ЯДЕРНАЯ ФИЗИКА, 2007, том 70, № 11, с. 1910-1924



© 2007 A. G. Magner*, A. M. Gzhebinsky, S. N. Fedotkin

Institute for Nuclear Research, Kyiv, Ukraine Received August 23, 2006; in final form, January 29, 2007

The modified shell-correction method is suggested for the calculation of transport coefficients in a slow nuclear collective dynamics. For the multipole low-lying vibrations near the spherical shape of nucleus the smooth transport coefficients corresponding to the extended Thomas—Fermi approach are used as a macroscopic background. The time-dependent mean field is approximated through the infinitely deep square-well potential for the calculation of the shell corrections. The significant shell effects in stiffness and inertia are found at small temperatures. These effects disappear approximately at the same large enough temperature as in the free energy. It is shown that the collective inertia is essentially larger than that of irrotational flow due to the consistency condition of particle density and potential variations. The collective vibration energies, reduced friction, and effective damping coefficients with accounting for the shell effects are in better agreement with experimental data than that found from the hydrodynamic model.

PACS: 21.60.Ev, 21.60.Cs, 24.10.Pa, 24.75.+i


For calculation of the static nuclear properties like the total binding and deformation energy, the famous shell-correction method (SCM) was suggested by Strutinsky [1] and successfully applied in many further works, see, for instance, [2, 3]. The nuclear energy was defined in [1] as a sum of the phenomenological macroscopic part given by the liquid-drop energy and shell correction. The SCM is based on the concept of existence of the quasiparticle spectrum near the Fermi surface by the Migdal theory of finite fermion systems with a strong interaction of particles [4]. Within this concept, the shell component of free energy can be considered perturbatively as a quasiparticle correction to the total nuclear free energy on the basis of the statistically averaged (macroscopic) background described phenomeno-logically through the liquid-drop model (LDM) or the extended Thomas—Fermi (TF) approach [3].

A first attempt to generalize these ideas to the collective dynamics has been made within the so-called liquid—particle (or gas—liquid) model [5] for time evolution of the one-body density matrix by extracting the quasiparticle effects of the quantum gas with nearly independent particles from the macroscopic liquid-drop background of the almost incompressible condensed matter. In this model of combined dynamics, the macroscopic quantities determined by means of their averaging over the particle phase space describe the short-length correlation (liquid) properties,

E-mail: magner@kinr.kiev.ua

as compared to the size of a heavy enough nucleus. Small nuclear quasiparticle excitations near the Fermi surface [4] are responsible for the long-length (gas) correlations. Thus, they arise as self-consistent corrections to the phenomenological LDM like for the zero sound in the Fermi-liquid theory by Landau. In the semiclassical approximation to the liquid—particle model [6], such a splitting into the two components described by the suitable "liquid" and "gas" properties is realized in the nuclear volume. The collective dynamics within a relatively small nuclear-surface layer is reduced self-consistently to the macroscopic boundary conditions for the Landau—Vlasov equation of motion inside the nucleus.

For the description of low-energy nuclear collective excitations [4, 7—10], more simple proposals were suggested in [11 — 14] by employing the response theory. The collective variables were introduced there explicitly as deformation parameters of a mean single-particle field. The nuclear excitations were parametrized in terms of the transport coefficients, such as the stiffness, the inertia, and the friction parameters defined through the adequate collective response functions. In analogy with the SCM, the response function was split into smooth macroscopic and fluctuative shell (quasiparticle) components. Its fluctuative part was calculated semiclassically within the Periodic Orbit Theory (POT) [3, 15-18], which is a powerful analytical tool for study of the shell effects in level densities and energy shell corrections.

A simple macroscopic derivation of the famous wall formula for the friction due to collisions of particles of the perfect Fermi gas inside the nucleus with


its slowly moving surface was suggested in [19]. From a quite general semiclassical and quantum starting point Koonin, Randrup, and Hatch have rederived this wall formula for the average friction in [20, 21 ]. Other independent derivations of the wall formula with focus on specific averaging procedures can be found, e.g., in [22, 23]. The SCM averaging procedure and TF approach for level density were applied in [23] for relating the wall formula to the averaged quantum friction coefficient.

The main scope of this work is a suggestion of a simpler version of the SCM splitting with applying it immediately to the transport coefficients for slow collective motion. As shown in [14], the averaged transport coefficients can be simplified analytically with help of the POT at the lowest orders in h in the nearly local approximation corresponding to the extended TF approach. We are going now to use this approach accounting for the consistency condition between the particle density and the potential variations [8, 14, 24], as a macroscopic background in the formulation of the modified SCM for the calculations of transport coefficients like the nuclear inertia in a slow collective dynamics. This work is mainly focused on the calculation of shell corrections to the transport parameters and their temperature dependence.

In Section 2, we begin with a general response-function formalism following basically [8]. In Section 3, we suggest to extend the SCM to the calculations of transport coefficients for a slow collective motion by making use of the consistent extended TF approximation [14] for the macroscopic background. For the calculations of their shell corrections, the infinitely deep spherical square-well potential is used as a mean field. Our SCM results for the temperature dependence of the transport coefficients as well as the quadrupole vibration energies, the reduced and effective friction parameters are compared in Section 4 with their counterpartners of the quantum independent-particle (cranking) model [8, 23] and experimental data discussed in [14, 25—27]. The conclusion remarks are given in Section 5. Some details of the semiclassical and quantum calculations can be found in Appendices.


Many-body collective excitations are conveniently described in terms of the nuclear response to an external harmonic perturbation, Vext = F q^e-^, where q5xt is a vibration amplitude, and F is some one-body operator [7, 8]. The nuclear Hamiltonian at q5xt = 0 is assumed to be dependent on a collective variable Q as the time-dependent deformation parameter of

mean-field potential V(Q). For the axially symmetric vibrations of nuclear surface with multipolarity L, frequency u, and radius R(9, Q) near the spherical shape (Q = 0) in the spherical coordinates r, 9, p one writes

R(d,Q) = R[1 + Q(t)YLo(d)],


Q(t) = Qu e


With help of the consistency condition, the collective response, xFF(w), can be expressed in terms of the so-called intrinsic response function x(w) [7, 8, 24],


xFF M = «

x(w) + k'


K = -x(0) - C(0), C(0) =


\9Q2 J


where k is the coupling constant, C(0) the stiffness, F the nuclear free energy, x(0) the isolated susceptibility. The "intrinsic" response x(u), see Eq. (2), can be expressed in terms of the one-body Green function G(ri, r2,£) [11,24]:

x(w) = — den(e) / dr\ x



x J dr2jP(ri)F(r2)ImG(ri, r2,e) x x [G*(ri, r2,e - hw) + G(ri, r2,e + hw)]

G(ri, r2,e) = Y^

e — ei + ir '

where n(e) is the Fermi occupation numbers at the energy e for temperature T, n(e) = {1 + exp[(e — — A)/T]}-1, A the chemical potential, A w eF = = h2kF/2m, eF and kF are, respectively, the Fermi energy and wave number, m is the nucleon mass. The factor 2 accounts for the spin degeneracy. For the Green function, G(r1, r2,e), we may use the spectral representation of Eq. (3) with eigenvalues ei, eigenfunctions ^(r), and r — +0 in the mean-field approximation (the asterisk means the complex conjugation). For the example of the infinitely deep spherical square-well potential V(Q), the single-particle operator F in Eq. (3) is given analytically by:

F(r) =


= —VoR5(r — R)YLo(d), (4)


*lVoG - ( d2°

h2 \dr1dr2 j

j = 1,2,


for Vo —> to with V0 being the potential well depth.


One dominating peak in the collective strength function, rc Imx^(w), in Eq. (2) at low excitation energies is assumed to be well separated from all other solutions of the secular equation x(w) + k = 0. As shown [8, 24] in this case, the response function Xqq (w) in the Q mode can be conveniently written in

an inverted form: 1 1

+ k & -Mw2 - iYw + C, (5)

Xqq(u) xqq(^



Here, the inverse collective response function, Xqq(w), for low frequencies is approximated by the corresponding response function of a damped harmonic oscillator with the stiffness C, the inertia M, and the friction 7 parameters, as seen from Eq. (5). According to Eq. (5), such self-consistent transport coefficients:


C =

1 +

c{ 0) x(0)J

C (0),

Y =

1 +

M =

1 +

g(0) x(0)

M (0) +




x(0 )

Y (0), (6)

were related in [8, 24] to the coefficients C(0), 7(0), M(0) in expansion of the "intrinsic" response function x(w), Eq. (3), in powers of w in the "zero-frequency limit" (w ^ 0), for slow enough collective motion, see [28],



M (0) =


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