Nuclear scissors with pairing and continuity equation
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NUCLEI Theory
Nuclear Scissors with Pairing and Continuity Equation* E. B. Balbutsev1), L. A. Malov1) , P. Schuck2), and M. Urban2) Received November 6, 2008; in final form, February 5, 2009
Abstract—The coupled dynamics of the isovector and isoscalar giant quadrupole resonances and low-lying modes (including scissors) is studied with the help of the Wigner-function-moment method generalized to take into account pair correlations. Equations of motion for collective variables are derived on the basis of the time-dependent Hartree–Fock–Bogoliubov equations in the harmonic-oscillator model with quadrupole–quadrupole residual interaction and a Gaussian pairing force. Special care is taken of the continuity equation. PACS numbers: 21.60.Ev, 21.60.Jz, 24.30.Cz DOI: 10.1134/S1063778809080067
1. INTRODUCTION
2. PHASE SPACE MOMENTS OF TDHFB EQUATIONS
An exhaustive analysis of the dynamics of the scissors mode and the isovector giant quadrupole resonance in a harmonic-oscillator model with quadrupole–quadrupole (QQ) residual interaction has been performed in [1]. The method of WignerFunction Moments (WFM) was applied to derive the dynamical equations for angular momentum and quadrupole moment. Analytical expressions for energies, B(M 1) and B(E2) values, sum rules, and flow patterns were found for arbitrary values of the deformation parameter. These calculations were performed without pair correlations. However, it is well known [2] that pairing is very important for the correct description of the scissors mode. A first attempt to include pairing into the WFM method was done in [3], where the description of qualitative and quantitative characteristics of the scissors mode was drastically improved. However, the variation of the gap during vibrations was neglected there, resulting in a violation of the continuity equation and in the appearance of an instability in the isoscalar channel. In the present work we suggest a generalization of the WFM method which takes into account pair correlations conserving the continuity equation.
The Time-Dependent Hartree–Fock–Bogoliubov (TDHFB) equations in matrix formulation are [4, 5]
∗
The text was submitted by the authors in English. Joint Institute for Nuclear Research, Dubna, Russia. 2) ´ Institut de Physique Nucleaire, CNRS and University ParisSud, France. 1)
iR˙ = [H, R] with
⎛
R=⎝
ρˆ −ˆ κ†
−ˆ κ 1−
ρˆ∗
⎞ ⎠,
⎛ H=⎝
(1)
ˆ h
⎞ ˆ ∆
ˆ∗ ˆ † −h ∆
⎠.
(2)
ˆ are The normal density matrix ρˆ and Hamiltonian h Hermitian, whereas the abnormal density κ ˆ and the ˆ† = ˆ are skew symmetric: κ κ∗ , ∆ pairing gap ∆ ˆ † = −ˆ ˆ ∗. −∆ The detailed form of the TDHFB equations is ˆ ρ − ρˆh ˆ − ∆ˆ ˆ †, ˆ κ† + κ ˆ∆ iρˆ˙ = hˆ
(3)
ˆ ∗ ρˆ∗ − ρˆ∗ h ˆ∗ − ∆ ˆ †κ ˆ ˆ+κ ˆ† ∆, −iρˆ˙ ∗ = h ˆκ − κ ˆ∗ + ∆ ˆ −∆ ˆ ρˆ∗ − ρˆ∆, ˆ −iκ ˆ˙ = −hˆ ˆh ˆ∗κ ˆ−∆ ˆ† + ∆ ˆ † ρˆ + ρˆ∗ ∆ ˆ †. ˆ† + κ ˆ† h −iκ ˆ˙ † = h We will work with the Wigner transformation [5] of these equations. The relevant mathematical details can be found in [3]. From now on, we will not specify the spin and isospin indices in order to mak
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