Quantum correlations in rotating nuclei

Using the Hamiltonian that consists of the separable quadrupole + pairing forces and the cranking term, we analyze the correlations associated with shape, orientation, and particle-number fluctuations in rotating nuclei. Quantum fluctuations around mean field solutions are treated in the random phase approximation (RPA), with special emphasis on the restoration of rotational symmetry and particle number conservation. The mean field calculations have been made within the self-consistent cranking model. The effect of the RPA correlation energy for the moment of inertia is studied with the integral representation method proposed.     

Collective excitations and a backbending phenomenon in <Superscript>156</Superscript>Dy

We propose a self-consistent practical method to study collective excitations in rotating nuclei within the cranking + random phase approximation approach. It consists in solving the cranking Hartree-Bogolyubov equations for the modified Nilsson potential + monopole pairing forces. Further, the mean field results are used to construct collective vibrations treated in the random phase approximation (RPA). Special attention is paid to fulfill all conservation laws in the RPA to separate spurious and physical solutions. We demonstrate that the backbending in ¹⁵⁶Dy can be explained as a result of the disappearance of collective γ vibrations of the positive signature in the rotating frame.     

Structure of symmetry restoring interactions: A modified BCS plus rotational approach for a schematic model space

Symmetry restoring interactions are constructed for pairing and cranking Hamiltonians in a deformed single particle basis. The structure of the interactions is explicitly shown and the effective Hamiltonian is treated in the framework of a modified BCS + RPA approach. Numerical results for a schematic single particle basis, which consists of a splitj=3/2 multiplet, are discussed.     

Quantization in the high-spin RPA

The problem of how to quantize the angular momentum of the self-consistent cranking model at high spin when small oscillations (RPA) about the steady rotation are included is reexamined, in view of a recent criticism by Reinhardt of an earlier treatment. This criticism is shown to be unfounded. On the other hand, it is shown that Reinhardt's quantization procedure leads to some serious problems, and the result that the vibrational frequencies differ in the rotating and lab frames is called into question.     

Microscopic description of collective states near the yrast line of nuclei with stable octupole deformation

Collective states near the yrast line in nuclei with stable octupole deformation are discussed in the framework of the random phase approximation (RPA) based on the cranking model. These vibrational states are characterized by the quantum number of generalized signature (eigenvalue of the operator S_x = PR_x^?1(π)). In the zero-octupole deformation limit the RPA equations of motion are reduced to the well-known ones characterized by both values of parity and signature, respectively. The connection of the translational and rotational symmetry of the model hamiltonian with the spurious solutions of the RPA equation of motion is discussed. Expressions for the reduced probabilities B(E1), B(E2) and B(E3) are obtained. These expressions confirm the conclusions of phenomenological models for the strong E1 and E3 intraband transitions in nuclei with stable octupole deformation.     

Adiabatic and non-adiabattc cranking models for the solution of the large amplitude time-dependent Hartree-Fock equations and the calculation of nuclear energy surfaces

An analysis is made of the device of forced vibrations to reduce the difficult large amplitude TDHF (time-dependent Hartree-Fock) problem to a simpler static problem. It is shown that cranking is a perfectly valid technique if the cranking field is defined by the criterion that the constrained static wave function be identical at a given instant of time to the freely oscillating TDHF wave function. On the basis of this criterion an exact general expression is derived for the cranking field and adiabatic and non-adiabatic cranking models are proposed as practical means of solving the large amplitude TDHF equations and calculating energy surfaces for fission and heavy ion reactions. The non-adiabatic model, which is favoured, corresponds to cranking the nucleus to move always in the direction of a local decoupled “normal mode” calculated from its unconstrained time-dependent equations of motion. The familiar Belyaev cranking model and the Baranger-Kumar model are discussed in the context of the new formalism.     

Microscopic description of giant resonances in highly excited nuclei

Giant resonances of general multipolarity in highly excited nuclei, which are produced in compound nuclear and deep inelastic heavy ion reactions, are described microscopically in the finite temperature linear response formalism. The linear response function is calculated in the finite temperature (FT) quasi-particle RPA approximation (FT-HFB-RPA) and is based on the corresponding self-consistent quasi-particle basis (FT-HFB). The theory is derived from the small amplitude limit of FT-TDHFB. The inclusion of cranking constraints allows the investigation of giant resonances in nuclei with large intrinsic excitation energy and high spin. A schematic model for the FT-HFB-RPA is developed and applied to the isovector giant dipole resonance in hot spherical nuclei. It is shown that the energy of the resonance depends only weakly on temperature in these systems. The experimentally observed lowering of the giant mode in highly excited nuclei is to be attributed to different effects. The descritpion of resonance damping lies beyond the scope of the random phase approximation. Possible extensions in this direction and qualitative features of the width of giant resonances at finite temperature are discussed.     

On the validity of the cranking model for 2+-vibrations

The relation of the cranking model to more fundamental methods of determining the generator of a collective state is investigated for the quadrupole case. We are able to show that the heuristic procedure starting with an adiabatic change in the nuclear deformation and splitting the cranking operator according to main quantum number selection rules approximates the true collective states very well. However, for the low-lying collective state the analytic form of the cranking term is completely different from any type of scaling ansatz.     

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 and M1 and E2 transition probabilities are carried out for the lowest bands. Very good agreement is found, except near band crossings. Various recipes to take into account finite K 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.     

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.     

Generalized cranking model for collective nuclear motion

The Inglis cranking model is generalized to take into account effects of any velocity dependence present in the single-particle potential and the reaction of the pairing field to the collective motion. The generalized model is applied to translations, rotations and some special types of vibrations. Some of our results and our numerical calculations are obtained with a harmonic-oscillator single-particle potential. Unlike the inertia calculated with the Inglis cranking model, the inertia calculated with the generalized cranking model is independent of the effective mass and approaches the irrotational value in the limit of large pairing.     

Collective gyromagnetic ratio from density dependent hartree-fock calculations: (II). Odd nuclei

The collective gyromagnetic ratio and moment of inertia of deformed odd-proton and oddneutron axially symmetric nuclei have been calculated in the cranking approximation using wave functions obtained with the Skyrme force SIII. Good agreement with experiment is found for g_R. Our parameter-free cranking results are better than those of Prior, Boehm and Nilsson where effective charges were used. The cranking formula leads to better results than the projection method (in which one simply takes the expectation value of the relevant operator in the deformed HF ground state, neglecting corrections of relative order 1/〈J²〉. In particular, the cranking results follow nicely the exceptionally large/small g_R for the odd-proton/neutron nuclei around mass 153–167.     

Collective gyromagnetic ratio and moment of inertia from density-dependent Hartree-Fock calculations

The collective gyromagnetic ratio and moment of inertia of deformed even-even axially symmetric nuclei are calculated in the cranking approximation using wave functions obtained with the Skyrme force S-III. Good agreement is found for g_R, while the moment of inertia is about 20 % too small. The cranking formula leads to better agreement than the projection method.     

Spin orientation in rotating nuclei

The orientation of the angular momentum in triaxial nuclei has been investigated in a three-dimensional cranked-HFB formalism. Some quasiparticle excitations are found to favour cranking about a non-principal axis for particular combinations of shell filling and triaxial shape. This effect also exists when the energy of the rotating core is added, at least for low spins, which is shown in a complete cranking calculation for ⁸⁴Y.     

原子核振动与转动模型（Ⅰ）推转玻尔─莫特逊哈密顿量和偶偶核正常转动谱分析

从推转壳模型出发，导出了转动频率未量子化的集体振动－转动哈密顿量，称为推转玻尔－莫特逊哈密顿量（ＣＢＭＨ）．引入合理的集体运动位势，由ＣＢＭＨ可以得到解析形式的转动谱公式．应用这一振动－转动模型，对偶偶变形核的正常转动能谱进行了分析，取得了满意的结果．     

Electromagnetic moments and transition rates from cranking model wavefunctions

Simplified formulas are derived for the angular-momentum projected matrix elements of the electromagnetic transition probabilities and spectroscopic moments using microscopic cranking model wavefunctions. They are compared with exact calculations.     

Sum-rules and Goldstone modes from extended random phase approximation theories in Fermi systems with spontaneously broken symmetries

The Self-Consistent RPA (SCRPA) approach is elaborated for cases with a continuouslybroken symmetry, this being the main focus of the present article. Correlations beyondstandard RPA are summed up correcting for the quasi-boson approximation in standard RPA.Desirable properties of standard RPA such as fulfillment of energy weighted sum rule andappearance of Goldstone (zero) modes are kept. We show theoretically and, for a modelcase, numerically that, indeed, SCRPA maintains all properties of standard RPA forpractically all situations of spontaneously broken symmetries. A simpler approximate formof SCRPA, the so-called renormalised RPA, also has these properties. The SCRPA equationsare first outlined as an eigenvalue problem, but it is also shown how an equivalent manybody Green’s function approach can be formulated.     

Vibrational and Rotational Motions Model of Nucleus(Ⅰ) Cranking Bohr-Mottelson Hamiltonian and Analyses of the Rotational Bands of Even-Even Nuclei

Starting from cranking shell model,a collective vibrational and rotational Hamiltonian(cranking Bohr-Mottelson Hamiltonian CBMH)is derived,in which the rotational frequency is not quantized.Introducing a reasonable collective potential,the formula for the rotational spectrum can be obtained.The formula is applied to analyze the rotational bands of even-even nuclei with satisfactory results.     

推转PNC波函数的角动量投影

研究原子核较高自旋态的特征,不仅需要考虑K混合,而且应该引入Coriolis相互作用项,亦即使用推转Hamilton量.推转项的引入破坏了时间反演对称性,单粒子角动量在内禀对称轴上的投影量子数已不再是好量子数.从表面上看,这会使得理论计算变得非常复杂,但考虑到原子核的某些固有对称性,将角动量投影技术应用于含推转成份的粒子数守恒波函数(推转PNC波函数)并不十分困难.在这一理论框架下详细推导计算原子核能谱及电磁性质的基本表达式,从而表明实施该方案的可行性.