ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 9, с. 1138-1163
ЯДРА
FROM CONVENTIONAL (NEUTRON-PROTON) NUCLEAR SCISSORS
TO SPIN NUCLEAR SCISSORS
©2014 E. B. Balbutsev1)*, I. V. Molodtsova1), P. Schuck2)
Received June 28, 2013
Investigations of the nuclear scissors mode in the frame of the Wigner Function Moments (WFM) method leading to the discovery of the new types of the nuclear collective motion are reviewed. It is demonstrated how the generalization of WFM method to take into account spin degrees of freedom allows one to reproduce all earlier described qualitative features of the conventional (neutron—proton) nuclear scissors (deformation dependence of the energy and transition probabilities, connection with isovector GQR implying the Fermi surface deformation, flows) and allows one to reveal a variety of new collective modes: isovector and isoscalar spin scissors, the relative motion of the orbital angular momentum and spin, isovector and isoscalar spin-vector GQR, spin-flip excitations.
DOI: 10.7868/S0044002714090025
1. INTRODUCTION
The nuclear scissors mode was predicted about forty years ago [1, 2] as a counter-rotation of protons against neutrons in deformed nuclei. Its experimental discovery [3] has initiated a cascade of theoretical studies. An excellent review of their twenty-year development was given by Zawischa [4]. Very briefly the situation can be described in the following way. All microscopic calculations with effective forces reproduce experimental data with respect to the position and the strength of the scissors mode, some of them [5] giving also reasonable fragmentation of its strength. However, the situation is more obscure in regard to simple phenomenological models whose aim is to explain the physics of the phenomenon and to interpret it in the most simple and transparent terms.
One of the essential problems here is the scissors mode collectivity that turned out to be small. From RPA results which were in qualitative agreement with experiment, it was even questioned whether this mode is collective at all [4, 5]. Purely phenomenological models (such as, e.g., the two rotors model [6]) did not clear up the situation in this respect. Finally, in a very recent review [7] it is concluded that the scissors mode is "weakly collective, but strong on the single-particle scale" and further: "The weekly-collective
1)1 Joint Institute for Nuclear Research, Dubna, Russia.
2) Institut de Physique Nuclêaire, IN2P3-CNRS, Université; Paris-Sud, Orsay, France; Laboratoire de Physique et Mode lisation des Milieux Condensé s, CNRS and Université Joseph Fourier, Grenoble, France.
E-mail: balbuts@theor.jinr.ru
scissors mode excitation has become an ideal test of models — especially microscopic models — of nuclear vibrations. Most models are usually calibrated to reproduce properties of strongly-collective excitations (e.g., of Jn = 2+ or 3" states, giant resonances, ...). Weekly-collective phenomena, however, force the models to make genuine predictions and the fact that the transitions in question are strong on the single-particle scale makes it impossible to dismiss failures as a mere detail, especially in the light of the overwhelming experimental evidence for them in many nuclei [8, 9]".
The Wigner Function Moments (WFM) or phase space moments method turns out to be very useful in this situation. On the one hand, it is a purely microscopic method, because it is based on the Time-Dependent Hartree—Fock (TDHF) equation. On the other hand, the method works with average values (moments) of operators which have a direct relation to the considered phenomenon and, thus, make a natural bridge with the macroscopic description. This makes it an ideal instrument to describe the basic characteristics (energies and excitation probabilities) of collective excitations such as, in particular, the scissors mode.
The full analysis of the scissors mode in the framework of a solvable model (harmonic oscillator with quadrupole—quadrupole residual interaction (HO + + QQ)) was given in [ 10]. The HO + QQ model is a very convenient ground for such kind of investigation, because most of the results can be obtained analytically. Analytical expressions for currents of both coexisting modes, their excitation energies, magnetic and electric transition probabilities were derived. Our
formulae for energies and transition probabilities turned out to be identical with those derived by Hamamoto and Nazarewicz [11] in the framework of the RPA. Our investigations have shown that already the minimal set of collective variables, i.e., phase space moments up to quadratic order, is sufficient to reproduce the most important property of the scissors mode: its inevitable coexistence with the IsoVector Giant Quadrupole Resonance (IVGQR) implying a deformation of the Fermi surface. Practically all qualitative features of the scissors mode were reproduced except one: the deformation dependence of its energy Esc and B(M 1)sc factor.
All the rather numerous experimental data demonstrate undoubtedly the 52 dependence of B(M 1)sc and the very weak deformation dependence of Esc. On the other hand, at the beginning of the nuclear scissors studies all theoretical models, starting from the first work by Suzuki and Rowe [12], predicted a linear 5 dependence for both, B(M 1)sc and Esc. Our method was not an exception obtaining the same result.
It turned out that the correct 5 dependence is supplied by the pairing correlations. The effects of the pairing interaction in the description of the scissors mode were evaluated for the first time by Bes and Broglia [13], who had shown that the influence of pairing is quite remarkable.
Further developments of the WFM method, namely, the switch from TDHF to TDHFB (Time-Dependent Hartree—Fock—Bogoliubov) equations, i.e., taking into account pair correlations, allowed us to obtain the correct 5 dependence of Esc and B(M1) sc and to improve considerably the quantitative description of the scissors mode [14, 15]: for rare earth nuclei the energies are reproduced with ^10% accuracy and transition probabilities were reduced about two times with respect to their non-superfluid values. However, they remain about two times too high with respect to experiment. It was natural to suspect that the reason of this last discrepancy is probably hidden in the spin degrees of freedom, which were so far ignored by the WFM method. One cannot exclude that, due to spin-dependent interactions, some part of the force of M1 transitions is shifted to the energy region of 5—10 MeV, where a 1 + resonance of spin nature is observed. That is why the generalization of the WFM method to take into account spin degrees of freedom was undertaken in [16]. In a first step, we include in the consideration only the spin—orbit interaction, as the most important one among all possible spin-dependent interactions because it enters into the mean field. Spin—spin residual interaction was included [17] at the next stage of investigations (see Conclusion). To study the
pure effect of spin degrees of freedom, without any admixtures, we have disregarded pair correlations. This allowed us to understand the structure of necessary modifications of the method avoiding too cumbersome calculations. In this way it becomes clear already on the level of the equations of motion for the new collective variables that we are faced with a new type of collective motion, namely the spin scissors mode. It turns out that the experimentally observed group of peaks in the energy interval 2—4 MeV corresponds very likely to two different types of motion: the conventional (neutron—proton) scissors mode and this new kind of mode, i.e., the spin scissors mode. The description of the essential points on the way to this discovery is the goal of this paper.
The paper is organized as follows. At the beginning of Section 2, we recall the principal points of the WFM formalism. The dynamical equations for collective variables are derived in Section 2.1. The expressions for isoscalar and isovector energies are found in Sections 2.2 and 2.3. The linear response theory in the frame of WFM method (Section 2.4) is applied in Sections 2.5, 2.6 to calculate magnetic and electric transition probabilities. Sum rules are analyzed in Section 2.7. The method of infinitesimal displacements is used in Section 2.8 to find the expressions for the nucleon currents of various modes. The TDHF equations with the spin degrees of freedom taken into account are considered in Section 3. Their Wigner transformation is written out in Section 3.1. The spin—orbit potential is analyzed in Section 3.2. The dynamical equations for spin-dependent collective variables are derived in Section 3.3, their linearized version being given in Section 3.4. The angular momentum conservation and derivation of energies, excitation probabilities, and flows are considered in Sections 3.5 and 3.6. The results of calculations are discussed in Section 3.7. The concluding remarks are contained in Section 4. Mathematical details are given in Appendices.
2. THE WFM METHOD
The basis of the method is the TDHF equation for the one-body density matrix [18] pT(ri, r2,t) =
= (ri|PT (i)|r2 >:
dp
lhlñ =
h t, pt
(1)
where HT is the one-body self-consistent mean field Hamiltonian depending implicitly on the density matrix and t is an isotopic spin index. We omit spin indices, because in this part of the paper we consider spin saturated system without the spin—orbit interaction. Spin degrees of freedom will be included later.
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It is convenient to modify equation (1) introducing integrating (5) with different tensor products of ra the Wigner transform of the density matrix (Wigner and pa. Here we consider the case n = 2.
function)
fT (r, p,t)= (2)
= J d3sexp(—ip • s/h)pT (r + |'r~|'i) > and of the Hamiltonian
2.1. Model Hamiltonian, Equations of Motion
The microscopic Hamiltonian of the model, harmonic
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