ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 10, с. 1293-1366
К 90-ЛЕТНЕМУ ЮБИЛЕЮ ^^^^^^^
СПАРТАКА ТИМОФЕЕВИЧА БЕЛЯЕВА
A SEMICLASSICAL COLLECTIVE RESPONSE OF HEATED, ASYMMETRIC, AND ROTATING NUCLEI
©2014 A. G. Magner1)*, D. V. Gorpinchenko1),2), J. Bartel3)
Received August 9, 2013; in final form, October 7, 2013
The quasiparticle Landau Fermi-liquid and periodic orbit theories are presented for the semiclassical description of collective excitations in nuclei, which are close to one of the main topics of the fruitful activity of S.T. Belyaev. Density—density response functions are studied at low temperatures within the temperature-dependent collisional Fermi-liquid theory in the relaxation time approximation. The isothermal, isolated (static) and adiabatic susceptibilities for nuclear matter show the ergodicity property. Temperature corrections to the response function, viscosity and thermal conductivity coefficients have been derived, also in the long wave-length (hydrodynamic) limit. The relaxation and correlation functions are obtained through the fluctuation—dissipation theorem and their properties are discussed in connection to the static susceptibilities and ergodicity of the Fermi systems. Transport coefficients, such as nuclear friction and inertia as functions of the temperature for the hydrodynamic (heat-pole and first sound) and Fermi-surface-distortion zero-sound modes are derived within the Fermi-liquid droplet model. They are shown to be in agreement with the semi-microscopical calculations based on the nuclear shell model (SM) for large temperatures. This kinetic approach is extended to the study of the neutron—proton correlations in asymmetric neutron-rich nuclei. The surface symmetry binding-energy constants are presented as functions of the Skyrme-force parameters in the approximation of a sharp-edged proton— neutron asymmetric nucleus and applied to calculations of the isovector giant dipole resonance. The energies, sum rules, and transition densities of these resonances obtained by using analytical expression for these surface constants in terms of the Skyrme-force parameters are in fairly good agreement with the experimental data. An analysis of the experimental data, in particular the specific structure of these resonances in terms of a main, and some satellite peaks, in comparison with our analytical approach and other theoretical semi-microscopical models, might turn out to be of capital importance for a better understanding of the values of the fundamental surface symmetry-energy constant. The semiclassical collective moment of inertia is derived analytically beyond the quantum perturbation approximation of the cranking model for any potential well as a mean field. It is shown that this moment of inertia can be approximated by its rigid-body value for the rotation with a given frequency within the ETF and more general periodic orbit theories in the nearly local long-length approximation. Its semiclassical shell-structure components are derived in terms of the periodic-orbit free-energy shell corrections. An enhancement of the moment of inertia near the symmetry-breaking bifurcation deformations was found. We obtained good agreement between the semiclassical and quantum shell-structure components of the moment of inertia for several critical bifurcation deformations for the completely analytically solved example of the harmonic oscillator mean field.
DOI: 10.7868/S0044002714090062
1. INTRODUCTION
Nuclear collective motion, such as fission, or multipole vibration and rotation excitation modes, was successfully studied by using several microscopic-macroscopic approximations to the description of the finite Fermi systems of the strongly interacting nucleons [1-6]. Many significant phenomena
''Institute for Nuclear Research, Kiev, Ukraine.
2)National Technical University of Ukraine, "KPI", Kyiv.
Institut Pluridisciplinaire Hubert Curien, CNRS/IN2P3,
Université; de Strasbourg, France.
E-mail: magner@kinr.kiev.ua
deduced from experimental data on nuclear fission, vibrations, and rotations were explained within the theoretical approaches based mainly on the cranking model [4, 7—9] and its extensions to the pairing correlations [10—13], including shell and temperature effects [14], and to non-adiabatic effects [5, 15—21], which were originally applied for the rotational modes (see also [22] for the review paper and references therein).
For the nuclear collective excitations within the general response-function theory [4,23, 24], the basic idea is to parametrize the complex dynamical problem of the collective motion of many strongly interact-
ing particles in terms of a few collective variables found from the physical meaning of the considered dynamical problem, for example the nuclear surface itself [25—27] or its multipole deformations [4]. We can then study the response to an external field of the dynamical quantities describing the nuclear collective motion in terms of these variables. Thus, we get important information on the transport properties of nuclei. For such a theoretical description of the collective motion it is very important to take into account the temperature dependence of the dissipative nuclear characteristics as the friction coefficient, as shown in [24, 28—30]. The friction depends strongly on the temperature and its temperature dependence can therefore not be ignored in the description of the collective excitations in nuclei. Concerning the temperature dependence of the nuclear friction, one of the most important problems is related to the properties of the static susceptibilities and ergodicity of the Fermi systems like nuclei.
However, the quantum description of dissipative phenomena in nuclei is rather complicated because we have to take into account the residual interactions beyond the mean-field approximation. Therefore, more simple models [26, 31—33] accounting for some macroscopic properties of the many-body Fermi-system are helpful to understand the global average properties of the collective motion. Such a model is based on the Landau Fermi-liquid theory [34—36], applied for the nuclear interior and simple macroscopic boundary conditions on the nuclear surface [26, 27, 33, 37—40] (see also macroscopic approaches with different boundary conditions [41 — 45]). In [32], the response-function theory can be applied to describe collective nuclear excitations as the isoscalar quadrupole mode. The transport coefficients, such as friction and inertia, are simply calculated within the macroscopic Fermi-liquid droplet model (FLDM) [31—33] and their temperature dependence can be clearly discussed (see also earlier works [27, 37, 46—49]). The asymmetry of heavy nuclei near their stability line and the structure of the isovector dipole resonances are studied in [33, 50—52] (see also [53, 54]). In this way, the giant multipole resonances were described, and, with increasing temperature [31, 32], a transition from zero-sound modes to the hydrodynamic first sound. The friction in [31, 32] is due to the collisions of particles, which were taken into account in the relaxation-time approximation [35, 36, 55—58] with a temperature and frequency dependence (retardation effects) [31, 34].
The most important results obtained in [32, 59] are related to the overdamped surface excitation mode for the low-energy region and its dissipative characteristics as friction. For the low excitation energy region these investigations can be completed
by the additional sources of the friction related to a more precise description of the heated Fermi liquids presented in [57, 58] for the infinite matter. Following [57], we should take into account the thermody-namic relations along with the dynamical Landau-Vlasov equation and introduce the local equilibrium distribution instead of the one of global statics, used earlier in [32, 59] for the linearization procedure of this equation. These new developments of the Landau theory are especially important for the further investigations of the temperature dependence of the friction. For the first step we have to work out in more details the theory [57] of the heated Fermi liquids for nuclear matter to apply then it for the dynamical description of the collective motion in the interior of nuclei in the macroscopic FLDM [31, 32]. Our purpose is also to find the relations to some general points of the response function theory and clarify them taking the example of the analytically solved model based on the non-trivial temperature-dependent Fermi-liquid theory. One of the most important questions which would be better to clarify is the above-mentioned ergodicity property, temperature dependence of the friction and coupling constant.
Another important extension of this macroscopic theory is to study the structure of the isovector giant dipole resonance (IVGDR) as a splitting phenomenon due to the nuclear symmetry interaction between neutrons and protons near the stability line [33, 40, 50—54]. The neutron skin of exotic nuclei with a large excess of neutrons is also still one of the exciting subjects of nuclear physics and nuclear astrophysics [2, 60—69]. Simple and accurate solutions for the isovector particle density distributions were obtained within the nuclear effective surface (ES) approximation [25—27, 39, 40]. It exploits the saturation of nuclear matter and a narrow diffuse-edge region in finite heavy nuclei. The ES is defined as the location of points of the maximum density gradient. The coordinate system, connected locally with the ES, is specified by the distance from the given point to the surface and by tangent coordinates at the ES. The variational condition for the nuclear energy with some additional fixed integrals of motion in the local energy-density theory [70, 71] is significantly simplified in these coordinates. In particular, in the extended Thomas—Fermi (ETF) approach [72, 73] (with Skyrme forces [74—79]) this can be done for any
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