= ЯДРА ^^
SHELL STRUCTURE AND ORBIT BIFURCATIONS IN FINITE FERMION SYSTEMS
©2011 A. G. Magner1)*, I. S. Yatsyshyn1), K. Arita2), M. Brack3)
Received December 1, 2010
We first give an overview of the shell-correction method which was developed by V.M. Strutinsky as a practicable and efficient approximation to the general self-consistent theory of finite fermion systems suggested by A.B. Migdal and collaborators. Then we present in more detail a semiclassical theory of shell effects, also developed by Strutinsky following original ideas of M.C. Gutzwiller. We emphasize, in particular, the influence of orbit bifurcations on shell structure. We first give a short overview of semiclassical trace formulae, which connect the shell oscillations of a quantum system with a sum over periodic orbits of the corresponding classical system, in what is usually called the "periodic orbit theory". We then present a case study in which the gross features of a typical double-humped nuclear fission barrier, including the effects of mass asymmetry, can be obtained in terms of the shortest periodic orbits of a cavity model with realistic deformations relevant for nuclear fission. Next we investigate shell structures in a spheroidal cavity model which is integrable and allows for far-going analytical computation. We show, in particular, how period-doubling bifurcations are closely connected to the existence of the so-called "superdeformed" energy minimum which corresponds to the fission isomer of actinide nuclei. Finally, we present a general class of radial power-law potentials which approximate well the shape of a Woods—Saxon potential in the bound region, give analytical trace formulae for it and discuss various limits (including the harmonic oscillator and the spherical box potentials).
1. INTRODUCTION
This paper is devoted to the memory of A.B. Migdal. Our first aim here is to review the shell-correction method (SCM) which was first introduced by Strutinsky on a phenomenological basis [ 1 ] and then microscopically founded [2] on Migdal's theory for strongly interacting finite fermion systems [3]. Our second aim is the discussion of a semiclassical theory of shell effects, using the so-called periodic orbit theory (POT) (see [4] for an introductory text book). It provides us with a nice tool for answering, sometimes even analytically, some fundamental questions asked [5— 7] by Strutinsky: Why are nuclei deformed? What are the physical origins of the double-humped fission barrier and, in particular, of the existence of the isomer minimum? His idea was to use the POT for a deeper understanding, based on classical pictures, of the origin of nuclear shell structure and its relation to a possible chaotic nature of the nucleons' dynamics. We shall present some applications of the POT to nuclear deformation energies and discuss in more
'-'Institute for Nuclear Research, National Academy of Sciences of Ukraine, Kyiv.
2)Department of Physics, Nagoya Institute of Technology, Japan.
3)Institute for Theoretical Physics, University of Regensburg, Germany.
E-mail: magner@kinr.kiev.ua
detail the relation of bifurcations of periodic orbits with pronounced shell effects.
According to the SCM, the oscillating part of the total energy of a finite fermion system, the so-called shell-correction energy 5E, is associated with an in-homogeneity of the single-particle (s.p.) energy levels near the Fermi surface. Its existence in dense fermion systems is a basic point of Landau's quasi-particle theory of infinite Fermi liquids [8, 9], as extended to self-consistent finite fermion systems by Migdal and collaborators [3, 10]. This is schematically illustrated in Fig. 1, where the s.p. level spectrum of a bound nu-
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Fig. 1. Schematic picture showing the relation of local level density at the Fermi energy to the binding of the system (after [11]). Left: low level density — more bound. Right: high level density — less bound.
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N 300
250
200
150
100
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Fig. 2. Neutron energy shell correction SEn(N,n) for a realistic nuclear Woods—Saxon potential (with spinorbit term), versus neutron number N and axis ratio n of the spheroidally deformed potential (equidistance: 2.5 MeV, areas with negative values are shaded). Dots indicate experimental deformations. The heavy bars are the semiclassical predictions for the loci of the ground-state minima using the leading classical periodic orbit families in a spheroidal cavity.
cleus is shown in two extremal situations. Depending on the level density at the Fermi energy — and with it the shell-correction energy 5E — being a maximum or a minimum, the nucleus is particularly unstable or stable, respectively. This situation varies with particle numbers and deformations of the nucleus. In consequence, the shapes of stable nuclei depend strongly on particle numbers and deformations. This is illustrated in Fig. 2. Here the shell correction 5E of the neutrons is shown as function of the neutron number N and the deformation parameter n of a Woods—Saxon potential [12] with spheroidal shape, n being the ratio of the semi-axes. if we fix the neutron number N, e.g., N = 150, and increase the deformation n, we meet the first minimum (ground state) at about n ~ ~ 1.25 and the next one (isomeric state) at much larger deformations n ~ 1.9—2.1. The experimental data corresponding to these deformations are shown in Fig. 2 by the heavy dots.
The SCM was successfully used to describe nuclear masses and deformation energies and, in particular, fission barriers of heavy nuclei. For an early review by Strutinsky's group, in which also the miscroscopic foundations of the SCM are discussed, see [11]. (We refer to Section 3.2 for a further discussion of fission barriers.)
In Section 2, we will give a short review of the SCM and its foundation on the basis of a self-
consistent theory of finite interacting fermion systems.
Section 3 is devoted to the semiclassical theory of shell effects. The POT is based on Gutzwiller's semiclassical trace formula for the level density for a Hamiltonian system with isolated orbits [13] and its extensions to systems with continuous symmetries [4, 5, 14—17]. It allows one to relate both the oscillating part of the level density and the shell-correction energy of a quantum system to the shortest periodic orbits (POs) of the corresponding classical Hamiltonian system. Thus, one can often explain pronounced shell effects by the role of particular short POs. As an early example, taken from [5], the heavy bars in Fig. 2 are the predictions of the POT for the loci of the ground-state minima, using the shortest POs in a spheroidal cavity. Bifurcations of POs under the variation of a deformation parameter or the (Fermi) energy can have noticeable effects for the shell structure [5-7, 18-21].
In Section 3.1, we will present the structure of semiclassical trace formula and discuss a general method of treating bifurcations in the POT, using the catastrophe theory of Fedoryuk and Maslov for caustic and turning-point problems [22-25].
In Section 3.2, we review the semiclassical description [26] of a typical nuclear fission barrier in terms of the shortest periodic orbits, employing a cavity model with the realistic shape parameterization developed in [11]. In particular, the effect of left-right asymmetric deformations on the height of the outer fission barrier will be discussed. Isochronous bifurcations of the shortest orbits are treated here in a uniform approximation employing a suitable normal form for the action function.
In Section 3.3, we use the spheroidal cavity [5, 19, 20, 27] as a simple integrable model that allows to study semiclassically the shell structure related to the "superdeformed" energy minimum which in realistic actinide nuclei corresponds to the fission isomers. Strutinsky's prediction concerning the importance of the enhancement of the shell structure owing to period-doubling bifurcations of three-dimensional POs from simple equatorial (EQ) orbits in the spheroidal cavity model [5] will be discussed.
In Section 3.4, we shall study a radial power-law potential [28], which is a good approximation to the familiar Woods-Saxon potential for nuclei in the spatial domain where the particles are bound. We shall establish generalized trace formulae for this potential and discuss various limits to other known potentials.
The paper is summarized in Section 4, where we also present some conclusions and plans for future research. Some technical details of our POT calculations are given in the Appendix.
2. THE SHELL-CORRECTION METHOD
In 1966, Strutinsky achieved a far-reaching break-through [29], following basically Migdal's theory of finite fermion systems [2, 3]. Until then, many attempts had been made to incorporate quantum shell effects in the calculation of nuclear deformation energies. But they all failed in reproducing the fission barriers of actinide nuclei and details such as, e.g., the left—right asymmetry of the nascent fission fragments [30]. Summing the s.p. energies of a deformed shell model (like the Nilsson model [31]) up to the Fermi energy failed at larger deformations. There was a need to renormalize the wrong average part of the s.p. energy sum. Knowing that the smooth part of the nuclear binding energy could be well described by the phenomenological liquid drop model (LDM) [32-34] (or droplet model [35]), Strutinsky wrote the total nuclear energy as [1, 11]
Etot = ELDM + $E, (1)
where ELDM is the LDM energy and 5E the so-called "shell-correction energy" which contains the fluctuating part of the s.p. energy sums for the neutrons:
N/2 IN/ 2 \
6E = J2 En En) , (2)
n=1 \n=1 /
and similarly for the protons. Both parts of the total energy (1) depend on the neutron and proton numbers N and Z, as well as on the nuclear deformation which has to be suitably parameterized both in the LDM and the
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