ЯДЕРНАЯ ФИЗИКА, 2011, том 74, № 10, с. 1453-1463
= ЯДРА ^^
THE LANDAU-MIGDAL PARAMETERS FROM THE BRUECKNER THEORY
©2011 D. Gambacurta1),2)'3), U. Lombardo^'2^, W. Zuo^
Received January 11, 2011
The zero-order Landau—Migdal parameters are discussed in the framework of the Brueckner—Hartree— Fock approximation with realistic two- and three-body forces. Their usefulness is proved in two applications. First, the structure functions in high-density nuclear matter are calculated within the linear response theory to weak probes and the neutrino mean free path is predicted for neutron stars. Second, the screening of the low-density neutron matter to the neutron pairing is calculated in the RPA without and with induced interaction. The extension of the calculation to symmetric nuclear matter reveals the anti-screening effect of the proton medium polarization.
1. INTRODUCTION
The Landau—Migdal (LM) parameters were introduced in the theory of Fermi liquids as a simple phenomenological description of the interaction between quasi-particles at the Fermi surface. Their relation with some physical observables provides important constraints on the properties of the residual interaction. In nuclear systems such observables are related to the excitation of spin and isospin collective states [1]. Recently the interest of the nuclear physics turned on exotic states of matter, such as nuclei far from the stability valley or astrophysical compact objects. In neutron-star physics, collective excitations in high-density and isospin nuclear matter have been studied in connection with the cooling process via the neutrino transport [2]. The onset of superfluid states responsible for the peculiar spin dynamics and thermodynamics of neutron stars demands for the study of the nucleon pairing screening due to such exotic environment. The use of the LM parameters can greatly simplify how to face the low density instabilities of the residual interaction [3]. Recently, the extension of the Skyrme force to include a tensor component for a better description of the spin— orbit interaction in nuclei raised the interest for the corresponding LM parameters H and H' [4].
At the present time there is a poor experimental knowledge of the physical observables that could constrain the LM parameters. Therefore, it is of primary importance their microscopic calculation within
^Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou, China.
2)Dipartimento di Fisica, Universita' di Catania, Italy
3)Sezione INFN di Catania, Italy
4)Laboratori Nazionali del Sud (INFN), Catania, Italy.
a many-body approach, based on a realistic nucleon— nucleon interaction. Of course there exist several investigations on such an issue [5—10]. But, due to the continuous progress of the microscopic models and approximations, more and more sophisticated predictions of the LM parameters are demanded for application to the exotic states of nuclear matter.
In this contribution we present calculations of the LM parameters in nuclear matter based on the most advanced version of the Brueckner—Hartree—Fock (BHF) (see [11] and references therein). Thanks to the introduction of a unified model of two- and three-body forces, the BHF approximation proved to give a satisfactory description of the empirical properties of nuclear matter at the saturation point.
In Section 2 we introduce the definition of the LM parameters and discuss their relation with the G matrix of the Brueckner theory. Due to the complexity of a complete derivation based on the diagrammatic expansion of the effective interaction, we present the alternative derivation requiring only the calculation of total energy of nuclear matter in various spin and isospin configurations. This procedure restricts the derivation of the LM parameters to only the zero-order components. The latter are in fact the most important with respect to their direct relation with physical quantities.
In Section 3 we shortly review the BHF theory and show the numerical prediction of the LM parameters from the energy of nuclear matter, including the symmetry energy and compression modulus. We also show how to remove the well-known low-density instability by a suitable renormalization of the Landau parameters based on the Babu—Brown theory [12].
Section 4 is devoted to two applications of the LM parameters. First, the linear response theory is
developed to describe the coupling between high-density and asymmetric nuclear matter and weak probes and the formalism is applied to calculate the neutrino mean free path in the neutron-star core. Second, the medium screening on the pairing coupling between nucleons in the 1S0 state is studied in the low-density neutron and nuclear matter. The induced interaction is calculated in the random-phase approximation (RPA) with the LM parameters renor-malized according to the Babu—Brown theory. The effect of neutron matter screening and nuclear matter anti-screening is illustrated with the calculation of the pairing gap.
2. DEFINITION
According to the Landau theory of Fermi liquids the excitation energy of the system is given by
5E = eP$nP + E f (P,Pr)$np5np,
pp'
where np are the occupation numbers of the quasi-particle states p = (p,a,r), defined by momentum, spin, and isospin, respectively, and ep are the corresponding energies. The effective interaction f (p,p'), in the Landau limit |p| = |p'| = pF, can be expressed in terms of the LM parameters as follows:
f (p,p') = Y; Pl(cos d)
+ (Gl<t • a' + ^Hls12(q)) +
related to the G-matrix elements in the particle-particle channel:
1
ST
f' = -y.(2s + l)(2t st,
ST
ST
= 2t - 1)[q1t - g0T],
ST
ST
W_ Pf 1
(1)
q ST
(5)
(6)
(7)
(8) (9)
(10)
fl + f'lt ■ r' + (2)
+ (G'La ■ a' + ^H'LSi2(q)j r • r'
where Su(q) is the tensor operator. In order to derive the LM parameters from the Brueckner theory, we extend the BHF energy functional
E = J2epnp + i J2{p,p'\G\p,p')anpnp/ (3) p pp'
replacing the step functions np with the quasi-particle occupation numbers. The quasi-particle effective interaction is obtained from the double functional derivative
52E
f(p,p') = ——= (4)
dnp dnp'
= {p,p'\G\p,p')a + ¡R(P,P')-
The rearrangement terms fR(p,p') come from the functional derivative of the G matrix, that implicitly depends on the occupation numbers. If we neglect the rearrangement terms, the LM parameters are simply
where G matrix is the G matrix with the total spin projected into the relative momentum direction. The derivation of Eqs. (9), (10) is shown in Appendix A (see also [8]). This procedure to evaluate the LM parameters was pursued by several authors [5—10], but the calculations, including the rearrangement terms, are rather involute and the approximations are not always well justified. If one restricts the calculation to the L = 0 LM components, that are in fact the most relevant for their direct relation with physical observables, the procedure is much simpler.
Let us consider symmetric nuclear matter with density p. From the expansion to the second order of the potential energy in Eq. (3) with respect to the variation of the occupation numbers 5np one obtains:
62u = i j2{p,p'\g\p,p')jnp6np/ pp'
(11)
Introducing the two-particle total spin S and isospin T, one gets in the Landau limit
^ = E c* (UUssz) * (i2)
aa' tt' SSZ TTZ
X C2 Qr, \r'\TT^j G°ST5paT5pa,T/ + higher,
where G°ST is the L = 0 component in the Landau angle expansion of G matrix, and higher L contributions are omitted. For spin and isospin saturated nuclear matter ôpaT = ôp/A so that
ó2u óp2
16
Y^(2S + 1)(2T + 1)Gst (13)
ST
p
L
1
which, according to Eq. (5), gives the isoscalar LM parameter F0 and also provides the relation with the nuclear compression modulus
o2 Tp
K = 9p2^ = 6eF(l + F0).
(14)
The value of this parameter can be extracted by the excitation energy of the giant monopole resonances [13].
In the case of isospin-asymmetric nuclear matter 5pn — -5pp — 1/2p(—1)1/2-a/, so that Eq. (11) yields
S2U = p2 + 1)(2T - 1 )G°ST5f32 + higher.
ST
(15)
Comparing with Eq. (6), one recovers the isovector parameter FO and also the relation with the symmetry energy appearing in the nuclei mass tables,
Esym —
d2EA dp2
— ep(1 + Fq )/3.
(16)
To study the spin and spin—isospin Landau parameters we consider isospin-asymmetric nuclear matter with fixed density p and asymmetry 3, and different states of spin polarization. Let us call jn (Yp) the spin polarization of neutrons (protons). The variation of the potential energy with respect to spin polarization 5paT — pT(—1)1/2-T5jT is
tt' TTz
(17)
x [GOt -G0t]pTPt' (-1)1/2-T (-1)1/2-
Defining
x 5yt5jt' + higher.
Go r = 1 S2U
TT' prpr'
(18)
E C*[\t,\ATTz
TTZ
Concerning the tensor LM parameters, it is well known that the tensor force gives rise to the coupling between different two particle channels, the most famous of which is the 3S1-3D1 introduced to explain the nonvanishing deuteron quadrupole moment. The previous procedure cannot be applied in this case, because the energy per particle in the BHP approximation is only contributed by diagonal matrix elements of G matrix in the two particle channels. Therefore the LM tensor parameters must be calculated directly from the diagrammatic expansion of the interaction. The lowest-order contribution is given by off-diagonal matrix elements of G matrix according to Eqs. (9), (10).
3. LANDAU PARAMETERS PROM BHP APPROXIMATION
As above mentioned, the microscopic derivation of the LM parameters has been extensively performed in the framework of the Brueckner theory. In the last years this has made a remarkable step forward, thanks to the introduction of the three-body force [15]. The latter was introduced not only to reproduce the empirical saturation properties of nuclear matter but also to extend hole-line expansion for the energy correlation up to high density.
3.1. Review of the BHP Approximation
The Brueckner theory with two- and three-
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