Symmetry Breaking and Bifurcations in the Periodic Orbit Theory. II: Spheroidal Cavity

Magner, A. G.; Arita, Ken–ichiro; Fedotkin, S. N.; Matsuyanagi, Kenichi

doi:10.1143/ptp.108.853

Cited by 28 publications

(118 citation statements)

References 23 publications

(49 reference statements)

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“…More specifically, we find a rather good convergence of the PO sums by using the 2DE orbits with n v ≤ 12, At η = √ 2, there occurs a bifurcation of the doubly repeated EQ diameter 2(2,1) with the emergence of the butterfly orbits (4,2,1)2DH, and the improved SPM becomes important in comparison with the quantum results, see the second panels of these figures. However, even at these deformations (η ∼ < 1.4), as shown in [20], these longer orbits do not influence much the gross-shell structure due to the leading period-one orbits mentioned above. Noticeable ISPM contributions of the bifurcating (4,2,1)2DH orbits are found in the lowest panels of Figs.…”

Section: Comparison With Quantum Resultsmentioning

confidence: 78%

“…For more exact higher-order expansions of the action (8), for example up to third-order terms ("3rd-order ISPM"), the convergence for the extension of the integration limits to an infinite region can be sufficiently good, even near the bifurcations, see [20]. As shown from the comparison of the 2nd and 3rd order ISPM approaches in Figs.…”

Section: Trace Formulae and Shell Structure Enhancementmentioning

confidence: 94%

“…The classical dynamics of particles in the spheroidal cavity can be solved in terms of action-angle variables related to the elliptic coordinates κ = v, u, ϕ in this Hamiltonian system. The Hamiltonian, H(I v , I u , I ϕ ), depends only on the partial actions [5,20],…”

Section: Classical Dynamicsmentioning

confidence: 99%

“…12 For the following it is useful to note the classical degeneracies K of the above-mentioned POs [15,18,20,21]. The meridional (planar) orbits (denoted by "2D" henceforth for both elliptic 2DE…”

Section: Classical Dynamicsmentioning

confidence: 99%

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Shell structure and orbit bifurcations in finite fermion systems

et al. 2011

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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 selfconsistent 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. 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 fargoing 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).

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Section: Comparison With Quantum Resultsmentioning

confidence: 78%

Section: Trace Formulae and Shell Structure Enhancementmentioning

confidence: 94%

Section: Classical Dynamicsmentioning

confidence: 99%

Section: Classical Dynamicsmentioning

confidence: 99%

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Shell structure and orbit bifurcations in finite fermion systems

et al. 2011

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“…The pronounced shell effects caused by the deformations have been discussed. Within the improved stationary-phase method [10][11][12][13][14][15][16] (ISPM), the divergences and discontinuities of the standard stationary-phase method (SSPM) [3-5, 8, 9] near the symmetry-breaking and bifurcation points were removed.…”

Section: Introductionmentioning

confidence: 99%

Semiclassical catastrophe theory of simple bifurcations

Magner

Arita

2017

Phys. Rev. E

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The Fedoriuk-Maslov catastrophe theory of caustics and turning points is extended to solve the bifurcation problems by the improved stationary phase method (ISPM). The trace formulas for the radial power-law (RPL) potentials are presented by the ISPM based on the second-and thirdorder expansion of the classical action near the stationary point. A considerable enhancement of contributions of the two orbits (pair of consisting of the parent and newborn orbits) at their bifurcation is shown. The ISPM trace formula is proposed for a simple bifurcation scenario of Hamiltonian systems with continuous symmetries, where the contributions of the bifurcating parent orbits vanish upon approaching the bifurcation point due to the reduction of the end-point manifold. This occurs since the contribution of the parent orbits is included in the term corresponding to the family of the newborn daughter orbits. Taking this feature into account, the ISPM level densities calculated for the RPL potential model are shown to be in good agreement with the quantum results at the bifurcations and asymptotically far from the bifurcation points.

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Microscopic-macroscopic level densities for low excitation energies

Magner

Sanzhur

Fedotkin

et al. 2022

Low Temperature Physics

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Level density [Formula: see text] is derived within the micro-macroscopic approximation (MMA) for a system of strongly interacting Fermi particles with the energy E and additional integrals of motion Q, in line with several topics of the universal and fruitful activity of A. S. Davydov. Within the extended Thomas Fermi and semiclassical periodic orbit theory beyond the Fermi-gas saddle-point method, we obtain [Formula: see text], where Iν ( S) is the modified Bessel function of the entropy S. For small shell-structure contribution, one finds ν = κ/2 + 1, where κ is the number of additional integrals of motion. This integer number is a dimension of Q, Q = { N, Z, …} for the case of two-component atomic nuclei, where N and Z are the numbers of neutrons and protons, respectively. For much larger shell structure contributions, one obtains ν = κ /2 + 2. The MMA level density ρ reaches the well-known Fermi gas asymptote for large excitation energies and the finite micro-canonical combinatoric limit for low excitation energies. The additional integrals of motion can also be the projection of the angular momentum of a nuclear system for nuclear rotations of deformed nuclei, number of excitons for collective dynamics, and so on. Fitting the MMA total level density ρ( E, Q) for a set of the integrals of motion Q = { N, Z}, to experimental data on a long nuclear isotope chain for low excitation energies, one obtains the results for the inverse level-density parameter K, which differs significantly from those of neutron resonances due to shell, isotopic asymmetry, and pairing effects.

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Symmetry Breaking and Bifurcations in the Periodic Orbit Theory. II: Spheroidal Cavity

Cited by 28 publications

References 23 publications

Shell structure and orbit bifurcations in finite fermion systems

Shell structure and orbit bifurcations in finite fermion systems

Semiclassical catastrophe theory of simple bifurcations

Microscopic-macroscopic level densities for low excitation energies

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