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Momentum dependence of the mean field and equation of state of nuclear matter
Nuclear Physics A 699 (2002) 770–794 www.elsevier.com/locate/npe
Momentum dependence of the mean field and equation of state of nuclear matter B. Behera ∗ , T.R. Routray, B. Sahoo, R.K. Satpathy Post Graduate Department of Physics, Sambalpur University, Jyoti Vihar, Burla, Sambalpur, Orissa, PIN-768 019, India Received 17 May 2001; revised 16 August 2001; accepted 22 August 2001
Abstract The momentum and density dependence of the mean field and other properties of nuclear matter are studied with finite-range effective interactions having different functional forms by evaluating the single-particle momentum distribution function self-consistently. In these calculations relativistic effects and supraluminous behaviour of nuclear matter are also taken into account. The equation of states obtained from these effective interactions are used to study liquid–gas phase transition in nuclear matter. 2002 Elsevier Science B.V. All rights reserved. PACS: 21.65.+f; 24.10.Pa Keywords: Momentum and density dependence; Mean field; Equation of state; Functional form; Nuclear matter; Finite-range effective interactions
1. Introduction The nuclear mean field is an important ingredient in many nuclear calculations, particularly, in the simulations of dynamical evolution of heavy-ion (HI) collisions at intermediate and high energies. Moreover, the mean field is directly connected with the calculations of nuclear equation of state (EOS). In the initial stage, analysis of heavyion collision data with momentum-independent mean fields demanded a stiff EOS with a high value of incompressibility [1,2]. However, it was later shown that if a reasonable momentum dependence is introduced in the nuclear mean field, a rather soft EOS with a lower value of incompressibility is favoured in the interpretation of experimental data [3–6]. Subsequently there were many important investigations [7–14] which strongly suggest that momentum dependence of the nuclear mean field is an unavoidable feature for a fundamental understanding of nuclear matter properties and for the successful interpretation of heavy-ion collision dynamics at intermediate and high energies. * Corresponding author.
0375-9474/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 1 ) 0 1 2 8 5 - 4
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Momentum and density dependence of the mean field in nuclear matter have been calculated microscopically using various approaches, such as (a) Dirac–Brueckner–Hartree–Fock (DBHF) calculations using realistic nucleon– nucleon interactions derived from relativistic meson-field theory [15–20], (b) Brueckner–Hartree–Fock (BHF) calculations with Reid soft core potential [21–24], (c) Brueckner–Bethe–Goldstone (BBG) calculations with the Paris potential [25,26], and (d) variational calculations using different combinations of two- and three-nucleon interactions [27,28]. On the other hand phenomenological effective interactions have been extensively used to study the mean field and other properties of nuclear matter both at zero and finite temperatures. Although quite less fundamental compared to the microscopic calculations mentioned above the use of effective interactions provide the simplest possible way to simulate momentum, density and temperature dependence of the mean field in nuclear matter. A major advantage of this approach is that it very often leads to analytical results for the mean field and other properties of nuclear matter at zero temperature. The Skyrme interactions and the Seyler–Blanchard type effective interactions which have been widely used in many nuclear calculations lead to a mean field in nuclear matter whose momentumdependent part is repulsive and has a quadratic dependence on the momentum. On the other hand, the momentum-dependent part of the mean field in nuclear matter derived from a general finite-range effective interaction is attractive and is strong at very low momenta. With increase in momentum, this part of the mean field weakens and vanishes asymptotically at very large momenta. This behaviour of the mean field with increase in momentum is an essential feature for a successful interpretation of heavy-ion collision data at intermediate and high energies [7–12,29,30]. A similar behaviour of the mean field in nuclear matter has also been observed in the microscopic DBHF calculations [19,20]. The purpose of this paper is to analyze the momentum, density and temperature dependence of the mean field in nuclear matter derived from finite-range effective interactions and to examine the influence of the functional form of the interaction on the high-momentum behaviour of the mean field. Emphasis will be given to use very simple parameterization of the effective interaction with a minimum number of adjustable parameters and yet capable of giving a good description of the mean field in nuclear matter over a wide range of momentum, density and temperature. The nature of density dependence of the effective interaction is chosen appropriately to prevent the supraluminous behaviour of nuclear matter at very high density [31]. Possible relativistic effects in calculating the equation of state of nuclear matter at high temperature and at high density are also approximately included with the use of relativistic relation between momentum and kinetic energy. The single-particle momentum distribution function is evaluated in a self-consistent way to obtain the single-particle energy, energy density, chemical potential and other thermodynamical properties of nuclear matter. The equation of state calculated with these effective interactions is used to study liquid–gas phase transition in nuclear matter.
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2. Mean field and equation of state of nuclear matter with phenomenological effective interactions Because of translational symmetry, the single-particle states of nuclear matter are given by eigenstates of the momentum operator and one can introduce a single-particle momentum distribution function fT (k) subject to the normalization, ρ = fT (k) d3 k, (1) where ρ is density of nuclear matter and T is the temperature. Using this single-particle momentum distribution function, the energy density HT (ρ) in nuclear matter derived from a nucleon–nucleon effective interaction can be expressed as 1/2 3 ρ2 fT (k) c2 h¯ 2 k 2 + m2 c4 d k+ HT (ρ) = νd (r) d3 r 2 1 (2) + fT (k)fT (k )ei(k−k )·r νex (r) d3 k d3 k d3 r. 2 The first term in this equation represents the kinetic energy density where we have used the relativistic relation between momentum and kinetic energy in order to include possible relativistic effects at high temperature and at high density in an approximate way. The second and third terms represent respectively the direct and exchange contributions to the potential energy density. The direct and exchange parts of the effective interaction are given by 3 se 3 1 9 ν (r) + ν te (r) + ν so (r) + ν to (r) and (3) 16 16 16 16 3 3 1 9 (4) νex (r) = ν se (r) + ν te (r) − ν so (r) − ν to (r). 16 16 16 16 The quantities ν se , ν te , ν so , and ν to , represent effective interactions in the specified states averaged over angles, spins and isospins of the two interacting nucleons. These effective interactions are functions of the relative separation between the two nucleons and may depend on local density ρ. The symbols e/o and t/s represent respectively the parity (even/odd) and the spin (triplet/singlet) of the two nucleon states. The single-particle energy εT (k, ρ) can be obtained as the functional derivative of the energy density HT (ρ) and is given by 1/2 + uT (k, ρ), (5) εT (k, ρ) = c2 h¯ 2 k 2 + m2 c4 νd (r) =
where uT (k, ρ) is the mean field or the single-particle potential in nuclear matter and can be written as 3 i(k−k )·r 3 3 fT (k )e νex (r) d k d r uT (k, ρ) = ρ νd (r) d r + +
ρ2 2
1 ∂νd 3 d r+ ∂ρ 2
fT (k)fT (k )ei(k−k )·r
∂νex 3 3 3 d kd k d r . ∂ρ (6)
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The single-particle potential has three distinct parts: a direct part, an exchange part which is momentum dependent and a rearrangement part arising out of any explicit dependence of the effective interaction on density ρ. At finite temperature the single-particle momentum distribution function fT (k) depends on the single-particle energy εT (k, ρ), and Eqs. (5) and (6) imply a self-consistent calculation of fT (k). For nuclear matter at thermal equilibrium at temperature T and density ρ, the single-particle momentum distribution function is given in terms of a Fermi– Dirac distribution: g nT (k), (7) fT (k) = (2π)3 where the occupation probability nT (k) is given by −1 nT (k) = exp (εT (k, ρ) − µT )/T + 1 .
(8)
Here g is the spin–isospin degeneracy and µT is the chemical potential. A method of successive iteration, starting from the single-particle energy ε0 (k, ρ) at zero temperature can be implemented to evaluate the single-particle momentum distribution function fT (k) self-consistently [32] at a given density ρ and temperature T . This procedure gives simultaneously the Fermi–Dirac distribution function fT (k), the single-particle energy εT (k, ρ) and the chemical potential µT . Using the self-consistent distribution function fT (k), the energy density HT (ρ) can be calculated. The pressure PT (ρ) in nuclear matter can be obtained as PT (ρ) = µT ρ − HT (ρ) + T S,
(9)
where S is the entropy density of nuclear matter at density ρ and temperature T and can be approximated by the relation g (10) nT (k) ln nT (k) + 1 − nT (k) ln 1 − nT (k) d3 k. S =− (2π)3 The incompressibility and the effective nucleon mass are two important quantities in the calculation of equation of state of nuclear matter. The incompressibility of nuclear matter at density ρ and temperature T is defined as KT (ρ) = 9
∂PT (ρ) . ∂ρ
(11)
The effective nucleon mass m∗ usually arises from the momentum dependence of the mean field uT (k, ρ) and is defined through the relation
1/2 1 ∂εT (k, ρ) 1 d 2 2 2 c h¯ k + m2 c4 . (12) = k dk k ∂k ∗ m=m Using the relation for εT (k, ρ), the ratio of effective nucleon mass to the actual mass can be written as
m∗ h¯ 2 k 2 −1/2 m ∂uT (k, ρ) −2 h¯ 2 k 2 1/2 = 1+ 2 2 + 2 − 2 2 . (13) m m c ∂k m c h¯ k
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The effective nucleon mass obtained from this equation is clearly a function of momentum k of the nucleon under consideration as well as density ρ and temperature T of nuclear matter.
3. Properties of zero-temperature nuclear matter and simple parameterizations of finite-range effective interactions Calculation of the mean field and equation of state of nuclear matter with effective interactions according to the formalism outlined in the previous section will crucially depend on the momentum and density dependence of the mean field in zero-temperature nuclear matter, since the temperature evolution of εT (k, ρ) is built upon ε0 (k, ρ) in the process of self-consistent calculation of the distribution function fT (k). At zero temperature, the single-particle momentum distribution function takes the form of a step function: g θ (k f − k), (14) f0 (k) = (2π)3 where kf is the Fermi momentum and according to the normalization in Eq. (1), it is related to the density ρ as 2kf3 . (15) 3π 2 Using the distribution function f0 (k) in the relation for HT (ρ) and integrating over k and k , the energy density in nuclear matter at zero temperature can be obtained in the form ρ2 3mc2ρ 3 2x u − x u − ln(x + u ) + νd (r) d3 r H0 (ρ) = f f f f f f 2 8xf3 2 j1 (kf r) 9ρ 2 νex (r) d3 r, (16) + 2 (kf r)2 ρ=
where j , represents spherical Bessel function of order and 1/2 . xf = (h¯ kf /mc) and uf = 1 + xf2
(17)
In a similar way, the single-particle energy ε0 (k, ρ) at zero temperature can be obtained in the form 1/2 + u0 (k, ρ), (18) ε0 (k, ρ) = c2 h¯ 2 k 2 + m2 c4 where the single-particle potential u0 (k, ρ) at zero temperature is found to be j1 (kf r) 3 3 νex (r) d r u0 (k, ρ) = ρ νd (r) d r + 3ρ j0 (kr) (kf r) 2 2 j1 (kf r) ∂νex 3 ρ ρ2 ∂νd 3 d r +9 d + r . 2 ∂ρ 2 (kf r)2 ∂ρ
(19)
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It may be noticed that the single-particle energy ε0 (k = k f , ρ) at the Fermi surface exactly satisfies the Hugenholtz–Van Hove (HV) theorem: ε0 (k = k f , ρ) = ε0 (ρ) + ρ
dε0(ρ) , dρ
(20)
where ε0 (ρ) = H0 (ρ)/ρ is the total energy per nucleon in zero-temperature nuclear matter. This can be verified from the relations of H0 (ρ) and u0 (k = k f , ρ). The chemical potential, pressure and incompressibility in zero temperature nuclear matter are found to be related to the total energy per nucleon ε0 (ρ) and are given by µ0 (ρ) = ε0 (ρ) + ρ
dε0 (ρ) , dρ
dε0 (ρ) and dρ d2 ε0 (ρ) dε (ρ) K0 (ρ) = 9ρ 2 + 18ρ 0 . 2 dρ dρ P0 (ρ) = ρ 2
(21) (22) (23)
The effective nucleon mass in zero-temperature nuclear matter is given by the relation
−2
m∗ h¯ 2 k 2 −1/2 3mρ h¯ 2 k 2 1/2 j1 (kr) j1 (kf r) 1+ 2 2 = νex (r)r 2 d3 r − 2 − 2 2 . m m c (kr) (kf r) m c h¯ (24) From Eq. (19) it may be noticed that momentum dependence of the mean field is simulated by the finite range of the exchange part of the effective interaction νex (r). For a short-range attractive interaction νex (r), the momentum-dependent part of the singleparticle potential is attractive and is strong at very low momenta. With increase in momentum this part of the mean field gradually weakens and vanishes asymptotically for very large momenta. Because of this behaviour of the mean field, the effective nucleon mass defined above will have a minimum value at k = 0 at a given density ρ and with increase in k it would gradually increase and would approach the bare mass m asymptotically for very large values of k. We now focus on the momentum dependence of u0 (k, ρ) defined in Eq. (19). A problem which frequently arises in comparing the momentum dependence of the mean fields derived from various effective interactions is the difference in the saturation curve of ε0 (ρ) for these interactions. In view of this it is desirable to separate out the saturation properties of the interaction from the single-particle potential u0 (k, ρ). For this purpose, we make use of the HV theorem and express u0 (k, ρ) as dε (ρ) − mc2 uf + uex (k, ρ), where u0 (k, ρ) = ε0 (ρ) + ρ 0 dρ j1 (kf r) νex (r) d3 r. j0 (kr) − j0 (kf r) uex (k, ρ) = 3ρ (kf r)
(25) (26)
In a similar way we can separate out the saturation properties of the interaction from the rearrangement part uR (ρ) of the single-particle potential u0 (k, ρ). The result is
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j12 (kf r) ∂νex 3 d r uR (ρ) = (kf r)2 ∂ρ dε (ρ) mc2 − uR 3xf uf + 2xf3uf − 3 ln(xf + uf ) − ε0 (ρ) + ρ 0 = ex (ρ), 2 dρ 4x where 3j1 (kf r) j1 (kf r) νex (r) d3 r. (ρ) = 3ρ j (k r) − uR 0 f ex (kf r) (kf r) ρ2 2
ρ2 ∂νd 3 d r +9 ∂ρ 2
(27)
(28)
Note that a zero-range interaction does not contribute to uex (k, ρ) and uR ex (ρ). In the simplest possible form, momentum and density dependence of uex (k, ρ) and density dependence of uR ex (ρ) can be simulated by a two-parameter short-range attractive interaction of conventional form such as gaussian, Yukawa or exponential. A major advantage of this approximation is that the exchange integrals appearing in the above ∗ equations for uex (k, ρ), uR ex (ρ) and m /m can be obtained analytically in terms of only two parameters although the results are not simple. For these interactions, the exchange integrals can be expressed as ρ j1 (kf r) νex (r) d3 r = εex Iα (k, ρ) and (29) 3ρ j0 (kr) (kf r) ρ0 2 j1 (kf r) ρ 9ρ νex (r) d3 r = εex Jα (ρ), (30) ρ0 (kf r)2 where ρ0 is standard nuclear matter density and the strength εex of the exchange interaction νex is given by εex = ρ0 νex (r) d3 r. (31) The functionals Iα (k, ρ) and Jα (ρ) can be calculated analytically [33] for the three different forms of exchange interactions, Yukawa, gaussian and exponential, in terms of momentum k, density ρ and the range α of these interactions. Since Iα (k, ρ) vanishes in the limit of large k, the single-particle potential u0 (k, ρ) will approach the asymptotic result u0 (k → ∞, ρ) = ε0 (ρ) + ρ
dε0 (ρ) εex − mc2 uf − ρIα (k = kf , ρ) dρ ρ0
(32)
for large k. The two parameters εex and α for the three different interactions considered here can not assume arbitrary values and must be determined by requiring to give a reasonable account of zero-temperature mean field u0 (k, ρ0 ) at normal nuclear matter density over a wide range of momentum as demanded by optical model fits to nucleon-nucleus scattering data. An important feature of optical model fits to nucleon–nucleus scattering data at intermediate energy is that u0 (k, ρ0 ) turns out to be repulsive [29,30] for momenta k > k300 where k300 corresponds to a kinetic energy of 300 MeV (excluding rest-mass energy). We, therefore, exploit the condition u0 (k = k 300 , ρ0 ) = 0 and write 1/2 2 2 2 c h¯ kf0 + m2 c4 − ε0 (ρ0 ) Iα (k = k300, ρ0 ) − Iα (k = kf0 , ρ0 ) = . (33) εex
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To ensure almost identical results of u0 (k, ρ0 ) for the three interactions in the range kf0 < k < k300 , we minimize the left-hand side of this equation with respect to the range α and obtain the two parameters εex and α simultaneously. For this purpose we assume standard values of mc2 = 939 MeV, ε0 (ρ0 ) = 923 MeV and (c2 h¯ 2 kf20 + m2 c4 )1/2 = 976 MeV (corresponding to ρ0 = 0.1658 fm−3 ). The values of the two parameters ∗ εex and α, together with the results of u0 (k = 0, ρ0 ), u0 (k → ∞, ρ0 ), mm (k = 0, ρ0 ), m∗ m (k = kf0 , ρ0 ) and uR (ρ0 ) which can be calculated analytically with the help of Eq. (25) are listed in Table 1 for the three interactions. For all these interactions, u0 (k = kf0 , ρ0 ) = ∗ −53 MeV and mm (k → ∞, ρ0 ) = 1. It may be noted that the results in Table 1 have been obtained by exploiting the condition that u0 (k, ρ0 ) becomes repulsive for k > k300 and no other assumptions were made except for standard values of mc2 , ε0 (ρ0 ) and ρ0 . In view of this the differences in the results for the three interactions are due to the specific choice of the functional form of the exchange interactions νex (r). A comparison of the values of u0 (k → ∞, ρ0 ) for the three interactions shows that at high momenta the mean field will be more repulsive for the Yukawa interaction and less repulsive for the gaussian interaction. Moreover, the exponential interaction gives results for the mean field, effective nucleon mass and rearrangement energy which are in between those of Yukawa and gaussian interactions. In view of this, in the remaining part of this work, we compare various results calculated with Yukawa and gaussian interactions. This will bring out the influence of the functional form of the interaction on the calculated results in a more clear manner. The single-particle potential u0 (k, ρ0 ) calculated analytically is shown in Fig. 1 as a function of k. It may be noted that calculation of u0 (k, ρ0 ) simply requires the numerical values of ε0 (ρ0 ), ρ0 , εex and α. It is seen that the results are almost same in the momentum range 0 < k < 5.5 fm−1 for Yukawa and gaussian interactions. However, at higher momenta u0 (k, ρ0 ) is more repulsive for Yukawa interaction than the gaussian interaction which is also reflected in the values of u0 (k → ∞, ρ0 ) for these two interactions. We have also verified that the momentum dependence of the single-particle potential u0 (k, ρ0 ) shown in Fig. 1 compares very well with that obtained from the momentum-dependent Yukawa interaction (MDYI) of Csernai et al. [30] in the momentum range k = 0−5.5 fm−1 . Table 1 Results of exchange parameters εex , α and properties of mean field at standard nuclear matter density Function interaction
Yukawa
Exponential
Gaussian
εex (MeV) α (fm) uR (ρ0 ) (MeV) m∗ (k = 0, ρ ) m 0 m∗ (k = k , ρ ) f0 0 m U (k = 0, ρ0 ) U (k → ∞, ρ0 )
–121.8 0.4044 16.56 0.6366 0.6698 –71.37 32.48
–106.4 0.2735 16.94 0.6563 0.6805 –70.18 21.05
–91.07 0.7204 17.56 0.6896 0.6991 –68.29 10.91
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Fig. 1. u0 (k, ρ0 ) shown as a function of k for finite-range Yukawa and gaussian interactions.
It may be emphasized here that the momentum dependence of u0 (k, ρ0 ) calculated from MDYI is in remarkable agreement with the real part of optical potential as extracted from experiments on nucleon–nucleus scattering both at low and high energies. The momentum dependence of uex (k, ρ) has been examined at three different densities ρ = 0.1, 0.3 and 0.5 fm−3 and this is shown in Fig. 2. The two exchange interactions lead to almost the same results in the momentum range k = 0 to k = 5.5 fm−1 . At higher momenta uex (k, ρ) is more repulsive for the Yukawa interaction than the gaussian one. We have also compared our results of uex (k, ρ), with those obtained by Wiringa [28] from a microscopic calculation using realistic Hamiltonians containing two-body and three-body interactions which fit nucleon–nucleon scattering data, binding energies of few-body systems and saturation properties of nuclear matter. In comparing our results of uex (k, ρ) with those obtained by Wiringa it is found that both of the results are quite similar over a wide range of momenta and densities for all the three realistic Hamiltonians, namely, UV14+TNI, UV14+UVII and AV14+UVII. However, our results are in better agreement with those of UV14+TNI and UV14+UVII. In Fig. 3 we have shown uex (k, ρ) for the Yukawa interaction and the UV14+UVII interaction of Wiringa as a function of momentum k at densities ρ = 0.1, 0.3 and 0.5 fm−3 . The two sets of results agree well over a wide range of momentum for the three densities considered. It may be mentioned that momentum and
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Fig. 2. uex (k, ρ) defined in Eq. (26) shown as a function of k at densities ρ = 0.1, 0.3 and 0.5 fm−3 for finite-range Yukawa and gaussian interaction.
density dependence of the single-particle potential u0 (k, ρ) for the interactions UV14+TNI and UV14+UVII agree well with those extracted from analysis of elliptic flow data in heavy-ion collisions [12]. In Fig. 4, the density dependence of uR ex (ρ) has been shown for Yukawa and gaussian interactions. The results are almost identical over a wide range of density. It is important to note here that momentum and density dependence of uex (k, ρ) and density dependence uR ex (ρ) simulated by a short-range attractive interaction νex (r) of Yukawa, gaussian or exponential forms remain unaffected if any zero-range interaction is added on to νex (r). This suggests that the simplest possible parameterization of finiterange effective interaction which can give a good description of the mean field in nuclear matter over a wide range of momentum and density can be given by
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Fig. 3. uex (k, ρ) as a function of k for Yukawa interaction is compared with the realistic UV14+UVII interaction of Wiringa.
γ t3 ρ(R) νeff (r) = t0 (1 + x0 Pσ )δ(r) + (1 + x3 Pσ ) δ(r) 6 1 + bρ(R) + (W + BPσ − H P − MPσ P )f (r).
(34)
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 have their usual meanings. This interaction is very similar to Skyrme-type interactions except for two differences. The first one is the short-range interaction (W + BPσ − H P − MPσ P )f (r) in place of the t1 and t2 terms in Skyrmetype interactions. This replacement is necessary to provide a description leading to vanishing attractive interaction between nucleons of very large relative momenta, a feature important for the successful interpretation of heavy-ion collision data at intermediate and high energies. The second modification is the denominator of the density-dependent term
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Fig. 4. uR ex (ρ) defined in Eq. (28) shown as a function of ρ for finite-range Yukawa and gaussian interaction.
of the effective interaction which is necessary to prevent the supraluminous behaviour of nuclear matter at very high density [31]. A major advantage of this simple parameterization of finite-range effective interaction is that it leads to analytical calculations of all zerotemperature nuclear matter properties with a minimum number of adjustable parameters. The energy density and the mean field in nuclear matter at temperature T and density ρ obtained from the simple effective interactions can be written as 1/2 3 ε ρ2 εγ ρ γ 2 fT (k) c2 h¯ 2 k 2 + m2 c4 d k+ d + γ +1 ρ HT (ρ) = 2 ρ0 2ρ 1 + bρ 0 εex 3 3 + (35) fT (k)fT (k )gex |k − k| d k d k 2ρ0 and uT (k, ρ) = εd
εγ ρ γ +1 ρ + γ +1 (1 + bρ + γ /2) ρ0 ρ 1 + bρ 0
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+
εex ρ0
fT (k )gex |k − k | d3 k .
The parameters εd , εγ and εex are given by
M H B 3t0 3 + W− − + εd = ρ0 f (r) d r , 4 2 2 2 t3 γ +1 and εγ = ρ0 8
H B W 3 + − εex = ρ0 M − f (r) d r . 4 2 2
(36)
(37) (38) (39)
The functional gex (k) appearing in HT (ρ) and uT (k, ρ) is the normalized Fourier transform of the short-range interaction f (r) and is given by 3 f (r) d3 r. (40) gex (k) = exp(ik · r)f (r) d r The results of gex (k) for Yukawa, gaussian and exponential form of f (r) can be obtained as −1 Y (k) = 1 + α 2 k 2 , (41) gex 2 2 G (42) gex (k) = exp −α k /4 , e 2 2 −2 gex (k) = 1 + α k . (43) It may be noticed that gex (k) vanishes in the limit of large k for all the short-range interactions f (r) but more slowly for the Yukawa and more rapidly for the gaussian interaction. The energy per nucleon and the mean field in zero-temperature nuclear matter can be calculated analytically and the results can be expressed as εd ρ εγ ρ γ 3mc2 3 2x u − x u − ln(x + u ) + + ρ ε0 (ρ) = f f f f f f 2 ρ0 2ρ γ +1 1 + bρ 8xf3 0 εex ρ Jα (ρ) (44) + 2ρ0 and u0 (k, ρ) = εd
εγ ρ γ +1 εex ρ ρ + γ +1 (1 + bρ + γ /2) + Iα (k, ρ). ρ0 ρ 1 + bρ ρ0 0
(45)
For nuclear matter calculations, there are altogether six adjustable parameters, namely, b, εex , α, εγ , εd and γ . The two parameters εex and α have already been determined for the three interactions, Yukawa, gaussian and exponential by requiring that the zerotemperature mean field u0 (k, ρ0 ) at standard nuclear matter density ρ0 becomes repulsive for (c2 h¯ 2 kf20 + m2 c4 ) > 1239 MeV. The parameter b appearing in the density-dependent part of the interaction is fixed by requiring to prevent the supraluminous behaviour of zerotemperature nuclear matter at high densities. An approximate result [31] can be obtained in the form
B. Behera et al. / Nuclear Physics A 699 (2002) 770–794
bρ0
mc2 Sf0 − ε0 (ρ0 )
1 γ +1
783
−1 −1
,
where
mc2 3 3xf uf + 2xf uf − 3 ln(xf + uf ) . Sf0 = 4xf3 ρ=ρ0
(46) (47)
Since density dependence of the form ρ γ have been very successful in predicting binding energies and radii of many nuclei, we take the lower limit of bρ0 in the above inequality. The remaining three parameters εd , εγ and γ are determined by requiring to give same binding energy per nucleon ε0 (ρ0 ) = 923 MeV and same incompressibility K0 (ρ0 ) = 210 MeV at standard nuclear matter density ρ0 = 0.1658 fm−3 . This would ensure similar results for the mean field u0 (k, ρ) over a wide range of density and momentum. The parameters of the three interactions obtained in this way are given in Table 2. The mean field u0 (k, ρ) which can be calculated analytically for these interactions is shown in Fig. 5 as a function of momentum at densities ρ = 0.1, 0.3 and 0.5 fm−3 . It is observed that the results are almost same in the momentum range k = 0 to 5.5 fm−1 . However, the highmomentum behaviour and the way in which u0 (k, ρ) approaches its asymptotic value at large k is determined by how gex (k) vanishes in the limit of large k. Thus, at high momenta,
Fig. 5. u0 (k, ρ) defined in Eq. (45) shown as a function of k at densities ρ = 0.1, 0.3 and 0.5 fm−3 for finite-range Yukawa and gaussian interaction.
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Table 2 Values of nuclear matter parameters of the effective interactions Function interaction
Yukawa
Exponential
Gaussian
εex (MeV) α (fm) γ bρ0 εd (MeV) εγ (MeV)
–121.8 0.4044 0.2362 0.0536 –134.2 151.8
–106.4 0.2735 0.2139 0.0506 –165.0 170.6
–91.07 0.7204 0.1883 0.0473 –205.0 199.8
u0 (k, ρ) for given ρ is more repulsive for Yukawa interaction than its gaussian counterpart. The effective interaction obtained in Eq. (34) has altogether eleven adjustable parameters. Analysis of the nuclear mean field and calculation of EOS of symmetric nuclear matter can be used to fix only six parameters. The remaining five parameters are to be determined from a study of asymmetric nuclear matter and binding energies as well as radii of finite nuclei (both N = Z and N = Z). However, since the primary purpose of the present work is to study the momentum and density dependence of the nuclear mean field and its temperature evolution we postpone the study of asymmetric nuclear matter and finite nuclei for a future work.
4. Properties of nuclear matter at finite temperature The analytical calculations of zero-temperature nuclear matter properties described earlier is no longer possible at finite temperature (T = 0). In this case, the singleparticle distribution function fT (k) is to be evaluated self-consistently to calculate various properties of nuclear matter. This has been done through an iterative procedure starting with the analytical result of the single-particle energy ε0 (k, ρ) at zero temperature. Using the self-consistently evaluated momentum distribution function, properties of nuclear matter such as energy density HT (ρ), mean field uT (k, ρ), entropy density S, incompressibility KT (ρ) and effective nucleon mass m∗ /m have been calculated. To examine the temperature evolution of the mean field as a function of momentum and density, we have calculated the quantity uex T (k, ρ) = uT (k, ρ) − u0 (k, ρ)
(48)
for Yukawa and gaussian interactions. These results are shown at temperatures T = 20, 40 and 80 MeV as functions of momentum k in Fig. 6, (a)–(c), at three different densities ρ = 0.1, 0.3 and 0.5 fm−3 , respectively. The two interactions give similar results. The functional uex T (k, ρ) for the Yukawa interaction approaches zero more slowly and has a longer tail than the gaussian interaction where it approaches zero rapidly at relatively lower value of k. This shows that the effect of temperature on the mean field is maximum at very low momenta and it gradually weakens with increase in momentum and ultimately vanishes at higher momenta. Thus, various curves of the mean field uT (k, ρ) at different temperatures but at
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(a)
(b) Fig. 6. (a), (b) and (c): uex T (k, ρ) defined in Eq. (48) shown as a function of k at temperatures T = 20, 40 and 80 MeV for the finite-range Yukawa and gaussian interaction densities ρ = 0.1, 0.3 and 0.5 fm−3 , respectively.
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(c) Fig. 6. —continued.
same density would merge together at higher values of k and then approach an asymptotic value u0 (k → ∞, ρ) with further increase in k. In this region, the mean field uT (k, ρ) can be approximated as ∼ u (k → ∞, ρ) + εex ρ gex (k), where uT (k, ρ) = (49) 0 ρ0 ε ρ εγ ρ γ +1 γ 1 + bρ + . (50) u0 (k → ∞, ρ) = d + γ +1 ρ0 1 + bρ 2 ρ 0
The high-momentum behaviour of the mean field and the way it approaches the asymptotic result u0 (k → ∞, ρ) is determined by how gex (k) vanishes for large values of k. Thus, for gaussian interaction the asymptotic result is reached rapidly at relatively lower values of k, whereas, for the Yukawa interaction this is reached very slowly at relatively much higher momenta. The effective nucleon mass m∗ /m is calculated as a function of momentum k at T = 0, 40 and 80 MeV and the results are shown in Fig. 7, (a) and (b), for densities ρ = ρ0 and ρ = 2ρ0 , respectively. The Yukawa and the gaussian interaction exhibit similar trends of
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(a)
(b) Fig. 7. (a) and (b): m∗ /m defined in Eq. (24) shown as a function of k at temperatures T = 0, 40 and 80 MeV for the finite-range Yukawa and gaussian interaction densities ρ = ρ0 and 2ρ0 , respectively.
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m∗ /m with increase in momentum. However, the gaussian interaction gives a larger value of m*/m at given density and temperature both in the low- and high-momentum region than the Yukawa interaction and this difference becomes more significant with increase in momentum in the high-momentum region. In both the cases, m∗ /m has a minimum value at k = 0. This minimum value of m∗ /m gradually increases with increase in temperature but decreases with an increase in density. As the momentum k increases m∗ /m gradually increases and ultimately approaches the asymptotic result m∗ /m = 1 with further increase in k. Thus, the effect of temperature on m∗ /m is maximum at low momenta and weakens with increase in momentum and ultimately vanishes at higher momenta. Because of this, various curves of m∗ /m corresponding to different temperatures but the same density merge together at higher values of k and then approaches the asymptotic result m∗ /m = 1 with further increase in k. In this region the effective mass can be approximated as
h¯ 2 k 2 −1/2 εex mρ 1 dgex −2 h¯ 2 k 2 1/2 m∗ 1+ 2 2 = + 2 − 2 2 . (51) m m c m c h¯ ρ0 k dk Thus, the high-momentum behaviour of m∗ /m and the way it approaches the asymptotic result is governed by how (1/k) (dgex /dk) vanishes for large k. Because of this m∗ /m approaches the asymptotic value very slowly for Yukawa interaction compared to gaussian interaction where the asymptotic value is reached quite rapidly at relatively lower value of k. The pressure PT (ρ) has been calculated by evaluating the energy density HT (ρ) and entropy density S. For stability of nuclear matter at given temperature T and density ρ, we must have ∂PT (ρ) 0. (52) PT (ρ) 0 and ∂ρ This stability condition is satisfied at all temperatures for densities ρ ρ0 . Below normal density ρ0 , nuclear matter is not stable at all temperatures and there exists a critical temperature Tc , such that for T Tc nuclear matter is stable at all densities. The value of Tc obtained with both Yukawa and gaussian interactions are found to be Tc = 14.9 MeV. Below Tc nuclear matter exists in liquid and gas phases in thermodynamical equilibrium at a temperature T provided P1 = P2
and µ1 = µ2 ,
(53)
where 1 and 2 refer to the first and second phases, respectively. It is found from graphical solutions that for both the forms of the finite-range interaction there exist solutions of Eq. (53) for temperature T 9 MeV and there is no solution for T < 9 MeV. Thus, nuclear matter exists only in one stable state i.e., in liquid phase, for T < 9 MeV. The transition from liquid to gas phase starts at T = 9 MeV and the two phases can coexist in equilibrium up to the critical temperature of Tc = 14.9 MeV. This is shown in Fig. 8 for the case of Yukawa interaction. For the gaussian interaction, the results are almost identical with those of Yukawa interaction. The critical temperature Tc has also been calculated to be 16.1 MeV for an equivalent zero-range effective interaction giving rise to a momentum-independent mean field and having the same saturation properties and
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Fig. 8. Pressure isotherms with phase boundaries for liquid and gas phases. Table 3 Values of nuclear matter parameters of the equivalent zero-range interaction Parameters
γ
bρ0
εd (MeV)
εγ (MeV)
Zero-range interaction
0.2762
0.0592
−257.99
184.24
incompressibility (ε(ρ0 ) = 923 MeV, (c2 h¯ 2 kf20 + m2 c4 )1/2 = 976 MeV and K0 (ρ0 ) = 210 MeV). This shows that the momentum dependence of the mean field brings down the critical temperature Tc . The parameters of the zero-range effective interaction are given in Table 3. Recent calculations by De et al. [34] using modified version of Seyler–Blanchard interaction and microscopic calculations of Baldo and Ferreira [35] have predicted the value of Tc to be 14.5 and 20 MeV, respectively. It has been verified with our finite-range interaction that raising the value of the parameter γ in the interaction will raise the value of Tc , but the nuclear matter incompressibility will have a higher value in that case.
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Fig. 9. Comparison of pressure isotherms for Yukawa and gaussian interaction at temperatures T = 0, 40 and 80 MeV.
The pressure PT (ρ) plotted as a function of ρ in Fig. 9 for both the forms of finite-range interaction at temperatures T = 0, 40 and 80 MeV are found to be identical over wide range of density and temperature. It is also found that the results of PT (ρ) for the equivalent zerorange effective interaction do not differ much from the corresponding values for the finiterange effective interactions over wide range of density and temperature. This is shown in Fig. 10 where the results of PT (ρ) as a function of ρ for the equivalent zero-range effective interaction has been compared with the results of the Yukawa form of effective interaction at T = 0, 40 and 80 MeV. However, unlike the pressure, the entropy density S as a function of density ρ and temperature T shows marked difference between the results for the finiterange and equivalent zero-range effective interaction. This is shown in Fig. 11 where the entropy density T S has been plotted as a function of ρ at temperatures T = 20, 40 and 80 MeV for both the forms of finite-range interaction and for the equivalent zero-range effective interaction. It may be seen that the results for both the finite-range interactions are identical over wide range of density and temperature, but their values are always less than the corresponding values for the equivalent zero-range interaction. This difference increases remarkably with the increase in density as well as in temperature.
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Fig. 10. Comparison of pressure isotherms of equivalent zero-range interaction with the finite-range Yukawa interaction.
The incompressibility KT (ρ) is calculated by using the relation KT (ρ) = 9 ∂PT (ρ)/∂ρ for both Yukawa and gaussian interactions and the results are shown as functions of density ρ at temperatures T = 0, 40 and 80 MeV in Fig. 12. Both the interactions give almost the same results of KT (ρ) over a wide range of density and temperature. The curve for KT (ρ) exhibit the strong dependence of incompressibility on temperature T and density ρ. For a given density ρ, the incompressibility increases rapidly with temperature and this rate of change decreases slowly with increase in density.
5. Conclusions In the simplest form momentum, density and temperature dependence of the mean field in nuclear matter can be simulated by effective interactions having a zero-range densitydependent part similar to Skyrme interaction and a finite-range density-independent part of conventional form such as Yukawa, gaussian or exponential. The parameters of these
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Fig. 11. Entropy density T S in MeV fm−3 defined in Eq. (10) as a function of ρ at temperatures T = 20, 40 and 80 MeV for finite-range Yukawa and gaussian, and equivalent zero-range interaction.
effective interactions can be constrained to give a mean field in nuclear matter which is almost independent of the functional form of the interaction in the momentum range k = 0 to 5.5 fm−1 and over a wide range of density. However, beyond this range of momentum, the functional form of the finite-range part of the interaction is important in determining the high-momentum behaviour of the mean field. This feature is clearly exhibited in the high-momentum behaviour of the mean field uT (k, ρ) and the effective mass m∗ /m and the way these quantities approach their asymptotic values at large momenta for Yukawa and gaussian form of the interaction. On the other hand, nuclear matter properties such as energy density, chemical potential, entropy, pressure and incompressibility calculated with different functional forms of the effective interaction are found to be almost same over a wide range of density and temperature.
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Fig. 12. The incompressibility KT (ρ) defined in Eq. (11) as a function of ρ at temperatures T = 0, 40 and 80 MeV for finite-range Yukawa and gaussian interaction.
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Momentum dependence of the mean field and equation of state of nuclear matter B. Behera ∗ , T.R. Routray, B. Sahoo, R.K. Satpathy Post Graduate Department of Physics, Sambalpur University, Jyoti Vihar, Burla, Sambalpur, Orissa, PIN-768 019, India Received 17 May 2001; revised 16 August 2001; accepted 22 August 2001
Abstract The momentum and density dependence of the mean field and other properties of nuclear matter are studied with finite-range effective interactions having different functional forms by evaluating the single-particle momentum distribution function self-consistently. In these calculations relativistic effects and supraluminous behaviour of nuclear matter are also taken into account. The equation of states obtained from these effective interactions are used to study liquid–gas phase transition in nuclear matter. 2002 Elsevier Science B.V. All rights reserved. PACS: 21.65.+f; 24.10.Pa Keywords: Momentum and density dependence; Mean field; Equation of state; Functional form; Nuclear matter; Finite-range effective interactions
1. Introduction The nuclear mean field is an important ingredient in many nuclear calculations, particularly, in the simulations of dynamical evolution of heavy-ion (HI) collisions at intermediate and high energies. Moreover, the mean field is directly connected with the calculations of nuclear equation of state (EOS). In the initial stage, analysis of heavyion collision data with momentum-independent mean fields demanded a stiff EOS with a high value of incompressibility [1,2]. However, it was later shown that if a reasonable momentum dependence is introduced in the nuclear mean field, a rather soft EOS with a lower value of incompressibility is favoured in the interpretation of experimental data [3–6]. Subsequently there were many important investigations [7–14] which strongly suggest that momentum dependence of the nuclear mean field is an unavoidable feature for a fundamental understanding of nuclear matter properties and for the successful interpretation of heavy-ion collision dynamics at intermediate and high energies. * Corresponding author.
0375-9474/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 1 ) 0 1 2 8 5 - 4
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Momentum and density dependence of the mean field in nuclear matter have been calculated microscopically using various approaches, such as (a) Dirac–Brueckner–Hartree–Fock (DBHF) calculations using realistic nucleon– nucleon interactions derived from relativistic meson-field theory [15–20], (b) Brueckner–Hartree–Fock (BHF) calculations with Reid soft core potential [21–24], (c) Brueckner–Bethe–Goldstone (BBG) calculations with the Paris potential [25,26], and (d) variational calculations using different combinations of two- and three-nucleon interactions [27,28]. On the other hand phenomenological effective interactions have been extensively used to study the mean field and other properties of nuclear matter both at zero and finite temperatures. Although quite less fundamental compared to the microscopic calculations mentioned above the use of effective interactions provide the simplest possible way to simulate momentum, density and temperature dependence of the mean field in nuclear matter. A major advantage of this approach is that it very often leads to analytical results for the mean field and other properties of nuclear matter at zero temperature. The Skyrme interactions and the Seyler–Blanchard type effective interactions which have been widely used in many nuclear calculations lead to a mean field in nuclear matter whose momentumdependent part is repulsive and has a quadratic dependence on the momentum. On the other hand, the momentum-dependent part of the mean field in nuclear matter derived from a general finite-range effective interaction is attractive and is strong at very low momenta. With increase in momentum, this part of the mean field weakens and vanishes asymptotically at very large momenta. This behaviour of the mean field with increase in momentum is an essential feature for a successful interpretation of heavy-ion collision data at intermediate and high energies [7–12,29,30]. A similar behaviour of the mean field in nuclear matter has also been observed in the microscopic DBHF calculations [19,20]. The purpose of this paper is to analyze the momentum, density and temperature dependence of the mean field in nuclear matter derived from finite-range effective interactions and to examine the influence of the functional form of the interaction on the high-momentum behaviour of the mean field. Emphasis will be given to use very simple parameterization of the effective interaction with a minimum number of adjustable parameters and yet capable of giving a good description of the mean field in nuclear matter over a wide range of momentum, density and temperature. The nature of density dependence of the effective interaction is chosen appropriately to prevent the supraluminous behaviour of nuclear matter at very high density [31]. Possible relativistic effects in calculating the equation of state of nuclear matter at high temperature and at high density are also approximately included with the use of relativistic relation between momentum and kinetic energy. The single-particle momentum distribution function is evaluated in a self-consistent way to obtain the single-particle energy, energy density, chemical potential and other thermodynamical properties of nuclear matter. The equation of state calculated with these effective interactions is used to study liquid–gas phase transition in nuclear matter.
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2. Mean field and equation of state of nuclear matter with phenomenological effective interactions Because of translational symmetry, the single-particle states of nuclear matter are given by eigenstates of the momentum operator and one can introduce a single-particle momentum distribution function fT (k) subject to the normalization, ρ = fT (k) d3 k, (1) where ρ is density of nuclear matter and T is the temperature. Using this single-particle momentum distribution function, the energy density HT (ρ) in nuclear matter derived from a nucleon–nucleon effective interaction can be expressed as 1/2 3 ρ2 fT (k) c2 h¯ 2 k 2 + m2 c4 d k+ HT (ρ) = νd (r) d3 r 2 1 (2) + fT (k)fT (k )ei(k−k )·r νex (r) d3 k d3 k d3 r. 2 The first term in this equation represents the kinetic energy density where we have used the relativistic relation between momentum and kinetic energy in order to include possible relativistic effects at high temperature and at high density in an approximate way. The second and third terms represent respectively the direct and exchange contributions to the potential energy density. The direct and exchange parts of the effective interaction are given by 3 se 3 1 9 ν (r) + ν te (r) + ν so (r) + ν to (r) and (3) 16 16 16 16 3 3 1 9 (4) νex (r) = ν se (r) + ν te (r) − ν so (r) − ν to (r). 16 16 16 16 The quantities ν se , ν te , ν so , and ν to , represent effective interactions in the specified states averaged over angles, spins and isospins of the two interacting nucleons. These effective interactions are functions of the relative separation between the two nucleons and may depend on local density ρ. The symbols e/o and t/s represent respectively the parity (even/odd) and the spin (triplet/singlet) of the two nucleon states. The single-particle energy εT (k, ρ) can be obtained as the functional derivative of the energy density HT (ρ) and is given by 1/2 + uT (k, ρ), (5) εT (k, ρ) = c2 h¯ 2 k 2 + m2 c4 νd (r) =
where uT (k, ρ) is the mean field or the single-particle potential in nuclear matter and can be written as 3 i(k−k )·r 3 3 fT (k )e νex (r) d k d r uT (k, ρ) = ρ νd (r) d r + +
ρ2 2
1 ∂νd 3 d r+ ∂ρ 2
fT (k)fT (k )ei(k−k )·r
∂νex 3 3 3 d kd k d r . ∂ρ (6)
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The single-particle potential has three distinct parts: a direct part, an exchange part which is momentum dependent and a rearrangement part arising out of any explicit dependence of the effective interaction on density ρ. At finite temperature the single-particle momentum distribution function fT (k) depends on the single-particle energy εT (k, ρ), and Eqs. (5) and (6) imply a self-consistent calculation of fT (k). For nuclear matter at thermal equilibrium at temperature T and density ρ, the single-particle momentum distribution function is given in terms of a Fermi– Dirac distribution: g nT (k), (7) fT (k) = (2π)3 where the occupation probability nT (k) is given by −1 nT (k) = exp (εT (k, ρ) − µT )/T + 1 .
(8)
Here g is the spin–isospin degeneracy and µT is the chemical potential. A method of successive iteration, starting from the single-particle energy ε0 (k, ρ) at zero temperature can be implemented to evaluate the single-particle momentum distribution function fT (k) self-consistently [32] at a given density ρ and temperature T . This procedure gives simultaneously the Fermi–Dirac distribution function fT (k), the single-particle energy εT (k, ρ) and the chemical potential µT . Using the self-consistent distribution function fT (k), the energy density HT (ρ) can be calculated. The pressure PT (ρ) in nuclear matter can be obtained as PT (ρ) = µT ρ − HT (ρ) + T S,
(9)
where S is the entropy density of nuclear matter at density ρ and temperature T and can be approximated by the relation g (10) nT (k) ln nT (k) + 1 − nT (k) ln 1 − nT (k) d3 k. S =− (2π)3 The incompressibility and the effective nucleon mass are two important quantities in the calculation of equation of state of nuclear matter. The incompressibility of nuclear matter at density ρ and temperature T is defined as KT (ρ) = 9
∂PT (ρ) . ∂ρ
(11)
The effective nucleon mass m∗ usually arises from the momentum dependence of the mean field uT (k, ρ) and is defined through the relation
1/2 1 ∂εT (k, ρ) 1 d 2 2 2 c h¯ k + m2 c4 . (12) = k dk k ∂k ∗ m=m Using the relation for εT (k, ρ), the ratio of effective nucleon mass to the actual mass can be written as
m∗ h¯ 2 k 2 −1/2 m ∂uT (k, ρ) −2 h¯ 2 k 2 1/2 = 1+ 2 2 + 2 − 2 2 . (13) m m c ∂k m c h¯ k
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The effective nucleon mass obtained from this equation is clearly a function of momentum k of the nucleon under consideration as well as density ρ and temperature T of nuclear matter.
3. Properties of zero-temperature nuclear matter and simple parameterizations of finite-range effective interactions Calculation of the mean field and equation of state of nuclear matter with effective interactions according to the formalism outlined in the previous section will crucially depend on the momentum and density dependence of the mean field in zero-temperature nuclear matter, since the temperature evolution of εT (k, ρ) is built upon ε0 (k, ρ) in the process of self-consistent calculation of the distribution function fT (k). At zero temperature, the single-particle momentum distribution function takes the form of a step function: g θ (k f − k), (14) f0 (k) = (2π)3 where kf is the Fermi momentum and according to the normalization in Eq. (1), it is related to the density ρ as 2kf3 . (15) 3π 2 Using the distribution function f0 (k) in the relation for HT (ρ) and integrating over k and k , the energy density in nuclear matter at zero temperature can be obtained in the form ρ2 3mc2ρ 3 2x u − x u − ln(x + u ) + νd (r) d3 r H0 (ρ) = f f f f f f 2 8xf3 2 j1 (kf r) 9ρ 2 νex (r) d3 r, (16) + 2 (kf r)2 ρ=
where j , represents spherical Bessel function of order and 1/2 . xf = (h¯ kf /mc) and uf = 1 + xf2
(17)
In a similar way, the single-particle energy ε0 (k, ρ) at zero temperature can be obtained in the form 1/2 + u0 (k, ρ), (18) ε0 (k, ρ) = c2 h¯ 2 k 2 + m2 c4 where the single-particle potential u0 (k, ρ) at zero temperature is found to be j1 (kf r) 3 3 νex (r) d r u0 (k, ρ) = ρ νd (r) d r + 3ρ j0 (kr) (kf r) 2 2 j1 (kf r) ∂νex 3 ρ ρ2 ∂νd 3 d r +9 d + r . 2 ∂ρ 2 (kf r)2 ∂ρ
(19)
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It may be noticed that the single-particle energy ε0 (k = k f , ρ) at the Fermi surface exactly satisfies the Hugenholtz–Van Hove (HV) theorem: ε0 (k = k f , ρ) = ε0 (ρ) + ρ
dε0(ρ) , dρ
(20)
where ε0 (ρ) = H0 (ρ)/ρ is the total energy per nucleon in zero-temperature nuclear matter. This can be verified from the relations of H0 (ρ) and u0 (k = k f , ρ). The chemical potential, pressure and incompressibility in zero temperature nuclear matter are found to be related to the total energy per nucleon ε0 (ρ) and are given by µ0 (ρ) = ε0 (ρ) + ρ
dε0 (ρ) , dρ
dε0 (ρ) and dρ d2 ε0 (ρ) dε (ρ) K0 (ρ) = 9ρ 2 + 18ρ 0 . 2 dρ dρ P0 (ρ) = ρ 2
(21) (22) (23)
The effective nucleon mass in zero-temperature nuclear matter is given by the relation
−2
m∗ h¯ 2 k 2 −1/2 3mρ h¯ 2 k 2 1/2 j1 (kr) j1 (kf r) 1+ 2 2 = νex (r)r 2 d3 r − 2 − 2 2 . m m c (kr) (kf r) m c h¯ (24) From Eq. (19) it may be noticed that momentum dependence of the mean field is simulated by the finite range of the exchange part of the effective interaction νex (r). For a short-range attractive interaction νex (r), the momentum-dependent part of the singleparticle potential is attractive and is strong at very low momenta. With increase in momentum this part of the mean field gradually weakens and vanishes asymptotically for very large momenta. Because of this behaviour of the mean field, the effective nucleon mass defined above will have a minimum value at k = 0 at a given density ρ and with increase in k it would gradually increase and would approach the bare mass m asymptotically for very large values of k. We now focus on the momentum dependence of u0 (k, ρ) defined in Eq. (19). A problem which frequently arises in comparing the momentum dependence of the mean fields derived from various effective interactions is the difference in the saturation curve of ε0 (ρ) for these interactions. In view of this it is desirable to separate out the saturation properties of the interaction from the single-particle potential u0 (k, ρ). For this purpose, we make use of the HV theorem and express u0 (k, ρ) as dε (ρ) − mc2 uf + uex (k, ρ), where u0 (k, ρ) = ε0 (ρ) + ρ 0 dρ j1 (kf r) νex (r) d3 r. j0 (kr) − j0 (kf r) uex (k, ρ) = 3ρ (kf r)
(25) (26)
In a similar way we can separate out the saturation properties of the interaction from the rearrangement part uR (ρ) of the single-particle potential u0 (k, ρ). The result is
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j12 (kf r) ∂νex 3 d r uR (ρ) = (kf r)2 ∂ρ dε (ρ) mc2 − uR 3xf uf + 2xf3uf − 3 ln(xf + uf ) − ε0 (ρ) + ρ 0 = ex (ρ), 2 dρ 4x where 3j1 (kf r) j1 (kf r) νex (r) d3 r. (ρ) = 3ρ j (k r) − uR 0 f ex (kf r) (kf r) ρ2 2
ρ2 ∂νd 3 d r +9 ∂ρ 2
(27)
(28)
Note that a zero-range interaction does not contribute to uex (k, ρ) and uR ex (ρ). In the simplest possible form, momentum and density dependence of uex (k, ρ) and density dependence of uR ex (ρ) can be simulated by a two-parameter short-range attractive interaction of conventional form such as gaussian, Yukawa or exponential. A major advantage of this approximation is that the exchange integrals appearing in the above ∗ equations for uex (k, ρ), uR ex (ρ) and m /m can be obtained analytically in terms of only two parameters although the results are not simple. For these interactions, the exchange integrals can be expressed as ρ j1 (kf r) νex (r) d3 r = εex Iα (k, ρ) and (29) 3ρ j0 (kr) (kf r) ρ0 2 j1 (kf r) ρ 9ρ νex (r) d3 r = εex Jα (ρ), (30) ρ0 (kf r)2 where ρ0 is standard nuclear matter density and the strength εex of the exchange interaction νex is given by εex = ρ0 νex (r) d3 r. (31) The functionals Iα (k, ρ) and Jα (ρ) can be calculated analytically [33] for the three different forms of exchange interactions, Yukawa, gaussian and exponential, in terms of momentum k, density ρ and the range α of these interactions. Since Iα (k, ρ) vanishes in the limit of large k, the single-particle potential u0 (k, ρ) will approach the asymptotic result u0 (k → ∞, ρ) = ε0 (ρ) + ρ
dε0 (ρ) εex − mc2 uf − ρIα (k = kf , ρ) dρ ρ0
(32)
for large k. The two parameters εex and α for the three different interactions considered here can not assume arbitrary values and must be determined by requiring to give a reasonable account of zero-temperature mean field u0 (k, ρ0 ) at normal nuclear matter density over a wide range of momentum as demanded by optical model fits to nucleon-nucleus scattering data. An important feature of optical model fits to nucleon–nucleus scattering data at intermediate energy is that u0 (k, ρ0 ) turns out to be repulsive [29,30] for momenta k > k300 where k300 corresponds to a kinetic energy of 300 MeV (excluding rest-mass energy). We, therefore, exploit the condition u0 (k = k 300 , ρ0 ) = 0 and write 1/2 2 2 2 c h¯ kf0 + m2 c4 − ε0 (ρ0 ) Iα (k = k300, ρ0 ) − Iα (k = kf0 , ρ0 ) = . (33) εex
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To ensure almost identical results of u0 (k, ρ0 ) for the three interactions in the range kf0 < k < k300 , we minimize the left-hand side of this equation with respect to the range α and obtain the two parameters εex and α simultaneously. For this purpose we assume standard values of mc2 = 939 MeV, ε0 (ρ0 ) = 923 MeV and (c2 h¯ 2 kf20 + m2 c4 )1/2 = 976 MeV (corresponding to ρ0 = 0.1658 fm−3 ). The values of the two parameters ∗ εex and α, together with the results of u0 (k = 0, ρ0 ), u0 (k → ∞, ρ0 ), mm (k = 0, ρ0 ), m∗ m (k = kf0 , ρ0 ) and uR (ρ0 ) which can be calculated analytically with the help of Eq. (25) are listed in Table 1 for the three interactions. For all these interactions, u0 (k = kf0 , ρ0 ) = ∗ −53 MeV and mm (k → ∞, ρ0 ) = 1. It may be noted that the results in Table 1 have been obtained by exploiting the condition that u0 (k, ρ0 ) becomes repulsive for k > k300 and no other assumptions were made except for standard values of mc2 , ε0 (ρ0 ) and ρ0 . In view of this the differences in the results for the three interactions are due to the specific choice of the functional form of the exchange interactions νex (r). A comparison of the values of u0 (k → ∞, ρ0 ) for the three interactions shows that at high momenta the mean field will be more repulsive for the Yukawa interaction and less repulsive for the gaussian interaction. Moreover, the exponential interaction gives results for the mean field, effective nucleon mass and rearrangement energy which are in between those of Yukawa and gaussian interactions. In view of this, in the remaining part of this work, we compare various results calculated with Yukawa and gaussian interactions. This will bring out the influence of the functional form of the interaction on the calculated results in a more clear manner. The single-particle potential u0 (k, ρ0 ) calculated analytically is shown in Fig. 1 as a function of k. It may be noted that calculation of u0 (k, ρ0 ) simply requires the numerical values of ε0 (ρ0 ), ρ0 , εex and α. It is seen that the results are almost same in the momentum range 0 < k < 5.5 fm−1 for Yukawa and gaussian interactions. However, at higher momenta u0 (k, ρ0 ) is more repulsive for Yukawa interaction than the gaussian interaction which is also reflected in the values of u0 (k → ∞, ρ0 ) for these two interactions. We have also verified that the momentum dependence of the single-particle potential u0 (k, ρ0 ) shown in Fig. 1 compares very well with that obtained from the momentum-dependent Yukawa interaction (MDYI) of Csernai et al. [30] in the momentum range k = 0−5.5 fm−1 . Table 1 Results of exchange parameters εex , α and properties of mean field at standard nuclear matter density Function interaction
Yukawa
Exponential
Gaussian
εex (MeV) α (fm) uR (ρ0 ) (MeV) m∗ (k = 0, ρ ) m 0 m∗ (k = k , ρ ) f0 0 m U (k = 0, ρ0 ) U (k → ∞, ρ0 )
–121.8 0.4044 16.56 0.6366 0.6698 –71.37 32.48
–106.4 0.2735 16.94 0.6563 0.6805 –70.18 21.05
–91.07 0.7204 17.56 0.6896 0.6991 –68.29 10.91
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Fig. 1. u0 (k, ρ0 ) shown as a function of k for finite-range Yukawa and gaussian interactions.
It may be emphasized here that the momentum dependence of u0 (k, ρ0 ) calculated from MDYI is in remarkable agreement with the real part of optical potential as extracted from experiments on nucleon–nucleus scattering both at low and high energies. The momentum dependence of uex (k, ρ) has been examined at three different densities ρ = 0.1, 0.3 and 0.5 fm−3 and this is shown in Fig. 2. The two exchange interactions lead to almost the same results in the momentum range k = 0 to k = 5.5 fm−1 . At higher momenta uex (k, ρ) is more repulsive for the Yukawa interaction than the gaussian one. We have also compared our results of uex (k, ρ), with those obtained by Wiringa [28] from a microscopic calculation using realistic Hamiltonians containing two-body and three-body interactions which fit nucleon–nucleon scattering data, binding energies of few-body systems and saturation properties of nuclear matter. In comparing our results of uex (k, ρ) with those obtained by Wiringa it is found that both of the results are quite similar over a wide range of momenta and densities for all the three realistic Hamiltonians, namely, UV14+TNI, UV14+UVII and AV14+UVII. However, our results are in better agreement with those of UV14+TNI and UV14+UVII. In Fig. 3 we have shown uex (k, ρ) for the Yukawa interaction and the UV14+UVII interaction of Wiringa as a function of momentum k at densities ρ = 0.1, 0.3 and 0.5 fm−3 . The two sets of results agree well over a wide range of momentum for the three densities considered. It may be mentioned that momentum and
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Fig. 2. uex (k, ρ) defined in Eq. (26) shown as a function of k at densities ρ = 0.1, 0.3 and 0.5 fm−3 for finite-range Yukawa and gaussian interaction.
density dependence of the single-particle potential u0 (k, ρ) for the interactions UV14+TNI and UV14+UVII agree well with those extracted from analysis of elliptic flow data in heavy-ion collisions [12]. In Fig. 4, the density dependence of uR ex (ρ) has been shown for Yukawa and gaussian interactions. The results are almost identical over a wide range of density. It is important to note here that momentum and density dependence of uex (k, ρ) and density dependence uR ex (ρ) simulated by a short-range attractive interaction νex (r) of Yukawa, gaussian or exponential forms remain unaffected if any zero-range interaction is added on to νex (r). This suggests that the simplest possible parameterization of finiterange effective interaction which can give a good description of the mean field in nuclear matter over a wide range of momentum and density can be given by
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Fig. 3. uex (k, ρ) as a function of k for Yukawa interaction is compared with the realistic UV14+UVII interaction of Wiringa.
γ t3 ρ(R) νeff (r) = t0 (1 + x0 Pσ )δ(r) + (1 + x3 Pσ ) δ(r) 6 1 + bρ(R) + (W + BPσ − H P − MPσ P )f (r).
(34)
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 have their usual meanings. This interaction is very similar to Skyrme-type interactions except for two differences. The first one is the short-range interaction (W + BPσ − H P − MPσ P )f (r) in place of the t1 and t2 terms in Skyrmetype interactions. This replacement is necessary to provide a description leading to vanishing attractive interaction between nucleons of very large relative momenta, a feature important for the successful interpretation of heavy-ion collision data at intermediate and high energies. The second modification is the denominator of the density-dependent term
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Fig. 4. uR ex (ρ) defined in Eq. (28) shown as a function of ρ for finite-range Yukawa and gaussian interaction.
of the effective interaction which is necessary to prevent the supraluminous behaviour of nuclear matter at very high density [31]. A major advantage of this simple parameterization of finite-range effective interaction is that it leads to analytical calculations of all zerotemperature nuclear matter properties with a minimum number of adjustable parameters. The energy density and the mean field in nuclear matter at temperature T and density ρ obtained from the simple effective interactions can be written as 1/2 3 ε ρ2 εγ ρ γ 2 fT (k) c2 h¯ 2 k 2 + m2 c4 d k+ d + γ +1 ρ HT (ρ) = 2 ρ0 2ρ 1 + bρ 0 εex 3 3 + (35) fT (k)fT (k )gex |k − k| d k d k 2ρ0 and uT (k, ρ) = εd
εγ ρ γ +1 ρ + γ +1 (1 + bρ + γ /2) ρ0 ρ 1 + bρ 0
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+
εex ρ0
fT (k )gex |k − k | d3 k .
The parameters εd , εγ and εex are given by
M H B 3t0 3 + W− − + εd = ρ0 f (r) d r , 4 2 2 2 t3 γ +1 and εγ = ρ0 8
H B W 3 + − εex = ρ0 M − f (r) d r . 4 2 2
(36)
(37) (38) (39)
The functional gex (k) appearing in HT (ρ) and uT (k, ρ) is the normalized Fourier transform of the short-range interaction f (r) and is given by 3 f (r) d3 r. (40) gex (k) = exp(ik · r)f (r) d r The results of gex (k) for Yukawa, gaussian and exponential form of f (r) can be obtained as −1 Y (k) = 1 + α 2 k 2 , (41) gex 2 2 G (42) gex (k) = exp −α k /4 , e 2 2 −2 gex (k) = 1 + α k . (43) It may be noticed that gex (k) vanishes in the limit of large k for all the short-range interactions f (r) but more slowly for the Yukawa and more rapidly for the gaussian interaction. The energy per nucleon and the mean field in zero-temperature nuclear matter can be calculated analytically and the results can be expressed as εd ρ εγ ρ γ 3mc2 3 2x u − x u − ln(x + u ) + + ρ ε0 (ρ) = f f f f f f 2 ρ0 2ρ γ +1 1 + bρ 8xf3 0 εex ρ Jα (ρ) (44) + 2ρ0 and u0 (k, ρ) = εd
εγ ρ γ +1 εex ρ ρ + γ +1 (1 + bρ + γ /2) + Iα (k, ρ). ρ0 ρ 1 + bρ ρ0 0
(45)
For nuclear matter calculations, there are altogether six adjustable parameters, namely, b, εex , α, εγ , εd and γ . The two parameters εex and α have already been determined for the three interactions, Yukawa, gaussian and exponential by requiring that the zerotemperature mean field u0 (k, ρ0 ) at standard nuclear matter density ρ0 becomes repulsive for (c2 h¯ 2 kf20 + m2 c4 ) > 1239 MeV. The parameter b appearing in the density-dependent part of the interaction is fixed by requiring to prevent the supraluminous behaviour of zerotemperature nuclear matter at high densities. An approximate result [31] can be obtained in the form
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bρ0
mc2 Sf0 − ε0 (ρ0 )
1 γ +1
783
−1 −1
,
where
mc2 3 3xf uf + 2xf uf − 3 ln(xf + uf ) . Sf0 = 4xf3 ρ=ρ0
(46) (47)
Since density dependence of the form ρ γ have been very successful in predicting binding energies and radii of many nuclei, we take the lower limit of bρ0 in the above inequality. The remaining three parameters εd , εγ and γ are determined by requiring to give same binding energy per nucleon ε0 (ρ0 ) = 923 MeV and same incompressibility K0 (ρ0 ) = 210 MeV at standard nuclear matter density ρ0 = 0.1658 fm−3 . This would ensure similar results for the mean field u0 (k, ρ) over a wide range of density and momentum. The parameters of the three interactions obtained in this way are given in Table 2. The mean field u0 (k, ρ) which can be calculated analytically for these interactions is shown in Fig. 5 as a function of momentum at densities ρ = 0.1, 0.3 and 0.5 fm−3 . It is observed that the results are almost same in the momentum range k = 0 to 5.5 fm−1 . However, the highmomentum behaviour and the way in which u0 (k, ρ) approaches its asymptotic value at large k is determined by how gex (k) vanishes in the limit of large k. Thus, at high momenta,
Fig. 5. u0 (k, ρ) defined in Eq. (45) shown as a function of k at densities ρ = 0.1, 0.3 and 0.5 fm−3 for finite-range Yukawa and gaussian interaction.
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Table 2 Values of nuclear matter parameters of the effective interactions Function interaction
Yukawa
Exponential
Gaussian
εex (MeV) α (fm) γ bρ0 εd (MeV) εγ (MeV)
–121.8 0.4044 0.2362 0.0536 –134.2 151.8
–106.4 0.2735 0.2139 0.0506 –165.0 170.6
–91.07 0.7204 0.1883 0.0473 –205.0 199.8
u0 (k, ρ) for given ρ is more repulsive for Yukawa interaction than its gaussian counterpart. The effective interaction obtained in Eq. (34) has altogether eleven adjustable parameters. Analysis of the nuclear mean field and calculation of EOS of symmetric nuclear matter can be used to fix only six parameters. The remaining five parameters are to be determined from a study of asymmetric nuclear matter and binding energies as well as radii of finite nuclei (both N = Z and N = Z). However, since the primary purpose of the present work is to study the momentum and density dependence of the nuclear mean field and its temperature evolution we postpone the study of asymmetric nuclear matter and finite nuclei for a future work.
4. Properties of nuclear matter at finite temperature The analytical calculations of zero-temperature nuclear matter properties described earlier is no longer possible at finite temperature (T = 0). In this case, the singleparticle distribution function fT (k) is to be evaluated self-consistently to calculate various properties of nuclear matter. This has been done through an iterative procedure starting with the analytical result of the single-particle energy ε0 (k, ρ) at zero temperature. Using the self-consistently evaluated momentum distribution function, properties of nuclear matter such as energy density HT (ρ), mean field uT (k, ρ), entropy density S, incompressibility KT (ρ) and effective nucleon mass m∗ /m have been calculated. To examine the temperature evolution of the mean field as a function of momentum and density, we have calculated the quantity uex T (k, ρ) = uT (k, ρ) − u0 (k, ρ)
(48)
for Yukawa and gaussian interactions. These results are shown at temperatures T = 20, 40 and 80 MeV as functions of momentum k in Fig. 6, (a)–(c), at three different densities ρ = 0.1, 0.3 and 0.5 fm−3 , respectively. The two interactions give similar results. The functional uex T (k, ρ) for the Yukawa interaction approaches zero more slowly and has a longer tail than the gaussian interaction where it approaches zero rapidly at relatively lower value of k. This shows that the effect of temperature on the mean field is maximum at very low momenta and it gradually weakens with increase in momentum and ultimately vanishes at higher momenta. Thus, various curves of the mean field uT (k, ρ) at different temperatures but at
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(a)
(b) Fig. 6. (a), (b) and (c): uex T (k, ρ) defined in Eq. (48) shown as a function of k at temperatures T = 20, 40 and 80 MeV for the finite-range Yukawa and gaussian interaction densities ρ = 0.1, 0.3 and 0.5 fm−3 , respectively.
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(c) Fig. 6. —continued.
same density would merge together at higher values of k and then approach an asymptotic value u0 (k → ∞, ρ) with further increase in k. In this region, the mean field uT (k, ρ) can be approximated as ∼ u (k → ∞, ρ) + εex ρ gex (k), where uT (k, ρ) = (49) 0 ρ0 ε ρ εγ ρ γ +1 γ 1 + bρ + . (50) u0 (k → ∞, ρ) = d + γ +1 ρ0 1 + bρ 2 ρ 0
The high-momentum behaviour of the mean field and the way it approaches the asymptotic result u0 (k → ∞, ρ) is determined by how gex (k) vanishes for large values of k. Thus, for gaussian interaction the asymptotic result is reached rapidly at relatively lower values of k, whereas, for the Yukawa interaction this is reached very slowly at relatively much higher momenta. The effective nucleon mass m∗ /m is calculated as a function of momentum k at T = 0, 40 and 80 MeV and the results are shown in Fig. 7, (a) and (b), for densities ρ = ρ0 and ρ = 2ρ0 , respectively. The Yukawa and the gaussian interaction exhibit similar trends of
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(a)
(b) Fig. 7. (a) and (b): m∗ /m defined in Eq. (24) shown as a function of k at temperatures T = 0, 40 and 80 MeV for the finite-range Yukawa and gaussian interaction densities ρ = ρ0 and 2ρ0 , respectively.
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m∗ /m with increase in momentum. However, the gaussian interaction gives a larger value of m*/m at given density and temperature both in the low- and high-momentum region than the Yukawa interaction and this difference becomes more significant with increase in momentum in the high-momentum region. In both the cases, m∗ /m has a minimum value at k = 0. This minimum value of m∗ /m gradually increases with increase in temperature but decreases with an increase in density. As the momentum k increases m∗ /m gradually increases and ultimately approaches the asymptotic result m∗ /m = 1 with further increase in k. Thus, the effect of temperature on m∗ /m is maximum at low momenta and weakens with increase in momentum and ultimately vanishes at higher momenta. Because of this, various curves of m∗ /m corresponding to different temperatures but the same density merge together at higher values of k and then approaches the asymptotic result m∗ /m = 1 with further increase in k. In this region the effective mass can be approximated as
h¯ 2 k 2 −1/2 εex mρ 1 dgex −2 h¯ 2 k 2 1/2 m∗ 1+ 2 2 = + 2 − 2 2 . (51) m m c m c h¯ ρ0 k dk Thus, the high-momentum behaviour of m∗ /m and the way it approaches the asymptotic result is governed by how (1/k) (dgex /dk) vanishes for large k. Because of this m∗ /m approaches the asymptotic value very slowly for Yukawa interaction compared to gaussian interaction where the asymptotic value is reached quite rapidly at relatively lower value of k. The pressure PT (ρ) has been calculated by evaluating the energy density HT (ρ) and entropy density S. For stability of nuclear matter at given temperature T and density ρ, we must have ∂PT (ρ) 0. (52) PT (ρ) 0 and ∂ρ This stability condition is satisfied at all temperatures for densities ρ ρ0 . Below normal density ρ0 , nuclear matter is not stable at all temperatures and there exists a critical temperature Tc , such that for T Tc nuclear matter is stable at all densities. The value of Tc obtained with both Yukawa and gaussian interactions are found to be Tc = 14.9 MeV. Below Tc nuclear matter exists in liquid and gas phases in thermodynamical equilibrium at a temperature T provided P1 = P2
and µ1 = µ2 ,
(53)
where 1 and 2 refer to the first and second phases, respectively. It is found from graphical solutions that for both the forms of the finite-range interaction there exist solutions of Eq. (53) for temperature T 9 MeV and there is no solution for T < 9 MeV. Thus, nuclear matter exists only in one stable state i.e., in liquid phase, for T < 9 MeV. The transition from liquid to gas phase starts at T = 9 MeV and the two phases can coexist in equilibrium up to the critical temperature of Tc = 14.9 MeV. This is shown in Fig. 8 for the case of Yukawa interaction. For the gaussian interaction, the results are almost identical with those of Yukawa interaction. The critical temperature Tc has also been calculated to be 16.1 MeV for an equivalent zero-range effective interaction giving rise to a momentum-independent mean field and having the same saturation properties and
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Fig. 8. Pressure isotherms with phase boundaries for liquid and gas phases. Table 3 Values of nuclear matter parameters of the equivalent zero-range interaction Parameters
γ
bρ0
εd (MeV)
εγ (MeV)
Zero-range interaction
0.2762
0.0592
−257.99
184.24
incompressibility (ε(ρ0 ) = 923 MeV, (c2 h¯ 2 kf20 + m2 c4 )1/2 = 976 MeV and K0 (ρ0 ) = 210 MeV). This shows that the momentum dependence of the mean field brings down the critical temperature Tc . The parameters of the zero-range effective interaction are given in Table 3. Recent calculations by De et al. [34] using modified version of Seyler–Blanchard interaction and microscopic calculations of Baldo and Ferreira [35] have predicted the value of Tc to be 14.5 and 20 MeV, respectively. It has been verified with our finite-range interaction that raising the value of the parameter γ in the interaction will raise the value of Tc , but the nuclear matter incompressibility will have a higher value in that case.
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Fig. 9. Comparison of pressure isotherms for Yukawa and gaussian interaction at temperatures T = 0, 40 and 80 MeV.
The pressure PT (ρ) plotted as a function of ρ in Fig. 9 for both the forms of finite-range interaction at temperatures T = 0, 40 and 80 MeV are found to be identical over wide range of density and temperature. It is also found that the results of PT (ρ) for the equivalent zerorange effective interaction do not differ much from the corresponding values for the finiterange effective interactions over wide range of density and temperature. This is shown in Fig. 10 where the results of PT (ρ) as a function of ρ for the equivalent zero-range effective interaction has been compared with the results of the Yukawa form of effective interaction at T = 0, 40 and 80 MeV. However, unlike the pressure, the entropy density S as a function of density ρ and temperature T shows marked difference between the results for the finiterange and equivalent zero-range effective interaction. This is shown in Fig. 11 where the entropy density T S has been plotted as a function of ρ at temperatures T = 20, 40 and 80 MeV for both the forms of finite-range interaction and for the equivalent zero-range effective interaction. It may be seen that the results for both the finite-range interactions are identical over wide range of density and temperature, but their values are always less than the corresponding values for the equivalent zero-range interaction. This difference increases remarkably with the increase in density as well as in temperature.
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Fig. 10. Comparison of pressure isotherms of equivalent zero-range interaction with the finite-range Yukawa interaction.
The incompressibility KT (ρ) is calculated by using the relation KT (ρ) = 9 ∂PT (ρ)/∂ρ for both Yukawa and gaussian interactions and the results are shown as functions of density ρ at temperatures T = 0, 40 and 80 MeV in Fig. 12. Both the interactions give almost the same results of KT (ρ) over a wide range of density and temperature. The curve for KT (ρ) exhibit the strong dependence of incompressibility on temperature T and density ρ. For a given density ρ, the incompressibility increases rapidly with temperature and this rate of change decreases slowly with increase in density.
5. Conclusions In the simplest form momentum, density and temperature dependence of the mean field in nuclear matter can be simulated by effective interactions having a zero-range densitydependent part similar to Skyrme interaction and a finite-range density-independent part of conventional form such as Yukawa, gaussian or exponential. The parameters of these
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Fig. 11. Entropy density T S in MeV fm−3 defined in Eq. (10) as a function of ρ at temperatures T = 20, 40 and 80 MeV for finite-range Yukawa and gaussian, and equivalent zero-range interaction.
effective interactions can be constrained to give a mean field in nuclear matter which is almost independent of the functional form of the interaction in the momentum range k = 0 to 5.5 fm−1 and over a wide range of density. However, beyond this range of momentum, the functional form of the finite-range part of the interaction is important in determining the high-momentum behaviour of the mean field. This feature is clearly exhibited in the high-momentum behaviour of the mean field uT (k, ρ) and the effective mass m∗ /m and the way these quantities approach their asymptotic values at large momenta for Yukawa and gaussian form of the interaction. On the other hand, nuclear matter properties such as energy density, chemical potential, entropy, pressure and incompressibility calculated with different functional forms of the effective interaction are found to be almost same over a wide range of density and temperature.
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Fig. 12. The incompressibility KT (ρ) defined in Eq. (11) as a function of ρ at temperatures T = 0, 40 and 80 MeV for finite-range Yukawa and gaussian interaction.
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