Nuclear scissors mode with pairing
ABSTRACT The coupled dynamics of the scissors mode and the isovector giant quadrupole resonance are studied using a generalized Wigner
function moments method, taking into account pair correlations. Equations ofmotion for angularmomentum, quadrupolemoment,
and other relevant collective variables are derived on the basis of the time-dependent Hartree-Fock-Bogolyubov equations.
Analytical expressions for energy centroids and transition probabilities are found for the harmonic-oscillator model with
the quadrupole-quadrupole residual interaction and monopole pairing force. Deformation dependences of energies and B(M1) values are correctly reproduced. The inclusion of pair correlations leads to a drastic improvement in the description of
qualitative and quantitative characteristics of the scissors mode.
arXiv:nucl-th/0701039v2 22 Jun 2007
NUCLEAR SCISSORS MODE WITH PAIRING
E.B. Balbutsev, L.A. Malov
Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia
P. Schuck, M. Urban,
Institut de Physique Nucleaire, CNRS and Univ. Paris-Sud, 91406 Orsay Cedex, France
X. Vi˜ nas
Departament d’Estructura i Constituents de la Mat´ eria Facultat de F´ ısica,
Universitat de Barcelona Diagonal 647, 08028 Barcelona, Spain
The coupled dynamics of the scissors mode and the isovector giant quadrupole res-
onance are studied using a generalized Wigner function moments method taking into
account pair correlations. Equations of motion for angular momentum, quadrupole mo-
ment and other relevant collective variables are derived on the basis of the time depen-
dent Hartree-Fock-Bogoliubov equations. Analytical expressions for energy centroids and
transitions probabilities are found for the harmonic oscillator model with the quadrupole-
quadrupole residual interaction and monopole pairing force. Deformation dependences
of energies and B(M1) values are correctly reproduced. The inclusion of pair correla-
tions leads to a drastic improvement in the description of qualitative and quantitative
characteristics of the scissors mode.
PACS: 21.60.Ev, 21.60.Jz, 24.30.Cz
An exhaustive analysis of the coupled dynamics of the scissors mode and the isovector giant
quadrupole resonance in a model of harmonic oscillator with quadrupole–quadrupole residual
interaction has been performed in . The Wigner Function Moments (WFM) method was
applied to derive the dynamical equations for angular momentum and quadrupole moment.
Analytical expressions for energies, B(M1)- and B(E2)-values, sum rules and flow patterns of
both modes were found for arbitrary values of the deformation parameter. The subtle nature
of the phenomenon and its peculiarities were clarified.
Nevertheless, this description was not complete, because pairing was not taken into account.
It is well known , that pairing is very important for the correct quantitative description of
the scissors mode. Moreover, its role is crucial for an explanation of the empirically observed
deformation dependence of Escand B(M1)sc.
The prediction of the scissors mode was inspired by the geometrical picture of a counterro-
tating oscillation of the deformed proton density against the deformed neutron density [3, 4].
Thus, as it is seen from its physical nature, the scissors mode can be observed only in deformed
nuclei. Therefore, quite naturally, the question of the deformation dependence of its properties
(for example, energy Escand B(M1)scvalue) arises. However, during the first years after its
discovery in156Gd  “nearly all experimental data were limited to nuclei of about the same
deformation (δ ≈ 0.20 − 0.25), and the important aspect of orbital M1 strength dependence
on δ has not yet been examined”, see ref. .
The first investigations of the δ-dependence of Esc and B(M1)sc were performed by W.
Ziegler et al. , who have studied the chain of isotopes148,150,152,154Sm, and by H. H. Pitz et
al.  and J. Margraf et al. , who have studied the chain of isotopes142,146,148,150Nd. They
found that the low-energy B(M1) strength exhibits approximately a quadratic dependence on
the deformation δ.
Shortly afterwards it was discovered [9, 10], that in even-even nuclei the total low-energy
magnetic dipole strength is closely related to the collective E2 strength of the 2+
thus, depends quadratically on the nuclear deformation parameter.
Later J. Enders et al.  made a theoretical analysis of experimental data on the scissors
mode in nuclei with 140 < A < 200. Investigating the sum rules S+1and S−1derived by E.
Lipparini and S. Stringari [12, 13], they found that the ratio ¯ ω = S+1/S−1is proportional to
Escwith very good accuracy: Esc= 0.44¯ ω. They also observed that the moment of inertia
Jgsbof the ground state rotational band and the moment of inertia for the irrotational flow
Jliq= δ2Jrig(where Jrigis the rigid body moment if inertia) differ by nearly a constant factor
(K ≈ 10) over the entire region. Using this fact and identifying the giromagnetic ratio and the
moment of inertia of the scissors mode with those of the ground state rotational band, they
found with the help of the S−1sum rule, that B(M1)scis proportional to δ2.
So, all the rather numerous experimental data demonstrate undoubtedly the δ2dependence
of B(M1)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 , predicted a linear δ-dependence for both, B(M1)scand Esc.
It turned out that the correct δ-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 . They assumed “for simplicity that only the two subsets
of levels which are closest to the Fermi level (n⊥ and n⊥+ 1) are affected by the pairing
interactions”. In this case B(M1) should be multiplied by the factor (un⊥vn⊥+1−un⊥+1vn⊥)2≈
energy, ∆ is the gap and µ is the chemical potential. The value of (en⊥/En⊥) was found by
n⊥)2= (en⊥/En⊥)2with Ei=
i+ ∆2and ei= ǫi− µ, where ǫiis a single particle
“making use of the fact that the moment of inertia is approximately 1/2 of the rigid body value
obtained in the absence of pairing”. Thus, in accordance with the Inglis formula one has
1/2 = J/Jrig= [(u2
n⊥)2/2En⊥]/[1/2en⊥] = (en⊥/En⊥)3
and en⊥/En⊥=0.79. As a result B(M1) is reduced by the factor (en⊥/En⊥)2= 0.62, i.e. the
influence of pairing is quite remarkable.
It was, however, noted by Hamamoto and Magnusson  that this result holds only for
well deformed nuclei, where the equality (1) is valid. In general it is necessary to take into
account the δ-dependence of the en⊥/En⊥- factor. This was done for the first time in ref. .
The authors applied “the method of averaging the position of the chemical potential between
the occupied subshell (N,n⊥) and the empty shell (N,n⊥+ 1)” to find that the δ-dependence
of B(M1) is determined by the function
Y = (δA4/3)
?√1 + 4x2+
2xln|√1 + 4x2+ 2x|
x√1 + x2ln|√1 + x2+ x|
with x = (¯ hω0δ)/(2∆). In the small deformation limit this function is proportional to δ2, while
for large δ it deviates remarkably from such a simple dependence. The authors performed also
a more realistic QRPA calculation for the Woods-Saxon potential with QQ and σσ residual
interactions, which confirmed their simplified analytical estimate.
In , N. Pietralla et al. established the δ-dependence of the en⊥/En⊥- factor phenomeno-
logically. They were first to perform the theoretical analysis of the experimental data of the
scissors mode in nuclei in the mass region 130 < A < 200. Following the idea of Bes and Broglia
they parametrized the en⊥/En⊥- factor as
1 + (bδ)2/(aδ).
The free parameters a and b were fixed by a fit to the experimental moments of inertia with
the help of a formula equivalent to (1)
(en⊥/En⊥)3= Jexp/Jrig, (3)
where Jexp= 3¯ h2/E(2+
1) is the effective moment of inertia of the ground state band. In this
way it was found that “the centers of gravity of the observed M1 strength distributions are
always close to 3 MeV”, i.e. “the data exhibit a weak dependence of the scissors mode on the
deformation parameter”. They also derived a semiempirical formula for the total M1 strength
of the scissors mode
sc) = 2.6c2
1 + (3δ)2
(cg= 0.8 is the scaling factor of the giromagnetic ratio), which describes very well the exper-
imental data and gives a deformation dependence “practically indistinguishable from the δ2
A direct way to demonstrate the δ-dependence of the en⊥/En⊥- factor was suggested in
. E. Garrido et al. have shown that it is possible to extract analytically the δ2factor from
the occupation coefficient Φαβ= (uαvβ−uβvα)2. Using the definitions of the uα,vαcoefficients,
it is easy to write Φαβas
(1 − Pαβ)(5)
the scissors mode eα−eβ≃ ¯ hωδ. The function (1−Pαβ)/(EαEβ) has a regular dependence on
δ (no poles), so the coefficient Φαβand, respectively, B(M1)scare obviously proportional to δ2.
2E2[z − (z2− 4e2
and E =1
2(eα+eβ), z = e2
For the IVGQR eα− eβ≃ 2¯ hω, but its B(M1) is proportional to δ2even without pairing due
to other reasons (see section 3).
In this paper we generalize the WFM method to take into account pair correlations. This
allows us to obtain the correct δ-dependence for Escand B(M1)scin a slightly different way
than in the papers, cited above.
The paper is organized as follows. In section 2 the moments of Time Dependent Hartree-
Fock-Bogoliubov (TDHFB) dynamical equations for normal and abnormal densities are cal-
culated and adequate approximations are introduced to obtain the final set of six dynamical
equations for the collective variables. In section 3 these equations are decoupled in the isoscalar
and isovector sets and the isovector excitation energies and transitions probabilities are calcu-
lated in the framework of the HO+QQ model. The results of calculations are discussed in
section 4. Concluding remarks are contained in section 5. Some mathematical details are given
2Phase space moments of TDHFB equations
The time dependent HFB equations in matrix formulation are [19, 20]
i¯ h˙R = [H,R] (7)
−ˆ κ†1 − ˆ ρ∗
− ˆ κ