Interpretation and quality of the tilted axis cranking approximation

Comparing with the exact solutions of the model system of one and two particles coupled to an axial rotor, the quality of the semi classical tilted axis cranking approximation is investigated. Extensive comparisons of the energies andM1 andE2 transition probabilities are carried out for the lowest bands. Very good agreement is found, except near band crossings. Various recipes to take into account finiteK within the frame of the usual principal axis cranking are included into the comparison. A set of rules is suggested that permits to construct the excited bands from the cranking configurations, avoiding spurious states.     

Progress on tilted axis cranking covariant density functional theory for nuclear magnetic and antimagnetic rotation

Magnetic rotation and antimagnetic rotation are exotic rotational phenomena observed in weakly deformed or near-spherical nuclei, which are respectively interpreted in terms of the shears mechanism and two shearslike mechanism. Since their observations, magnetic rotation and antimagnetic rotation phenomena have been mainly investigated in the framework of tilted axis cranking based on the pairing plus quadrupole model. For the last decades, the covariant density functional theory and its extension have been proved to be successful in describing series of nuclear ground-states and excited states properties, including the binding energies, radii, single-particle spectra, resonance states, halo phenomena, magnetic moments, magnetic rotation, low-lying excitations, shape phase transitions, collective rotation and vibrations, etc. This review will mainly focus on the tilted axis cranking covariant density functional theory and its application for the magnetic rotation and antimagnetic rotation phenomena.     

Antimagnetic rotation in ~(108,110)In with tilted axis cranking relativistic mean-field approach

Based on tilted axis cranking relativistic mean-field theory within point-coupling interaction PC-PK1,the rotational structure and the characteristic features of antimagnetic rotation for △I=2 bands in~(108,110)In are studied.Tilted axis cranking relativistic mean-field calculations reproduce the experimental energy spectrum well and are in agreement with the experimental I~ω plot,although the calculated spin overestimates the experimental values.In addition,the two-shears-like mechanism in candidate antimagnetic rotation bands is clearly illustrated and the contributions from two-shears-like orbits,neutron(gd)orbits above Z=50 shell and Z=50,N=50 core are investigated microscopically.The predicted B(E2),dynamic moment of inertia J~((2)),deformation parameters β and γ,and J~((2))/B(E2) ratios in tilted axis cranking relativistic mean-field calculations are discussed and the characteristic features of antimagnetic rotation for the bands before and after alignment are shown.     

Shape evolution in <Superscript>76,78</Superscript>Kr nuclei at high spins in tilted axis cranking Hartree-Fock-Bogoliubov approach

A two-dimensional tilted axis cranking Hartree-Fock-Bogoliubov (CHFB) calculation is performed for ⁷⁶Kr and ⁷⁸Kr nuclei up to high spins J = 30 employing a pairing-plus-quadrupole (PPQ) model interaction Hamiltonian. Intricate details of the evolution of single particle structures and shapes as a function of spin have been investigated. The results show the existence of energy levels with high K quantum numbers lying close to the yrast line in both the nuclei. Such high K states should exhibit isomeric characteristics due to the K-selection rules for the γ-decays. Moreover, in ⁷⁸Kr a new band with J = 20–30 lying below the observed ground band is predicted.      

The behavior of quasi-particle levels with allowance for neutron-proton correlations within the HFB cranking approximation

The cranking model taking into account the residual neutron-proton correlations with Gaussian interaction in the Hartree-Fock-Bogolyubov approximation has been formulated. The behavior of quasi-particle levels as a function of the rotation frequency of even-even and odd-odd nuclei with A = 158 has been investigated within this approach.     

An approximation for rotation-projected expectation values of the energy for deformed nuclei and a derivation of the cranking variational equation

An approximation is given for the expectation value of the Hamiltonian with functions: where ¦?〉 is a HFB wave function of a deformed nucleus. This is achieved by expanding the overlap function of the energy 〈?¦ HD(Ω)¦?〉 in derivatives of the overlap function of the norm 〈? D(Ω)¦?〉. The cranking equations for even nuclei are yielded by varying the approximate energy.     

Polar anchoring strength of a tilted nematic: confirmation of the dual easy axis model

The polar anchoring strength coefficient W and polar pretilt angle theta0 were measured simultaneously for the liquid crystal pentylcyanobiphenyl at a rubbed polyimide alignment layer that is ordinarily used for vertical alignment. It was found that W proportional theta(2)0 over the range 0 degrees < or =theta0 less or similar to 35 degrees . The results provide a confirmation of the dual easy axis model, wherein the liquid crystal director adopts an equilibrium orientation theta0 at the substrate that is determined by competition between a pair of preferred orientation directions.     

Stability of steady thermal convection in a tilted rectangular cavity in low-mode approximation

The model system of ordinary differential equations [1, 2] governing the behavior of a non-uniformly heated fluid in a tilted cavity is used for studying the stability of steady regimes of thermal convection at arbitrary (not small) tilting of the rectangular cavity. The bifurcation curve is constructed, which separates the region of parameters (the Rayleigh number — the cavity tilting angle) into two regions — the internal and external ones. In the external region, the system has one stable steady solution, and in the internal region, it has three steady solutions. One of them is always unstable in a monotone way, and two others may be both stable and unstable. The neutral curves are constructed, which determine the boundaries of the incipience of the oscillatory and monotone instabilities. The work was financially supported by the Russian Foundation for Basic Research (Grant No. 07-01-96070).     

Analytical relationship for the cranking inertia

The wave functions of a spheroidal harmonic oscillator without spin-orbit interaction are expressed in terms of associated Laguerre and Hermite polynomials. The pairing gap and Fermi energy are found by solving the BCS system of two equations. Analytical relationships for the matrix elements of inertia are obtained as a function of the main quantum numbers and potential derivative. They may be used to test complex computer codes developed in a realistic approach of the fission dynamics. Results given for the ²⁴⁰Pu nucleus are compared with a hydrodynamical model. The importance of taking into account the correction term due to the variation of the occupation number is stressed.     

Spin alignment in the particle-rotor and cranking models

It is shown that the cranking model normally gives a smaller rotation-aligned spin for an odd quasiparticle than the particle-rotor model, especially at low rotational frequencies. The basic reason is found to be that the rotational frequency vector of the cranking model is “sharp”. This is an unphysical model property, and in the presence of a particle whose rotational motion is partly decoupled from the rotational motion of the average field its consequences become serious. A “sharp” rotational frequency corresponds to a neglect of the recoil effect that establishes coherence between the motion of the decoupled nucleon and the other nucleons and therefore is a prerequisite for the conservation of angular momentum. In conclusion the cranking model cannot be invoked to explain the so-called “Coriolis attenuation”, relative to the particle-rotor model, that is observed experimentally. Particle-rotor calculations are carried out into the backbending region of some well-deformed rare-earth nuclei, and the results indicate that the “Coriolis attenuation” effect is weak or absent at high rotational frequencies. However, the experimental $\text{i}{}_{\mathrm{<mtext>13</mtext><mtext>2</mtext>}}$, unfavoured band of ¹⁶⁷Yb is found to exhibit an anomalous “downbending” behaviour.     

Thickness vibration of piezoelectric plates of 6mm crystals with tilted six-fold axis and two-layered thick electrodes

We perform a theoretical analysis of thickness vibrations in piezoelectric plates of crystals with 6 mm symmetry. The six-fold axis is tilted with respect to the plate surfaces. The major surfaces of the plate are covered with two layers of electrodes of different metals. The equations of linear piezoelectricity are used for the crystal plate. The electrodes are modeled by the equations of elasticity. Thickness vibrations frequencies and modes as well as impedance are calculated and examined.     

Solution of the cranking model to all orders

For a system with an energy gap a solution to the time dependent Schrödinger equation is obtained in form of a systematic expansion in powers of an inverse, collective time scale. The close connection to a model with a changing self consistent field is discussed as well as the applicability to nuclear fission. The fourth order correction to the almost adiabatic solution to the cranking model is given explicitly.     

The behavior of the temperature-dependent cranking mass parameter

The temperature dependence of the mass parameter is analyzed starting from the Inglis cranking formula. At temperatures higher than the critical temperature corresponding to a vanishing pairing gap, the mass parameter is found to approach that of the incompressible liquid drop value, at least qualitatively. This general result is tested for several simple cases where we can get almost analytical results. In these cases clear relations between the cranking mass and the incompressible liquid drop value are obtained.     

Interpretation of experimental data based on the piecewise linear approximation measurement model

Methods for estimating the input signal of a measuring device are considered. Information about the device is contained in the measurement results of a set of known test signals and the measurements were obtained in natural or in computational experiments. A method is proposed for the approximation of the measurement model by means of a piecewise linear model that is consistent with the results of measurements and the method of evaluation of the input signal and its accuracy. An example is given of the solution of a problem of the interpretation of measurements based on the approximation of a model of the photosynthetic system, in which the measured parameters are the values of saturation ΔpH and the rate of ATP synthesis and the estimated parameters are the concentration of photosystem-2 and light intensity.     

Theory of non-adiabatic cranking and nuclear collective motion

The cranking model is extended to the case of a general non-adiabatic motion. The time-dependent many-body Schrödinger equation is solved, where the time dependence of the collective motion is determined by the classical Lagrange equations of motion. The Lagrangian is obtained from the expectation value of the energy. In the case of one collective degree of freedom the condition that the expectation value of the energy is constant in time is sufficient to determine the collective motion. An iteration procedure is applied, of which the zeroth order is shown to be the common cranking formula. In an alternative approach the energy conservation is expressed in differential form. This leads in the case of one collective degree of freedom to a set of coupled, non-linear first-order differential equations in time for the expansion coefficients of the many-body wave function and for the collective variable. As an illustrative example we solve the case of two coupled linear harmonic oscillators.     

Angular momentum fluctuation energy in the cranking model

Angular momentum is approximately projected from Hartree-Fock-Bogoliubov cranked (HFBC) wave functions. At each J the projected energy is $\text{E}{}_{\mathrm{<mtext>proj</mtext>}}\text{≈ E}{}_{\mathrm{<mtext>HFBC</mtext>}}\text{? (Δ}\text{J}\text{)}{}^2\text{/2}\text{J}{}_{\mathrm{<mtext>HFBC</mtext>}}$. The spin-dependent fluctuation $\text{Δ}\text{J}$ includes contributions from J_y and J_z as well as J_x. There are no correlations in the three angular momentum components. Projected energies are calculated for ^{168, 170}Yb and ¹⁷⁴Hf. When compared to experimental energies, the projected spectra are less compressed than the HFBC spectra. At low spins the projected and experimental energies are in good agreement.     

On a method for estimating the quality of the Fermi approximation

The adequacy of substituting the true neutron-nucleus potential with the Fermi pseudopotential was studied for all elements where elastic neutron scattering is dominant.     

Inclusion of pairing into strutinsky-type cranking calculations

The shell correction method for rotating nuclei was generalized to include pairing correlations. This extends the applicability of the cranked Strutinsky-type calculations to the low-spin region. As an illustrative example the yrast-line of²⁴Mg is studied.