Semiclassical shell-structure moment of inertia within the phase-space approach
The moment of inertia for nuclear
collective rotations was derived
within the semiclassical
approach based on the cranking model and the
Strutinsky shell-correction method by using
the non-perturbative periodic-orbit theory in
the phase space variables.
This moment of inertia for adiabatic
(statistical-equilibrium) rotations can be approximated by
the generalized rigid-body moment of inertia
accounting for the shell corrections of the
particle density. A semiclassical phase-space trace formula
allows to express quite accurately
the shell components of the moment of inertia
in terms of the free-energy shell corrections for integrable and
partially chaotic Fermi systems,
in good agreement with the quantum calculations.
Keywords: Nuclear collective rotations, cranking model, shell corrections, periodic orbit theory.
Many significant phenomena deduced from experimental data on nuclear rotations were explained within theoretical approaches based mainly on the cranking model [2, 3, 4, 5] and the Strutinsky shell-correction method [6, 7] extended by Pashkevich and Frauendorf [8, 9]. For a deeper understanding of the correspondence between classical and quantum physics of the shell components of the moment of inertia (MI), it is worth to analyze them within the semiclassical periodic-orbit theory (POT) [10, 11, 12, 13, 14, 15, 16]. See also Refs. [17, 18] for the semiclassical description of the so called “classical rotation” as an alignment of the particle angular momenta along the symmetry axis, but also the magnetic susceptibilities in metallic clusters and quantum dots . The semiclassical perturbation expansion of Creagh  was used in the POT calculations  of the MI shell corrections for a spheroidal cavity mean field. The non-perturbative Gutzwiller POT extended to the bifurcation phenomena [21, 22, 23, 24] was applied in  within the cranking model for the adiabatic collective rotations (around an axis perpendicular to the symmetry axis) in the case of the harmonic-oscillator mean field. The MI for the collective adiabatic (statistical-equilibrium) rotations was described as a sum of the Extended Thomas-Fermi (ETF) MI  and shell corrections [26, 27]).
In the present work, we obtain a semiclassical phase-space trace formula for the MI shell components in terms of the free-energy shell corrections for integrable Hamiltonians, including those of the harmonic-oscillator mean field [26, 28] and a spheroidal cavity [22, 24], as well as for partially chaotic Fermi systems .
Within the cranking model, the nuclear collective rotation of the independent-particle Fermi system is associated with a many-body Hamiltonian (Routhian)
and its eigenvalue equation is given by
The frequency (Lagrange multiplier) is determined by the angular momentum projection onto the axis (perpendicular to the symmetry axis) through the constraint for ,
where and are respectively the particle angular-momentum projection onto the axis and the occupation numbers,
is the temperature. In the following is the chemical potential, and the spin (spin-isospin) degeneracy. Brackets, like in indicate a quantum average and a bar above the functions the complex conjugation. For simplicity, the spin degree of freedom is included here only through the degeneracy factor , and then, the moment of inertia can be considered as the following susceptibility:
is the energy of the yrast line due to the constraint (3). For the particle number conservation, one has
In the coordinate representation, Eq. (5) can be re-written for the adiabatic rotations through the one-body Green’s function :
This representation is useful within the semiclassical POT,
weakening the criterium
of the quantum perturbation approximation: The rotation
excitation energy becomes smaller
than the distance between major
with the Fermi energy ). This is in contrast to the quantum criterium
of smallness of the excitation energies with respect to
the nearest neighbor single-particle (s.p.) level spacing around
the Fermi surface.
For the s.p. Green’s function , we shall use the semiclassical Gutzwiller trajectory expansion ,
where the sum runs over the classical trajectories (CT) from point to point with the particle energy (see Fig. 1). is the amplitude depending on the CTs degeneracy (in the case ) and their stability propagator, is the particle action, the Maslov phase determined by the caustic and turning catastrophe points along the CT [11, 12, 13, 16, 23].
There are several reasons leading to oscillations of the MI in Eq. (8), local and non-local, where only the local part is related to the shell effects, whereas non-local contributions could have their origin in reflections of the particle from the boundary. The nearly local approximation () is valid after a statistical averaging over many microscopic quantum states. Then, the small parameter, which is the product of the two dimensionless quantities and , used in  for applying the perturbation approach of Creagh to the classical dynamics  becomes in our approach more weak, , under the usual semiclassical condition , where is the Fermi momentum in units and is the mean nuclear radius. According to (Semiclassical shell-structure moment of inertia within the phase-space approach), one can split the Green’s function into a contribution coming from the direct path between the two points and a contribution that contains the contributions from all other trajectories involving reflections
where is the component related to the trajectory CT for which the action disappears in the limit . One obtains
with the nucleon mass , the particle momentum
and the potential .
with . Such a splitting (12) is associated with that of the spatial particle density ,
and the one of equation (10) in terms of its ETF particle density and its shell correction ,
The standard decomposition of the occupation numbers into the smooth and fluctuating parts is used as usually in the shell correction method .
Substituting (15) and (Semiclassical shell-structure moment of
inertia within the phase-space approach) into the integrand
(12), for the total semiclassical MI ,
one obtains the phase-space trace formula:
and where now . Here is a semiclassical Green’s function in the mixed phase-space representation obtained by the Fourier transformation of the Green’s function , Eq. (Semiclassical shell-structure moment of inertia within the phase-space approach),
where is the Jacobian for the transformation from the perpendicular-to-CT momentum to the one. We inserted formally the additional integral over with the function in (12) and transformed the spacial coordinates and to the phase-space variables and [23, 27]. Using then the Fourier transformation of this function from the coordinate difference to a new momentum and integrating the MI in such a phase space representation over perpendicular-to-CT components of by the stationary phase method, one arrives at (19). Note that the ETF component of the Green’s function CT (10) under the nearly local approximation ( and ) is related to the Thomas-Fermi spectral function (20)
Equation (19) looks similar (the same besides of the factor ) to the semiclassical s.p. energy ,
where the brackets mean the average over the phase space variables , and energy with a weight ,
where is the free-energy shell correction. Here we used the stationary phase (periodic orbit (PO)) conditions for the evaluation of integrals over the phase-space variables and . Within the POT, at a given temperature , one has the PO sum [12, 13, 17, 19, 26, 27]
where is the PO component of the energy shell-correction,
Here and are the period of the particle motion along the PO (accounting for its repetition number) and the PO component of the oscillating (shell) correction to the level density, taken both at the chemical potential for , which, at zero temperature, is equal to the Fermi energy . This PO component can be represented as
with the amplitude of the density oscillations depending on the PO classical degeneracy and stability, and the action along the PO [11, 12, 13, 16, 27, 24]. In (25), is the smooth ETF component and the semiclassical oscillating contribution  where the latter can be expressed in terms of the PO sum, Eqs. (31) and (32). The POs appear through the stationary phase (PO) condition for the calculation of the integrals over and in (24) by the improved stationary phase method [22, 23, 24] as mentioned above. For the phase space average , Eq. (27), one again obtains approximately, through the Green’s function (21), a decomposition into an ETF and a shell-correction contributions:
In Figs. 2 and 3, we show the quantum and semiclassical energy shell corrections for the spheroidal cavity at small (almost no contributions from PO bifurcations) and large deformations (important contributions of PO bifurcations). At zero temperature, the energy shell correction corresponds to the free-energy shell correction . With increasing temperature , one observes an exponential decrease of the free-energy shell corrections [17, 19, 26, 27]. Minima of the energy shell corrections are related to magic particle numbers.
(25) and the TF spectral function (22) corresponding to the CT component of the Green’s function (21) in the nearly local approximation, one can simply neglect corrections (surface and curvature terms) and shell effects. For the spheroidal cavity potential, one obtains from (27) and (22)
where and are the semi-axises of the spheroidal cavity, , is the radius of the equivalent sphere. Using this estimate, one may evaluate the MI shell correction for two selected deformations (see Fig. 4 and 5).
One notices that there is a large supershell effect in both and . The shell structure is much enhanced by the bifurcations of the shorter 3-dimensional (3D) orbits from the parent equatorial (EQ) orbits  at . Both kinds of PO families yield the essential contributions through , in contrast to the classical perturbation results of  where the EQ orbits do not contribute and 3D PO contributions are not considered. Our non-perturbation results for the MI shell corrections can be applied for larger rotational frequencies and larger deformations () where the bifurcations play the dominating role like in the case of the deformed harmonic oscillator .
We derived the shell components of the moment of inertia in terms of the free-energy shell correction within the non-perturbative extended Gutzwiller POT for any effective mean-field potential using the phase-space variables. For the deformed spheroidal cavity and the harmonic oscillator potentials, we found a good agreement between the semiclassical POT and quantum results for and using the Thomas-Fermi approximation for at several critical deformations and temperatures. For smaller temperatures a remarkable interference of the dominant short three-dimensional and parent equatorial orbits and their bifurcations in the superdeformed region are shown. For larger temperatures, the shorter EQ orbits are dominant in this comparison. An exponential decrease of the shell corrections with increasing temperature is analytically demonstrated.
We thank S. Aberg, K. Arita, R. K. Bhaduri, M. Brack, F. A. Ivanyuk, S. N. Fedotkin, S. Frauendorf, M. Matsuo, K. Matsuyanagi, V. O. Nesterenko, V. V. Pashkevich, V. A. Plujko, K. Pomorski, and A. I. Sanzhur for many helpful and stimulating discussions. One of us (A.G.M.) is also very grateful for a nice hospitality and financial support during his working visits of the National Centre for Nuclear Research in Otwock-Swierk and Warsaw, the Institut Pluridisplinaire Hubert Curien in Strasbourg, the Nagoya Institute of Technology, and the Japanese Society of Promotion of Sciences, ID No. S-14130.
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