A Relativistic Thomas-Fermi Description of Collective Modes in Droplets of Nuclear Matter

Isoscalar collective modes in a relativistic meson-nucleon system are investigated in the framework of the time-dependent Thomas-Fermi method. The energies of the collective modes are determined by solving consistently the dispersion relations and the boundary conditions. The energy weighted sum rule satisfied by the model allows the identification of the giant ressonances. The percentage of the energy weighted sum rule exhausted by the collective modes is in agreement with experimental data, but the energies come too high.


I. INTRODUCTION
Renormalizable relativistic quantum field theories of hadronic degrees of freedom, called quantum hadrodynamics (QHD), have been studied for some time [1,2]. At the level of the mean-field theory (MFT) and one-loop approximation, these models have proven to be a powerful tool for describing the bulk properties of nuclear matter. The binding energy of nuclear matter in MFT arises from a strong cancellation between repulsive vector and attractive scalar potentials. Such potentials are comparable to those suggested by Dirac phenomenology [3,4], Brueckner calculations [4], and finite-density QCD sum rules [5]. Therefore, it is not obvious that QHD would be able to reproduce the spectrum of finite nuclei, involving energies of the order of tens of MeV. However, it has been shown that it can realistically describe densities, single-particle energies and the spectrum of collective excitations of finite systems [1,2,[6][7][8][9].
Collective modes of a relativistic many-body system are characterized as poles of the meson propagator. However, in the one-loop approximation, the meson propagators have also poles at space-like momenta, which arise from polarization effects of the Dirac sea [10][11][12][13]. While the existence of these poles does not rule out meson-nucleon field theories as useful descriptions of nuclear systems at low q, it may restrict the range of validity of several approximations to these theories. To avoid this problem, in this work we will study collective excitations of finite nuclear systems in a semiclassical approximation to the Walecka model.
In refs. [14,15] a semiclassical approximation to the Walecka model was introduced to study collective modes in nuclear matter at zero and finite temperature. It was found that the results obtained are compatible with microscopic calculations of the meson propagators [16,17]. We want to generalize this semiclassical approach to the description of collective modes of finite nuclei by using a nuclear fluid-dynamical model [18,19], which incorporates monopole and quadrupole distortions of the Fermi surface. This nuclear fluid-dynamical model has recently been applied with success to the description of temperature effects in collective excitations of finite nuclei [20].
In ref. [9] (which is a generalization of the works presented in refs. [7,8]) isovector and isoscalar collective modes were calculated in the Walecka model, by introducing local hydrodynamic variables to describe the nucleon fluids with the assumption of irrotational flow and in the limit of large masses for the vector mesons. As suggested in ref. [9], we lift these restrictions and in this work we calculate the isoscalar collective modes in the Walecka model in the framework of the time-dependent Thomas-Fermi method.
In Sec. II we extend the formalism developed in refs. [18,19] to the Walecka model. Collective modes are described by allowing the meson-fields and the nucleon densities to acquire a time dependence. The nucleon motion modifies the source terms in the meson field equations producing corresponding time-dependent changes in the meson fields. Since the nucleon dynamics is in turn specified by the meson fields, collective modes of nuclear motion arise naturally in this approach. In Sec. III we derive the equations of motion, boundary conditions and orthogonality relations that the normal modes must satisfy. The dispersion relations, which solved consistently with the boundary conditions, determine the eigenvalues, are presented in Sec. IV. In this section the sum rule satisfied by the model is also given. We identify two rather collective monopole modes at 28 MeV and 35 MeV. These large values are expected since the isoscalar monopole excitation is a compression mode and, therefore, its energy is related to the compressibility of nuclear matter [21], which is known to be too low in the Walecka model. For the other multipolarities, we also observe that the most collective states come at higher energies than the experimentally observed giant resonances. It is true that our lowest modes coincide with the modes obtained by [9]. However, these modes only carry a small percentage of the energy weighted sum rule and therefore should not be identified with the giant resonances. Finally, in Sec. V we give our numerical results and conclusions.

II. FLUID-DYNAMICAL MODEL
In a classical approximation to the Walecka model the energy of a nuclear system is given by [14] where the distribution function, f (x, p, t), is restricted by the requirements and its time evolution is described by the Vlasov equation Using the Vlasov equation, Eq.(2.4), it can be easily shown that the four-current satisfies the continuity equation, and that the components of the vector field are related through [14]: Therefore, the second term on the right-hand side of Eqs.(2.5b,2.5c) vanishes. In our calculations we will assume that the density of a spherical nucleus in the groundstate is constant inside the nucleus and zero outside, and is given by , p F is the nuclear matter Fermi momentum, and R 0 is the nuclear radius. The ground-state distribution function f 0 is determined by the particle number A and by the minimization of the energy and the equilibrium nuclear matter density, ρ 0 , is calculated from equations (2.11) and (2.10) Giant resonances manifest themselves as small amplitude highly collective modes. Therefore, they are described at the microscopic level by the RPA equations. In the classical limit, these equations are obtained by the linearization of the Vlasov equation. In this context we begin by expanding the distribution function around its equilibrium value f 0 (x, p): where S(x, p, t) is a generating function which describes small deviations from equilibrium. In its more general form, the distribution function, f (x, p, t), should include static as well as dynamic deformations of the nuclear system. For this reason we decompose the infinitesimal generator S(x, p, t) into a time-even and a time-odd part S(x, p, t) = P (x, p, t) + Q(x, p, t) , (2.13b) P (x, p, t) = −P (x, −p, t) . (2.13c) The time-even generator, Q(x, p, t), takes into account the dynamic deformations. The static deformations are described by the time-even distribution function, which includes the fields responsible for the deformations of the Fermi surface. In the present approach, it is expressed in terms of the time-odd generator P (x, p, t) (2.14) The scalar field, W (x, t), is related to the deformations which preserve the spherical form of the Fermi surface. The tensor field, χ ij (x, t), introduces deformations in the Fermi sphere. Hopefully, the scalar and tensor fields will provide an adequate description of the monopole and quadrupode deformations of the Fermi sphere. In Eq.(2.14), , and σ 0 (x) and V 0 0 (x) are, respectively, the equilibrium values of the fields σ and V 0 . The Fermi momentum is related to λ through The introduction of the generator Q(x, p, t) destroys the time reflexion invariance of the equilibrium distribution function. It will allow for the appearance of transverse flow [22] in the nucleus. The simplest choice which includes this possibility is given by [19] Q(x, p, t) = ψ(x, t) where ψ(x, t) and φ ij (x, t) are, respectively, scalar and symmetrical tensor fields. The time evolution of the generator S and the field fluctuations are determined by the appropriate Lagrangian. For small deviations from equilibrium it is enough to consider the quadratic Lagrangian Using the ansatz Eqs.(2.12), (2.14) and (2.16), decomposing the boson fields into a static (ground-state) contribution and a small time-dependent increment and imposing the barion number conservation, we get The surface integrals in the above equations take into account possible surface displacements parametrized by a vector field, δR(x). Our choice of the even distribution function allows explicitly for this effect. In Eq.

III. EQUATIONS OF MOTION, BOUNDARY CONDITIONS AND ORTHOGONALITY RELATIONS
The equations of motion and boundary conditions that specify the dynamics of the fields are obtained from Eq.(2.17) through the Euler-Lagrange equations. We get It is worth mentioning that Eqs.(3.1a) to (3.1i) are valid only in the interior of the nucleus. Therefore, we replace p F , ǫ F and ρ 0 in these equations by their equilibrium values. At the surface, the variational fields satisfy the following boundary conditions In order to ensure that the current density is not singular at the surface, the following boundary condition has also to be imposed [19] x k φ kj | r=R 0 = 0 .
In Eq.(3.2e), ξ is a vector Lagrange multiplier that takes into account the restriction (3.3).
We look for normal-mode solutions where all the fields oscillate harmonically in time. This means that the fields are described by a superposition of the real parts of where all the quantities within the braces are only functions of x. This normal-mode analysis leads to the RPA coupled equations for the eigenmodes: It is clear form Eq.(3.4h) that χ ij and φ ij are canonically conjugate fields. The solutions of the the above equations satisfy the following orthogonality relation

A. Dispersion Relations
The eletric modes are described by the same kind of solutions as constructed in ref. [19], i. e., by two kinds of transverse fields and by one longitudinal tensor field The advantage of using the above combination of the four linearly independent angular tensor functions: , is that all solutions given above are traceless. In particular, the transverse fields also verify the relations This is the same relation as obtained in ref. [19]. This should be expected since the meson fields, which are the new ingredients in the model used here, do not couple to the solution of kind 1. For solutions of kind 2, we still have [δR i ] 2 = 0, since, from Eq.(3.2a), the vector field δR is directly related to the scalar field δσ. However, the vector fields [δV i ] 2 and [Π V i ] 2 are coupled to the tensor fields. We get where and m * v 2 = m 2 v + ∆m 2 v . Using the solutions of kind 2 in the normal mode equations we get which give us two different solutions for k 2 2 . For g v = 0, one of the solutions is exactly the same which is obtained in ref. [19]. This solution is now modified and a new solution appears, due to the coupling between the vector meson field and the fields introduced to describe the nuclear deformations.
The longitudinal solutions, [φ ij ] 3 , couple to all other fields and give plus the corresponding solutions to the canonically conjugated fields. In the above equations we have introduced the functions and with G v (k) defined in Eq.(4.7). The dispersion relation obeyed by these solutions is There are four solutions of kind 3, two more than the number of this kind of solutions found in [19]. This should be expected since, besides the vector meson field, the scalar meson field also couples to the longitudinal solution [φ ij ] 3 . It is easy to show that for g s = 0 and g v = 0 one recovers the two solutions of ref. [19]. Therefore, the Walecka model leads to the appearance of 7 different values for k for a fixed frequency ω, in contrast with the model of ref. [19], which gives only 4 different values.
There is still a fourth kind of solution for the tensor fields, which can be chosen to be coupled to the scalar fields (4.14) and to the meson fields where F (r) is an arbitrary function. This solution is not trivial because of the boundary condition Eq. (3.3). The general solution, for each normal mode, is a linear combination of the eight particular solutions: 16) with similar expressions for the other fields.
To avoid zero frequency modes linked to the surface motion, we introduce in the model a surface energy which, in a classical approximation, is given by where σ sup is the surface tension coefficient. This term does not alter the equations of motion and, therefore, the dispersion relations. It only changes the boundary condition Eq.(3.2f) tȯ

B. Sum Rules
Sum rules can be regarded as a test to the validity of a particular nuclear model. Suppose that a nucleus is excited from its ground state |0 to an excited state |n , with an energy E n , due to interactions with an external field. One can define momenta, weighted in energy, of the excitation strength distribution whereÔ is the one-body hermitian operator, responsible for the excitation. In the above expression, k = 0, ±1, ±2, ... and |n stands for a set of eigenstates of the hamiltonian of the system. A sum rule is obtained when it is possible to relate a momentum with a known quantity.
The energy weighted sum rule (EWSR), m 1 , is obtained through the calculation of the expectation value of a double commutator In the present problem, the general solution for the variational fields is given by the real part of where the coefficients a n are determined by the initial conditions. In order to derive the EWSR for the electric modes we consider the following initial condition with D(x) to be specified. We then expand the fields ψ(x, 0), Π σ (x, 0), φ ij (x, 0) and δV(x, 0) as ϕ(x, 0) = n a n ϕ (n) (x), where, from the orthogonality relation Eq.(3.5), we get The coefficients a n are related to the expectation value of the transition operator, a n = √ 2 n|Ô|0 . Therefore, the EWSR can be written as m 1 = n |a n | 2 ω n = 2E (2) , (4.24) and, for the initial condition given in Eq.(4.24), the EWSR reads n |a n | 2 ω n = d 3 x ρ 0 ǫ F ∇D.∇D .  [23], is σ sup = 1.017 MeV/fm 2 . The results were calculated for a nucleus with A = 208. The radius R 0 is obtained from the value of p F corresponding to the chosen set of parameters.
For the excitation operator introduced in 4.22a we will use (5.1b) Table I shows the energies of the normal modes together with the corresponding percentage of the exhausted energy weighted sum rule (EWSR), for the two sets given above and for different multipolarities. The EWSR is fragmented over the whole range of energies and only the nuclear modes which exhaust more than 0.1% of the sum rule are given. The distribution of the EWSR between the nuclear modes and the mesonic modes (energies larger than the meson masses) agrees with the results obtained in [14], where it is shown that in infinite nuclear matter and for small momentum transfer about 62% of the EWSR is exhausted by the continuum nuclear modes and about 38% by the vector meson modes. For instance, for l = 2 + and for set I, we find a vector meson mode athw i = 984.56MeV which exhausts 27.30% of the EWSR. The other mesonic modes are not as collective as this one and are distributed over a large range of energies. This pattern is reproduced for the two sets of parameters and for all multipolarities. The EWSR is fulfilled considering all the nuclear and mesonic modes. In non-relativistic calculations using the same nuclear fluid-dynamical model used here [18][19][20], the mesonic modes are not present and, therefore, the EWSR is distributed only through the nuclear modes. From this Table we can see that for set II the collective modes come at a slightly higher energy than in set I and that the strength is more concentrated at higher energies.
In table II we give for set I and for l π = 0 + , 2 + , 3 − , 4 + the energy of the normal modes with energy below 100 MeV (first column) and the corresponding percentage of the energy weighted sum rule (EWSR) (third column). In the second column we present a renormalized percentage of the EWSR, renormalizing the strength distributed among states with energy below 130 MeV to 1. The renormalizing factor is m ′ 1 (l) = 0.56, 0.60, 0.61, 0.56, respectively, for l = 0, 2, 3, 4. This is done so that we can compare more easily the results obtained in the present work with previous results obtained in a non-relativistic fluid-dynamical model, [20], (columns 4 and 5) and experimental data (columns 6 and 7) [24]. Looking at the modes with energy below 100 MeV, we may immediately conclude that there is a certain correspondence between the states obtained in the present approach and the ones of [20], if we identify the states by the percentage of the exhausted EWSR. However, the corresponding states come, in the present relativistic approach, at higher energies. For instance, the quadrupole low lying mode and giant resonance come, respectively, at 10 and 20 MeV and exaust 8% and 77% of the EWSR while the experimental modes come at 4 and 11 MeV and exhaust 15% and 70% of the EWSR. Another possible way of identifying the modes is done by comparing the current transition density (2.8) and the transition density (2.7) for these two modes with the ones of ref. [20]. In figure 1 and 2 we plot j + , j − , j div (arbitrary units) defined by the equations: The function j div is related to the transition density δρ (ρ = −∇ · j). For the 10.03 MeV mode, j + and j − have oposite signs and j div is close to zero, characteristic of a surface mode. These are typical properties of a low lying mode. For the 20.15 MeV mode, j + and j − have the same sign and j div comes diferent from zero for r/R 0 > 0.5. This behaviour is closer to the behaviour expected from a giant resonance. We conclude the identification we have done is correct. We note that our modes with the lowest energy have energies similar to the ones obtained in ref. [9], however, these are not the states that exhaust the largest percentage of the EWSR and, therefore, they should not be identified with the giant resonances. The breathing mode comes at a very high energy, but this was expected owing to the high incompressibility of the model. While in ref. [9] only the lowest modes were determined, we have found all the modes that exhaust a significant fraction of the corresponding EWSR (which we also derived). Furthermore, we have shown that the lowest modes are not the most collective ones.

VII. ACKNOWLEDGMENTS
This work was partially supported by CNPq, JNICT and Fundação Calouste Gulbenkian. MN acknowledges the warm hospitality and congenial atmosphere provided by the Centro de Física Teórica of the University of Coimbra during her stay in Portugal. TABLE II. Comparison between the energies and fractions of the energy-weighted sum rule obtained in the present work (first column), in [20] (second column) and experimental data (third column) [24].