ЯДЕРНАЯ ФИЗИКА, 2010, том 73, № 8, с. 1433-1441
= ЯДРА ^^
MODERN ENERGY DENSITY FUNCTIONAL FOR NUCLEI AND THE NUCLEAR MATTER EQUATION OF STATE
® 2010 S. Shlomo*
Cyclotron Institute, Texas A&M University, College Station, USA
Received December 28, 2009
We discuss a method of determining a modern energy density functional (EDF) in nuclei. We adopt a Skyrme type EDF and fit the Skyrme parameters to an extensive set of experimental data on the ground-state binding energies, radii, and the breathing mode energies of a wide range of nuclei. We further constrain the values of the Skyrme parameters by requiring positive values for the slope of the symmetry energy S, the enhancement factor к, associated with the isovector giant dipole resonance, and the Landau parameter G'0. This is done within the approaches of Hartree—Fock (HF) and HF with the inclusion of correlation effects, using a simulated-annealing based algorithm for minimizing x2. We also present results of HF based random phase approximation for the excitation strength function of the breathing mode and discuss the current status of the nuclear matter incompressibility coefficient.
1. INTRODUCTION
Density functional theory is based on a theorem [1] for the existence of a universal energy-density functional (EDF) that depends on the densities of the constituents and their derivatives. The main challenge is to find the EDF. The development of a modern EDF is very important since it provides a powerful approach for theoretical calculations of properties of many-body systems. Starting from the EDF associated with the Skyrme effective nucleon— nucleon interaction [2—6], we will present below a more realistic EDF with improved predictive power for properties of nuclei at and away from the valley of stability.
Since the pioneering work of Brink and Vau-therin [4], continuous efforts have been made to readjust the parameters of the Skyrme-type effective nucleon—nucleon (NN) interaction to better reproduce experimental data [7]. Most of the parameters of the Skyrme interactions available in the literature were obtained by fitting the Hartree—Fock (HF) results to experimental data on bulk properties of a few stable closed-shell nuclei. Thus it is desirable to have a unified interaction which includes the merits of several families of the Skyrme interactions and is fitted to a more extensive data set. One can further enhance the applicability of the Skyrme-type effective nucleon—nucleon interaction by imposing certain constraints, as subsequently discussed.
It should be emphasized that in earlier attempts to determine a more realistic interaction, the main objective has been that the experimental data on ground
E-mail: shlomo@comp.tamu.edu
state properties of nuclei be reproduced by the mean-field (HF) theory. In our approach we use the effective interaction to determine the properties of giant resonances within the random-phase approximation (RPA) theory and thus introduce RPA correlations into the ground state. Therefore, for consistency, we must require that the comparison with data be made after the inclusion of the effects of RPA correlations in the ground states of the nuclei, i.e., going beyond mean-field theory [8].
We have recently determined, within the approaches of the HF approximation and the HF with the inclusion of effects of correlations, new and more realistic Skyrme interactions (named KDE0 and KDEX) by fitting [9, 10] a set of extensive data on binding energies, "bare" single-particle energies, charge radii, and radii of valence nucleon density distributions of ground states of nuclei. For the first time we have included in the fit the data on the energies of the isoscalar giant monopole resonances (ISGMR) of nuclei and imposed additional constraints, such as a non-negative value for the slope of the symmetry energy density at high nuclear matter (NM) density (at three times the saturation density of NM) and the Landau stability constraints on nuclear matter. We have implemented, for the first time, the simulated annealing method (SAM) together with an advanced least-square method to search for the global minima.
In the next section we briefly outline the form of the Skyrme NN effective interaction and the corresponding EDF adopted in the present work and provide feasible strategies for the calculations of Coulomb displacement energy (CDE) and the center
of mass (CM) corrections to the total binding energy and charge radii. In Section 3 we describe the SAM algorithm. The set of the experimental data along with the theoretical errors and the constraints used in the fit to determine the values of the Skyrme parameters are given in Section 4. In Section 5 we present our results for the two different fits carried out in this work. In Section 6 we outline the Green's function formalism for the nuclear response, and present the results of HF-based RPA for the ISGMR and discuss the value of the NM incompressibility coefficient. Finally, in Section 7 we summarize our main results and discuss the scope for the further improvement of the present work.
2. SKYRME ENERGY-DENSITY FUNCTIONAL
We adopt the following form for the Skyrme-type effective NN interaction [4]:
V12 = to (1+ xoP&) 5(ri - r2) +
(1)
+ \и{1+Х1РЪ)
^2
к 12S(ri - Г2) +
+ á(ri - Г2) к 12
+ t2 (1 + X2PI2) X
x к i2ô(ri - r2) к 12 + -и (1 + Ж3РГ2) X
+ iWo k I2^(ri - r2)( 1 + 2) k 12,
where ti, Xi, a, and W0 are the parameters of the interaction and Pi2 is the spin exchange operator,
~ai is the Pauli spin operator, k 12 = —i(V1 — V2)/2,
and k 12 = —i(V1 — V2)/2. Here, the right and left arrows indicate that the momentum operators act on the right and on the left, respectively. The total energy E of the system is given by
E = (Ф|Я|Ф) = ^ H(r)d3r,
(2)
where H and are the many-body Hamiltonian and wave function, respectively. The Skyrme EDF H(r), obtained using Eq. (1), is given by [4, 5],
H = K + Ho + H3 + Heff + Hfin + (3)
+ Hso + Hsg + HCouli
where JC = ^r is the kinetic-energy term, Ho is the zero-range term, H3 the density-dependent term, Heff an effective-mass term, Hfin a finite-range term, Hso a spin—orbit term, Hsg is a term that is due to tensor coupling with spin and gradient, and HCoul is the
contribution to the energy-density that is due to the Coulomb interaction. For the Skyrme interaction of Eq. (1), we have
Ho = [(2 + xo)p2 - (2xo + 1 )(P2P + pl)] , (4)
H3 = —t3pa [(2 + x3)p2 - (2x3 + 1) x (5)
X(Pp + P2n)]
1
He5 = -[tl(2 + xl)+t2(2 + x2)]Tp+ (6)
1
+ - [i2(2x2 + 1) - ii(2a;i + 1)] (Tppp + Tnpn),
H,n = — [3ii(2 + Xl) - t2(2 + x2)\ (Vp)2 - (7) --^[3*1(2Ж1 + 1)+*2(2Ж2 + 1)] x
x [(Vpp)2 + (Vpn)2] ,
Hs
Wc
[J • Vp + xw (Jp • Vpp +
+ Jn • mVpn)],
Hsg = --j-(tixi +t2x 2)J2 +
16
(8) (9)
1
+ -^(h-t2) [J2 + J2].
Here, p = pp + pn,t = Tp + Tn, and J = Jp + J„ are the particle number density, kinetic-energy density, and spin-density with p and n denoting the protons and neutrons, respectively. Note that the additional parameter xw, introduced in Eq. (8), allows us to modify the isospin dependence of the spin—orbit term.
The contribution to the energy-density Eq. (3) from the Coulomb interaction can be written as a sum of a direct and an exchange terms. Here we only include the Coulomb direct term:
H
dir Coul
(r) = ¿e2pp(r) J
pp(r')d3r'
r — r'
(10)
We note that by neglecting the Coulomb exchange term one obtains that calculated values of the CDE agree with experimental data [9, 11]. The HF approach applied to finite nuclei violates the transla-tional invariance, introducing a spurious CM motion. Thus, one must extract the contributions of the CM motion to the binding energy, radii, and other observables. To account for the CM correction to the total binding energy, one must subtract from it the so-called CM energy, given as
ecm =
1
2mA
(P 2),
2
2
where I3 = J] pi = —ihY,f=i Vi is the total linear momentum operator. The corresponding mean-square charge radius to be fitted to the experimental data is obtained as
{r^h) = (rp)HF -
3
2vA
+ (r\ + §(r2)n+ (12)
+
1
h
Z V mc
nljr
where (r2)p and (r2)n are the mean-squared radii of the proton and neutron charge distributions, respectively. The second term in Eq. (12) is due to the CM motion, where v = mu/h is the harmonic oscillator size parameter. The last term in Eq. (12) is due to the electromagnetic spin—orbit effect. We have used (r2)n = -0.12 fm2 and the recent value of (r2)p = = 0.801 fm2.
2.1. Hartree—Fock Equations
The HF equations are derived by applying the variational principle to minimize the energy E, taken from Eqs. (2) and (3),
5
öp0
E - ei Po,t dr
(13)
5E ¿Ei fif Po,tdr]
ôp0
ôp0
0.
Here, ei are the Lagrange multipliers, and 5paT is the variation in the density p which is given in terms of the single-particle wave functions. After carrying out the minimization of energy, we obtain the HF equations for the spherical case:
h2 l~K(r) +
2m* (r ) d hc
- (14)
hc
dr \2m* (r) J a I r dr 2m*(r)
ja (ja
+
+
X Ra(r) = 6aRa(r),
where m* (r), UT(r), and WT(r) are the effective mass, the central potential, and the spin—orbit potential. They are given in terms of the Skyrme parameters and the nuclear densities (see [4]). With an initial guess of the single-particle wave functions (i.e., harmonic oscillator wave functions), we can determine m*(r), U(r), and W(r) and solve the HF equation to get a set of A new single-particle wave functions; then one can repeat these steps until convergence is reached. The resulting single-particle wave functions
are then used to calculate expectation values of ground-state properties of the nucleus under consideration. We note that one should start close to the solution to insure convergence. In the nonsymmetric case one should solve the HF equations in three dimensions, where the convergence and accuracy of the calculations are difficult problems.
2.2. Determination of the Critical Density
The Landau stability conditio
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