matter as well as neutron star properties. It is found that only 27 of the interactions pass the test which have an increasing behaviour of Jτ (ρ) over a wide range of density. Microscopic calculations based on realistic nucleon–nucleon interactions as well as RMF calculations also predict such an increasing behaviour of Jτ (ρ). This suggests that high density behaviour of Jτ (ρ) can be constrained to some extent and an approximate lower limit of the gradient Jτ (ρ0 ) for the curves in Fig. 9 can be obtained on the basis of their predictions on proton fraction in beta-stable matter as well as neutron star properties. However, the actual high density behaviour of nuclear symmetry energy can only be ascertained from a detailed analysis of experimental observables sensitive to the differences between neutron and proton flow data in highly asymmetric dense nuclear matter [4–6].

5. Summary and conclusions In this paper we have analyzed the momentum and density dependence of the isospin part of nuclear mean field uτ (k, ρ) using simple density dependent finite range effective interactions which are very suitable in order to clarify the problem. The behaviour of uτ (k, ρ) around the Fermi momentum k = kf is found to be connected to the density dependence of nuclear symmetry energy Jτ (ρ) and nucleon effective mass M ∗ (k = kf , ρ)/M in symmetric nuclear matter. The existing uncertainty in the values of these two quantities even at normal nuclear matter density explains the large uncertainty in the value of the Lane

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potential noticed earlier from nucleon–nucleus scattering and reactions as well as BBG calculations with realistic nucleon–nucleon interactions. The momentum dependence of uτ (k, ρ) is separated out in terms of a simple functional uex τ (k, ρ) which vanishes at the Fermi momentum k = kf and involves only the finite range parts of the exchange interactions between like and unlike nucleons. The extremely divergent and contradicting behaviour of the functional uex τ (k, ρ) with momentum is examined for an Yukawa form of exchange interaction having the same range but different strengths for interactions between two like and unlike nucleons. Two different trends of momentum dependence are noticed; one in which uτ (k, ρ) increases and the other where it decreases with increase in momentum. An increasing trend in momentum implies proton effective mass going above the neutron one and the other way around for a decreasing trend. The EOS of asymmetric nuclear matter and the density dependence of nuclear symmetry energy are analysed by constraining the additional parameters involved on the basis that pure neutron matter should not be predicted to be bound by any reasonable nuclear interaction. It should be remarked here that properties of asymmetric nuclear matter such as energy per particle, neutron and proton chemical potentials, pressure and the equilibrium proton fraction in beta-stable n + p + e + µ matter follow in a trivial way once the density dependence of nuclear symmetry energy is fixed. The high density behaviour of Jτ (ρ) can also be constrained to certain extent on the basis of predictions of equilibrium proton fraction in beta-stable matter as well as neutron star properties. To conclude, it can be mentioned that the existing controversies in the prediction of high density behaviour of nuclear symmetry energy and the two opposite types of splitting of neutron and proton effective masses by different theoretical calculations can only be resolved through continuous analysis of experimental observables sensitive to the differences between neutron and proton flow data in highly asymmetric dense nuclear matter.

Acknowledgements This work is supported by a research project No. F.10-5/2001 (SR-I) dated 11.3.2001 sanctioned by the University Grants Commission, Government of India.

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