Microscopic description of large-amplitude shape-mixing dynamics with inertial functions derived in local quasiparticle random-phase approximation

1.9 MB·English

by  Nobuo Hinohara

Microscopic description of large-amplitude shape-mixing dynamics with inertial
functions derived in local quasiparticle random-phase approximation
Nobuo Hinohara,1 Koichi Sato,2,1 Takashi Nakatsukasa,1 Masayuki Matsuo,3 and Kenichi Matsuyanagi1,4
1Theoretical Nuclear Physics Laboratory, RIKEN Nishina Center, Wako 351-0198, Japan
2Department of Physics, Graduate School of Science, Kyoto University, 606-8502 Kyoto, Japan
3Department of Physics, Faculty of Science, Niigata University, Niigata 950-2181, Japan
4Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
(Dated: January 12, 2011)
On the basis of the adiabatic self-consistent collective coordinate method, we develop an effi-
cient microscopic method of deriving the five-dimensional quadrupole collective Hamiltonian and
illustrateitsusefulnessbyapplyingittotheoblate-prolateshapecoexistence/mixingphenomenain
proton-rich68,70,72Se. Inthismethod,thevibrationalandrotationalcollectivemasses(inertialfunc-
1
tions) are determined by local normal modes built on constrained Hartree-Fock-Bogoliubov states.
1
Numericalcalculationsarecarriedoutusingthepairing-plus-quadrupoleHamiltonianincludingthe
0
quadrupole-pairinginteractionwithinthetwomajor-shellactivemodelspacesbothforneutronsand
2
protons. It is shown that the time-odd components of the moving mean-field significantly increase

n the vibrational and rotational collective masses in comparison with the Inglis-Belyaev cranking
a masses. Solving the collective Schr¨odinger equation, we evaluate excitation spectra, quadrupole
J
transitions and moments. Results of the numerical calculation are in excellent agreement with re-

1 cent experimental data and indicate that the low-lying states of these nuclei are characterized as
1 anintermediatesituationbetweentheoblate-prolateshapecoexistenceandtheso-calledγ unstable
situation where large-amplitude triaxial-shape fluctuations play a dominant role.
]
h
PACSnumbers: 21.60.Ev,21.10.Re,27.50.+e
t
-
l
c
u I. INTRODUCTION These theories of large-amplitude collective motion have
n been tested for schematic solvable models and applied
[ to heavy-ioncollisions and giantresonances[18,26]. For
The major purpose of this paper is to develop an effi-
2 cientmicroscopicmethodofderivingthefive-dimensional nucleiwithpairingcorrelations,DobaczewskiandSkalski
v studied the quadrupole vibrational mass with use of the
(5D) quadrupole collective Hamiltonian [1–4] and illus-
4 adiabatictime-dependentHartree-Fock-Bogoliubov(AT-
trate its usefulness by applying it to the oblate-prolate
4 DHFB)theoryandconcludedthatthecontributionsfrom
5 shape coexistence/mixing phenomena in proton-rich Se
the time-odd components of the moving mean-field sig-
5 isotopes [5–8]. As is well known, the quadrupole collec-
nificantly increase the vibrational mass compared to the
. tiveHamiltonian,alsocalledthegeneralBohr-Mottelson
4 IB cranking mass [16]. Somewhat surprisingly, however,
Hamiltonian,containssixcollectiveinertiamasses(three
0 to the best of our knowledge, the ATDHFB vibrational
0 vibrational masses and three rotational moments of in-
masses have never been used in realistic calculations for
1 ertia) as well as the collective potential. These seven
low-lying quadrupole spectra of nuclei with superfluid-
: quantities are functions of the quadrupole deformation
v ity. Forinstance,inrecentmicroscopicstudies[29–34]by
Xi vtrairaixaiballeitsyβofantdheγq,uwahdircuhporeleprdeesfeonrtmtahteiomn,agrensiptuecdteivaenlyd. means of the 5D quadrupole Hamiltonian, the IB crank-
ing formula are still used in actual numerical calcula-
r Therefore, we also call the collective inertial masses ‘in-
a tionforvibrationalmasses. Thissituationconcerningthe
ertial functions.’ They are usually calculated by means
treatment of the collective kinetic energies is in marked
of the adiabatic perturbation treatment of the moving
contrastwiththeremarkableprogressinmicroscopiccal-
mean field [9], and the version taking into account nu-
culationofthecollectivepotentialusingmoderneffective
clear superfluidity [10] is called the Inglis-Belyaev (IB)
interactions or energy density functionals (see Ref. [35]
cranking mass or the IB inertial function. Its insuffi-
for a review).
ciencyhasbeenrepeatedlyemphasized,however(seee.g.,
Refs. [11–14]). The mostseriousshortcomingis thatthe In this paper, on the basis of the adiabatic self-
time-odd terms induced by the moving mean field are consistent collective coordinate (ASCC) method [36],
ignored, which breaks the self-consistency of the theory we formulate a practical method of deriving the 5D
[15, 16]. In fact, one of the most important motives of quadrupole collective Hamiltonian. The central con-
constructing microscopic theory of large-amplitude col- cept of this approach is local normal modes built on
lective motion was to overcome such a shortcoming of constrainedHartree-Fock-Bogoliubov(CHFB)states[37]
the IB cranking mass [15]. defined at every point of the (β,γ) deformation space.
As fruits of long-term efforts, advanced microscopic These local normal modes are determined by the lo-
theories of inertial functions are now available (see Refs. cal QRPA (LQRPA) equation that is an extension of
[15–26]fororiginalpapersandRefs. [27,28]forreviews). the well-known quasiparticle random-phase approxima-
2
tion (QRPA) to non-equilibrium HFB states determined neticenergiesofvibrationalandrotationalmotions,while
by the CHFB equations. We therefore use an abbrevia- V(β,γ) represents the collective potential. The veloci-
tion ‘CHFB+LQRPA method’ for this approach. This ties of the vibrational motion are described in terms of
method may be used in conjunction with any effec- the time-derivatives (β˙, γ˙) of the quadrupole deforma-
tive interaction or energy density functional. In this tion variables(β, γ) representingthe magnitude and the
paper, however, we use, for simplicity, the pairing- triaxiality of the quadrupole deformation, respectively.
plus-quadrupole (P+Q) force [38, 39] including the The three components ω of the rotational angular ve-
k
quadrupole-pairing force. Inclusion of the quadrupole- locity are defined with respect to the intrinsic axes asso-
pairingforceisessentialbecauseitproducesthetime-odd ciated with the rotating nucleus. The inertial functions
component of the moving field [40]. for vibrational motions (vibrational masses), D , D ,
ββ βγ
To examine the feasibility of the CHFB+LQRPA and D , and the rotational moments of inertia are
γγ k
J
method, we apply it to the oblate-prolate shape coex- functions of β and γ.
istence/mixing phenomena in proton-rich 68,70,72Se [5–
8, 41, 42]. These phenomena are taken up because we As seen in the recent review by Pr´ochniak and Ro-
obviously need to go beyond the traditional framework hozin´ski [4], there are numerous papers on microscopic
of describing small-amplitude vibrations around a single approachestothe5DquadrupolecollectiveHamiltonian;
HFB equilibrium point to describe them; that is, they among them, we should quote at least early papers by
are very suitable targets for our purpose. We shall show Belyaev [2], Baranger-Kumar [43, 44], Pomorski et al.
in this paper that this approach successfully describes [12, 13], and recent papers by Girod et al. [33], Nikˇsi´c
large-amplitude collective vibrations extending from the et al. [29, 30], and Li et al. [31, 32]. In all these works,
oblate to the prolate HFB equilibrium points (and vice the IB cranking formula is used for the vibrational iner-
versa). In particular, it will be demonstrated that we tial functions. Below, we outline the procedure of deriv-
candescribeverywellthetransitionalregionbetweenthe ing the vibrational and rotational inertial functions on
oblate-prolate shape coexistence and the γ unstable sit- the basis of the ASCC method.
uation where large-amplitude triaxial-shape fluctuations
play a dominant role.
This paper is organized as follows. In Sec. II, we
formulate the CHFB+LQRPA as an approximation of
the ASCC method and derive the 5D quadrupole col-
lective Hamiltonian. In Sec. III, we calculate the vibra-
tionalandrotationalmassesbysolvingtheLQRPAequa-
tions, and discuss their properties in comparison with
those calculated by using the IB cranking formula. In B. Basic equations of the ASCC method
Sec. IV, we calculate excitation spectra, B(E2), and
spectroscopic quadrupole moments of low-lying states
in 68,70,72Se and discuss properties of the oblate-prolate
To derive the 5D quadrupole collective Hamiltonian
shape coexistence/mixing in these nuclei. Conclusions startingfromamicroscopicHamiltonianHˆ,weuse
coll
are given in Sec. V. H
theASCCmethod[36,45]. Thismethodenablesustode-
termine a collective submanifold embedded in the large-
dimensional TDHFB configuration space. We can use
II. MICROSCOPIC DERIVATION OF THE 5D
thismethodinconjunctionwithanyeffectiveinteraction
QUADRUPOLE COLLECTIVE HAMILTONIAN
orenergydensityfunctionaltomicroscopicallyderivethe
collectivemassestakingintoaccounttime-oddmean-field
A. The 5D quadrupole collective Hamiltonian
effects. For our present purpose, we here recapitulate
a two-dimensional (2D) version of the ASCC method.
Our aim in this section is to formulate a practical We suppose existence of a set of two collective coordi-
method of microscopically deriving the 5D quadrupole nates(q1,q2)thathasaone-to-onecorrespondencetothe
collective Hamiltonian [1–4] quadrupoledeformationvariableset(β,γ)andtrytode-
termine a 2D collective hypersurface associatedwith the
=T +T +V(β,γ), (1)
coll vib rot
H large-amplitude quadrupole shape vibrations. We thus
1 1
T = D (β,γ)β˙2+D (β,γ)β˙γ˙ + D (β,γ)γ˙2, assume that the TDHFB states can be written on the
vib ββ βγ γγ
2 2 hypersurface in the following form;
(2)
3
1
T = (β,γ)ω2, (3)
rot 2 Jk k
Xk=1 φ(q,p,ϕ,n) =e−iPτϕ(τ)Ne(τ) φ(q,p,n)
| i | i
starting from an effective Hamiltonian for finite many-
nucleon systems. Here, T and T denote the ki- =e−iPτϕ(τ)Ne(τ)eiGˆ(q,p,n) φ(q) , (4)
vib rot | i
3
with and
Gˆ(q,p,n)= p Qˆi(q)+ n(τ)Θˆ(τ)(q), (5) ∂2V ∂V
i C (q)= Γk (12)
i=1,2 τ=n,p ij ∂qi∂qj − ij∂qk
X X k
Qˆi(q)=QˆA(q)+QˆB(q) X
with
= [QA (q)a†a† +QA∗(q)a a
αβ α β αβ β α 1 ∂B ∂B ∂B
αβ Γk(q)= Bkl( li + lj ij). (13)
X ij 2 ∂qj ∂qi − ∂ql
+QBαβ(q)a†αaβ], (6) Xl
Θˆ(τ)(q)= [Θ(ατβ)A(q)a†αa†β +Θ(ατβ)A∗(q)aβaα]. The infinitesimal generators Pˆi(q) are defined by
αβ
X
(7) Pˆ(q) φ(q) =i ∂ φ(q) , (14)
i | i ∂qi | i
For a gauge-invariant description of nuclei with super-
fluidity, we need to parametrize the TDHFB state vec- with
tors, as above, not only by the collective coordinates
q = (q1,q2) and conjugate momenta p = (p1,p2) but Pˆi(q)=i [Piαβ(q)a†αa†β −Pi∗αβ(q)aβaα], (15)
also by the gauge angles ϕ = (ϕ(n),ϕ(p)) conjugate to αβ
X
the number variables n = (n(n),n(p)) representing the
and determined as solutions of the moving-frame QRPA
pairing-rotational degrees of freedom (for both neutrons
equations.
andprotons). Intheaboveequations,Qˆi(q)andΘˆ(τ)(q)
The collectiveHamiltonian is givenas the expectation
are infinitesimal generators which are written in terms
valueofthe microscopicHamiltonianwithrespecttothe
of the quasiparticle creation and annihilation operators
TDHFB state:
(a†,a )locallydefinedwithrespecttothemoving-frame
α α
HFB states φ(q) . Note that the number operators are (q,p,n)= φ(q,p,n) Hˆ φ(q,p,n)
| i
tdieofinnveadluaessN(N(τ0()n≡),NNˆ0((pτ)))−ofNth0(τe)nseuubttrroanctainndg pthroeteoxnpneuctma-- H =Vh(q)+ |12B|ij(q)pipj +i λ(τ)(q)n(τ),
bers at φ(qe) . In this paper, we use units with ~=1. Xij Xτ
| i (16)
Themoving-frameHFBstates φ(q) andtheinfinites-
| i
imal generatorsQˆi(q) are determined as solutions of the
where
moving-frame HFB equation,
V(q)= (q,p,n) , (17)
δ φ(q) HˆM(q) φ(q) =0, (8) H p=0,n=0
h | | i 
∂2 
and the moving-frame QRPA equations, Bij(q)= H  , (18)
∂pi∂pj p=0,n=0
1 
δhφ(q)|[HˆM(q),Qˆi(q)]− i Bik(q)Pˆk(q) λ(τ)(q)= ∂H  , (19)
k ∂n(τ) p=0,n=0
X

+ 1 ∂V Qˆk(q),Qˆi(q) φ(q) =0, (9) representthecollectivepotential,inverseofthecollective
2" ∂qk #| i mass,andthechemicalpotential,respectively. Notethat
k
X the last term in Eq. (10) can be set to zero adopting the
QRPA gauge-fixing condition, dλ(τ)/dqi =0 [45].
1
δ φ(q) [Hˆ (q), Pˆ(q)] C (q)Qˆj(q) The basic equations of the ASCC method are invari-
M i ij
h | i −
antagainstpointtransformationsofthecollectivecoordi-
j
X nates(q1,q2). TheBij(q)andC (q)canbediagonalized
ij
1 Hˆ (q), ∂V Qˆk(q) , B (q)Qˆj(q) simultaneously by a linear coordinate transformation at
− 2" M ∂qk # ij  eachpoint ofq =(q1,q2). We assume that we canintro-
k j
X X duce the collective coordinate system in which the diag-
∂λ(τ)  onalformis keptglobally. Then, we canchoose,without
N(τ) φ(q) =0, (10)
− ∂qi | i losing generality and for simplicity, the scale of the col-
Xτ lective coordinates q =(q1,q2) such that the vibrational
e
which are derived from the time-dependent variational masses become unity. Consequently, the vibrational ki-
principle. Here,Hˆ (q)isthemoving-frameHamiltonian netic energyinthe collectiveHamiltonian(16)iswritten
M
given by as
HˆM(q)=Hˆ − λ(τ)(q)N(τ)− ∂∂qViQˆi(q) (11) Tvib = 21 (pi)2 = 21 (q˙i)2. (20)
Xτ Xi iX=1,2 iX=1,2
e
4
C. CHFB+LQRPA equations Here the infinitesimal generators, Qˆi(β,γ) and Pˆ(β,γ),
i
are local operators defined at (β,γ) with respect to the
ThebasicequationsoftheASCCmethodcanbesolved CHFB state φ(β,γ) . These equations are solved at
| i
with an iterative procedure. This task was successfully each point of (β,γ) to determine Qˆi(β,γ), Pˆi(β,γ), and
carriedout for extracting a one-dimensional(1D) collec- Ci(β,γ) =ωi2(β,γ). Note that these equations are valid
tive path embedded in the TDHFB configuration space also for regions with negative curvature (Ci(β,γ) < 0)
[46, 47]. To determine a 2D hypersurface, however, where the QRPA frequency ωi(β,γ) takes an imagi-
the numericalcalculationbecomes toodemanding atthe nary value. We call the above equations ‘local QRPA
present time. We therefore introduce practical approxi- (LQRPA) equations’. There exist more than two solu-
mations as follows: First, we ignore the curvature terms tionsofLQRPAequations(27)and(28),andweneedto
(the third terms in Eqs. (9) and (10)), which vanish at select relevant solutions. A useful criterion for selecting
the HFB equilibrium points where dV/dqi = 0, assum- twocollectivemodesamongmanyLQRPAmodeswillbe
ing that their effects are numerically small. Second, we given in subsection IIIC with numerical examples. Con-
replace the moving-frame HFB Hamiltonian Hˆ (q) and cerning the accuracy of the CHFB+LQRPA approxima-
M
the moving-frame HFB state φ(q1,q2) with a CHFB tion, some arguments will be given in subsection IIIF.
Hamiltonian Hˆ (β,γ) and a CHFB state φ(β,γ) ,
CHFB (cid:12) (cid:11) | i
respectively,onthe assumption(cid:12)thatthe lattersaregood
D. Derivation of the vibrational masses
approximations to the formers.
The CHFB equations are given by
OncetheinfinitesimalgeneratorsQˆi(β,γ)andPˆ(β,γ)
i
δ φ(β,γ) Hˆ (β,γ) φ(β,γ) =0, (21) are obtained, we can derive the vibrational masses ap-
CHFB
h | | i
pearinginthe5DquadrupolecollectiveHamiltonian(1).
We rewrite the vibrational kinetic energy T given by
vib
Hˆ (β,γ)=Hˆ λ(τ)(β,γ)N(τ) Eq.(20) interms of the time-derivatives, β˙ andγ˙, ofthe
CHFB
− quadrupole deformation variables in the following way.
τ
X
µ (β,γ)Dˆe(+) (22) We first note that an infinitesimal displacement of the
− m 2m collectivecoordinates(q1,q2)bringsaboutacorrespond-
m=0,2
X ing change,
with four constraints
∂D(+)
dD(+) = 2m dqi, (m=0,2), (29)
φ(β,γ) Nˆ(τ) φ(β,γ) =N(τ), (τ =n,p) (23) 2m ∂qi
h | | i 0 iX=1,2
φ(β,γ) Dˆ(+) φ(β,γ) =D(+), (m=0,2) (24)
h | 2m | i 2m in the expectation values of the quadrupole operators.
The partial derivatives can be easily evaluated as
where Dˆ(+) denotes Hermitian quadrupole operators,
2m
Dˆ20 and (Dˆ22 +Dˆ2−2)/2 for m = 0 and 2, respectively ∂D2(+0) = ∂ φ(β,γ) Dˆ(+) φ(β,γ)
(see Ref. [46] for their explicit expressions). We define ∂qi ∂qi h | 20 | i
the quadrupole deformation variables (β,γ) in terms of 1
= φ(β,γ) [Dˆ(+), Pˆ(β,γ)] φ(β,γ) , (30)
the expectation values of the quadrupole operators: h | 20 i i | i
βcosγ =ηD(+) =η φ(β,γ) Dˆ(+) φ(β,γ) , (25) ∂D2(+2) = ∂ φ(β,γ) Dˆ(+) φ(β,γ)
20 h | 20 | i ∂qi ∂qi h | 22 | i
1
βsinγ =ηD(+) =η φ(β,γ) Dˆ(+) φ(β,γ) , (26) 1
√2 22 h | 22 | i =hφ(β,γ)|[Dˆ2(+2), iPˆi(β,γ)]|φ(β,γ)i, (31)
where η is a scaling factor (to be discussed in subsection without need of numerical derivatives. Accordingly, the
IIIA). vibrational kinetic energy can be written
ThemovingframeQRPAequations,(9)and(10),then
1 1
reduce to T = M [D˙(+)]2+M D˙(+)D˙(+)+ M [D˙(+)]2,
vib 2 00 20 02 20 22 2 22 22
δ φ(β,γ) [Hˆ (β,γ),Qˆi(β,γ)] (32)
CHFB
h |
1Pˆ(β,γ) φ(β,γ) =0, (i=1,2) (27) with
i
− i | i
∂qi ∂qi
and Mmm′(β,γ)= ∂D(+)∂D(+). (33)
i=1,2 2m 2m′
X
1
δ φ(β,γ) [HˆCHFB(β,γ), Pˆi(β,γ)] Taking time-derivative of the definitional equations of
h | i
(β,γ),Eqs.(25)and(26),wecanstraightforwardlytrans-
C (β,γ)Qˆi(β,γ) φ(β,γ) =0. (i=1,2) (28)
i form the above expression (32) to the form in terms of
− | i
5
(β˙,γ˙). Thevibrationalmasses(D ,D ,D )arethen where
ββ βγ γγ
obtainedfrom(M ,M ,M )throughthefollowingre-
00 02 22
lations: Tˆ = 1 1 ∂ β2 RD ∂
vib − 2√WR(β4 " β rW γγ β!
D =η−2 M cos2γ+√2M sinγcosγ
ββ 00 02
R
+ 1(cid:16)M sin2γ , (34) −∂β β2rWDβγ∂γ!#
22
2
(cid:19)
1 R
Dβγ =βη−2[−M00sinγcosγ +β2sin3γ −∂γ W sin3γDβγ∂β
" r !
1 1
+ M (cos2γ sin2γ)+ M sinγcosγ ,
√2 02 − 2 22 R
(cid:21) +∂ sin3γD ∂ , (41)
(35) γ W ββ γ
r !#)
Dγγ =β2η−2 M00sin2γ √2M02sinγcosγ 3 Iˆ2
− Tˆ = k (42)
+ 1M(cid:16)22cos2γ . (36) rot Xk=12Jk
2
(cid:19) with
R(β,γ)=D (β,γ)D (β,γ)D (β,γ), (43)
1 2 3
E. Calculation of the rotational moments of inertia
W(β,γ)= D (β,γ)D (β,γ) [D (β,γ)]2 β−2.
ββ γγ βγ
−
(44)
Wecalculatetherotationalmomentsofinertia (β,γ) (cid:8) (cid:9)
k
J
usingtheLQRPAequationforthecollectiverotation[46] The collective wave function in the laboratory frame,
at each CHFB state, Ψ (β,γ,Ω), is a function of β, γ, and a set of three
αIM
Euler angles Ω. It is specified by the total angular mo-
δ φ(β,γ) [Hˆ ,Ψˆ ] 1( )−1Iˆ φ(β,γ) =0, (37) mentumI,itsprojectionontothez-axisinthelaboratory
h | CHFB k − i Jk k| i frame M, and α that distinguishes the eigenstates pos-
φ(β,γ) [Ψˆk(β,γ),Iˆk′] φ(β,γ) =iδkk′, (38) sessingthe same values ofI andM. With the rotational
h | | i wave function I (Ω), it is written as
DMK
whereΨˆ (β,γ)andIˆ representtherotationalangleand
k k Ψ (β,γ,Ω)= Φ (β,γ) ΩIMK , (45)
the angular momentum operators with respect to the αIM αIK h | i
principal axes associatedwith the CHFB state φ(β,γ) . KX=even
| i
ThisisanextensionoftheThouless-Valatinequation[48] where
for the HFB equilibrium state to non-equilibriumCHFB
states. The three moments of inertia can be written as 2I+1
ΩIMK = [ I (Ω)+( )I I (Ω)].
h | i s16π2(1+δk0) DMK − DM−K
(β,γ)=4β2D (β,γ)sin2γ (k =1,2,3) (39)
Jk k k (46)
with γk =γ (2πk/3). If the inertialfunctions Dk(β,γ) The vibrational wave functions in the body-fixed frame,
−
above are replaced with a constant, then Jk(β,γ) re- ΦαIK(β,γ), are normalized as
duce to the well-known irrotational moments of inertia.
In fact, however, we shall see that their (β,γ) depen-
dence is very important. We call k(β,γ) and Dk(β,γ) dβdγ|ΦαI(β,γ)|2|G(β,γ)|12 =1, (47)
J Z
determined by the above equation ‘LQRPA moments of
inertia’ and ‘LQRPA rotational masses’, respectively. where
Φ (β,γ)2 Φ (β,γ)2, (48)
αI αIK
| | ≡ | |
K=even
X
F. Collective Schr¨odinger equation
and the volume element G(β,γ) 12dβdγ is given by
| |
Quantizing the collective Hamiltonian (1) with the
Pauli prescription, we obtain the collective Schr¨odinger G(β,γ) 21dβdγ =2β4 W(β,γ)R(β,γ)sin3γdβdγ.
| |
(49)
equation [2]
p
Thoroughdiscussionsoftheirsymmetriesandthebound-
Tˆ +Tˆ +V Ψ (β,γ,Ω)=E Ψ (β,γ,Ω),
vib rot αIM αI αIM aryconditionsforsolvingthecollectiveSchr¨odingerequa-
{ }
(40)
tion are given in Refs. [1–3].
6
III. CALCULATION OF THE COLLECTIVE Each mesh point (β ,γ ) is represented as
i j
POTENTIAL AND THE COLLECTIVE MASSES
β =(i 0.5) 0.01, (i=1, 60), (50)
i
− × ···
A. Details of numerical calculation γj =(j 0.5) 1◦, (j =1, 60). (51)
− × ···
Oneoftheadvantagesofthepresentapproachisthatwe
The CHFB+LQRPA method outlined in the preced- can solve the CHB + LQRPA equations independently
ing section may be used in conjunction with any effec- ateachmeshpointonthe(β,γ)plane,sothatitissuited
tive interaction, e.g., density-dependent effective inter- to parallel computation.
actions like Skyrme forces, or modern nuclear density
Finally, we summarize the most important differences
functionals. In this paper, as a first step toward such
between the present approach and the Baranger-Kumar
calculations, we use a version of the P+Q force model
approach[43]. First,asrepeatedlyemphasized,weintro-
[38, 39] that includes the quadrupole-pairing in addi-
ducetheLQRPAcollectivemassessinplaceofthecrank-
tion to the monopole-pairing interaction. Inclusion of
ingmasses. Second,wetakeintoaccountthequadrupole-
the quadrupole-pairing is essential, because neither the
pairingforce(inadditiontothemonopole-pairingforce),
monopole-pairingnor the quadrupoleparticle-holeinter-
whichbringsaboutthe time-odd effects onthe collective
action contributes to the time-odd mean-field effects on
masses. Third, we exactly solve the CHB self-consistent
the collective masses [16]; that is, only the quadrupole-
problem, Eq. (21), at every point on the (β,γ) plane us-
pairing induces the time-odd contribution in the present
ing the gradient method, while in the Baranger-Kumar
model. Note that the quadrupole-pairing effects were
works the CHB Hamiltonian is replaced with a Nilsson-
not considered in Ref. [16]. In the numerical calculation
like single-particle model Hamiltonian. Fourth, we do
for68,70,72Se presentedbelow,weuse the samenotations
not introduce the so-calledcore contributions to the col-
and parameters as in our previous work [47]. The shell
lective masses, although we use the effective charges to
model space consists of two major shells (N =3,4) for
sh renormalize the core polarization effects (outside of the
neutronsandprotonsandthesphericalsingle-particleen-
model space consisting of two major shells) into the
ergies are calculated using the modified oscillatorpoten-
quadrupole operators, We shall see that we can well re-
tial [49, 50]. The monopole-pairinginteractionstrengths
produce the major characteristics of the experimental
(for neutrons and protons), G(τ), and the quadrupole-
0 data without introducing such core contributions to the
particle-holeinteractionstrength,χ,aredeterminedsuch
collective masses. Fifth, most importantly, the theoret-
thatthemagnitudesofthequadrupoledeformationβand
ical framework developed in this paper is quite general,
the monopole-pairing gaps (for neutrons and protons)
thatis,itcanbeusedinconjunctionwithmoderndensity
at the oblate and prolate local minima in 68Se approxi-
functionals going far beyond the P+Q force model.
matelyreproducethoseobtainedintheSkyrme-HFBcal-
culations [51]. The interaction strengths for 70Se and
72Se are then determined assuming simple mass-number
dependence [39]; G(0τ) ∼ A−1 and χ′ ≡ χb4 ∼ A−53
(b denotes the oscillator-length parameter). For the B. Collective potentials and pairing gaps
quadrupole-pairing interaction strengths (for neutrons
and protons), we use the Sakamoto-Kishimoto prescrip-
WeshowinFig.1thecollectivepotentialsV(β,γ)cal-
tion [52] to derive the self-consistent values. Following
culated for 68,70,72Se. It is seen that two local minima
the conventional treatment of the P+Q model [53], we
always appear both at the oblate (γ = 60◦) and prolate
ignore the Fock term, so that we use the abbreviation
(γ =0◦)shapes,and,inallthese nuclei,theoblatemini-
HB (Hartree-Bogoliubov) in place of HFB in the follow-
mumislowerthantheprolateminimum. Theenergydif-
ing. In the case ofthe conventionalP+Qmodel, the HB
ference between them is, however, only several hundred
equation reduces to a simple Nilsson + BCS equation
keV and the potential barrier is low in the direction of
(see, e.g., Ref. [37]). The presence of the quadrupole-
triaxial shape (with respect to γ) indicating γ-soft char-
pairing interaction in our case does not allow such a re-
acterofthesenuclei. InFig.1wealsoshowthecollective
duction,however,andwedirectlysolvetheHBequation.
paths (connecting the oblate and prolate minima) deter-
In the P+Q model, the scaling factor η in Eqs. (25) and
(26) is given by η = χ′/~ω b2, where ω denotes the minedbyusingthe1DversionoftheASCCmethod[47].
0 0
Itisseenthattheyalwaysrunthroughthetriaxialvalley
frequency of the harmonic-oscillator potential. Effective
and never go through the spherical shape.
charges, (e ,e ) = (0.4,1.4), are used in the calculation
n p
of quadrupole transitions and moments. In Fig. 2, the monopole- and quadrupole-pairing gaps
calculated for 68Se are displayed. They show a signifi-
To solve the CHB + LQRPA equations on the (β,γ) cant(β,γ)dependence. Broadlyspeaking,the monopole
plane, we employ a two-dimensional mesh consisting of pairing decreases while the quadrupole pairing increases
3600 points in the region 0<β <0.6 and 0◦ <γ <60◦. as β increases.
7
FIG. 2: (Color online) Monopole- and quadrupole-pairing
gaps for neutrons of 68Se are plotted in the (β,γ) deforma-
tion plane. (upper left) Monopole pairing gap ∆(n). (lower
0
left) Quadrupole pairing gap ∆(n). (lower right) Quadrupole
20
pairing gap ∆(n). See Ref. [46] for definitions of ∆(n),∆(n),
22 0 20
and ∆(n).
22
those of other modes. Therefore we adopt the two low-
estfrequencymodestoderivethecollectiveHamiltonian.
Thisresultofnumericalcalculationsupportsourassump-
tion that there exists a 2D hypersurface associated with
large-amplitude quadrupole shape vibrations, which is
approximately decoupled from other degrees of freedom.
The situation changes when the collective potential en-
ergy exceeds about 5 MeV and/or the monopole-pairing
gapbecomessmall. Atypicalexampleispresentedinthe
bottom panel of Fig. 3. It becomes hard to identify two
collective modes well-separated from other modes when
β >0.4,wherethecollectivepotentialenergyishigh(see
Fig.1)andthemonopole-pairinggapbecomessmall(see
Fig.2). Inthisexample,thesecond-lowestLQRPAmode
in the 0.4<β <0.5 region has pairing-vibrational char-
acter but becomes non-collective for β > 0.5. In fact,
many non-collective two-quasiparticle modes appear in
FIG. 1: (Color online) Collective potential V(β,γ) for
68,70,72Se. The regions higher than 3 MeV (measured from its neighborhood. This region in the (β,γ) plane is not
the oblate HB minima) are drawn by rosybrown color. One- important, however, because only tails of the collective
dimensionalcollectivepathsconnectingtheoblateandprolate wave function enter into this region.
localminimaaredeterminedbyusingtheASCCmethodand
It may be useful to set up a prescription that works
depicted with bold red lines.
even in a difficult situation where it is not apparenthow
tochoosetwocollectiveLQRPAmodes. Wefindthatthe
followingprescriptionalwaysworkswellforselectingtwo
C. Properties of the LQRPA modes
collectivemodes amongmany LQRPAmodes. This may
be called a minimal metric criterion. At each point on
In Fig. 3 the frequencies squared, ω2(β,γ), of various the (β,γ) plane, we evaluate the vibrational part of the
i
LQRPA modes calculated for 68Se are plotted as func- metric W(β,γ) given by Eq. (44) for all combinations of
tions of β and γ. In the region of the (β,γ) plane where twoLQRPAmodes,andfindthepairthatgivesthemini-
the collective potential energy is less than about 5 MeV, mumvalue. WeshowinFig.4howthisprescriptionactu-
we can easily identify two collective modes among many ally works. In this figure, the W(β,γ) values are plotted
LQRPA modes, whose ω2(β,γ) are much lower than as functions of β and γ for many pairs of the LQRPA
i
8
107
14
2V) 1102 -2V) 106
Me 8 Me105
2ωβγ (,) ( 246 2βγβW(,)/ (110034
LQRPA
0
IB
-2 102
0 10 20 30 40 50 60 0 10 20 30 40 50 60
γ (deg) [β =0.305] γ(deg) [β =0.305]
0 0
107
14
2V) 1102 -2V) 106
Me 8 Me105
2βγ (,) ( 246 2βγβ,)/ (104
ω 0 W(103
LQRPA
-2
IB
-4 102
0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6
β [γ =0.5(deg)] β [γ =0.5(deg)]
0 0
107
14
2V) 1102 -2V) 106
Me 8 Me105
2βγ (,) ( 246 2βγβ,)/ (104
ω 0 W(103
LQRPA
-2
IB
-4 102
0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6
β [γ =30.5(deg)] β [γ =30.5(deg)]
0 0
FIG. 3: Frequencies squared ω2 of the LQRPA modes calcu- FIG.4: Dependenceonβ andγ ofthevibrationalpartofthe
latedfor68Seareplottedasfunctionsofβ orγ. TheLQRPA metricW(β,γ)calculatedfor68Se. (top)Dependenceonγ at
modes adopted for calculation of the vibrational masses are β = 0.3. (middle) Dependence on β along the γ = 0.5◦ line.
connectedwithsolidlines. (top)Dependenceonγ atβ =0.3. (bottom)Dependenceonβalongtheγ =30.5◦ line. Thecross
(middle) Dependence on β along the γ =0.5◦ line. (bottom) symbols indicate values of the vibrational metric calculated
◦
Dependenceon β along theγ =30.5 line. for various choices of two LQRPA modes from among the
lowest 40 LQRPA modes; the lowest mode is always chosen
and the other is from the remaining 39 modes. The smallest
vibrational metric is shown by solid line. For reference, the
modes. In the situations where the two lowest-frequency vibrationalmetriccalculatedusingtheIBvibrationalmassis
LQRPAmodesarewellseparatedfromothermodes,this indicated bybroken lines.
prescription gives the same results as choosing the two
lowest-frequencymodes (see the topand middle panels).
Ontheotherhand,apairoftheLQRPAmodesdifferent rection of the infinitesimal displacement associated with
from the lowest two modes is chosen by this prescrip- thepairoftheLQRPAmodeshasalargeprojectiononto
tion in the region mentioned above (the bottom panel). the(β,γ)plane. Therefore,thisprescriptionmaybewell
This choice may be better than that using the lowest- suited to our purpose of deriving the collective Hamilto-
frequency criterion, because we often find that a nor- nianfor the (β,γ) variables. It remains as an interesting
mal mode of pairing vibrational character becomes the open question for future to examine whether or not the
second lowest LQRPA mode when the monopole-pairing explicitinclusionofthepairingvibrationaldegreeoffree-
gap significantly decreases in the region of large β. The dom as another collective variable will give us a better
smallvaluesofthevibrationalmetricimpliesthatthedi- description in such situations.
9
FIG. 6: (Color online) Ratios of the LQRPA vibra-
tional masses to the IB vibrational masses, D /D(IB) and
ββ ββ
Dγγ/Dγ(IγB), calculated for 68Se.
evaluated using the well-known formula:
∂Hˆ ∂Hˆ
CHB CHB
µν¯ 0 0 µν¯
h | ∂ξ | ih | ∂ξ | i
D(IB)(β,γ)=2 i j ,
ξiξj [Eµ(β,γ)+Eν¯(β,γ)]3
µν¯
X
(52)
(ξ =β or γ)
i
FIG. 5: (Color online) Vibrational masses, D (β,γ),
ββ
Dβγ(β,γ)/β, and Dγγ(β,γ)/β2,in unitof MeV−1 calculated where Eµ(β,γ), 0 , and µν¯ denote the quasiparticle
for 68Se. energy,theCHBs|taite φ(β|,γ)i andthetwo-quasiparticle
| i
state a†a† φ(β,γ) , respectively (see Ref. [46] for the
µ ν¯| i
meaning of the indices µ and ν¯).
D. Vibrational masses The vibrational masses calculated for 70,72Se exhibit
behaviors similar to those for 68Se.
InFig.5the vibrationalmassescalculatedfor68Seare
displayed. We see that their values exhibit a significant
variation in the (β,γ) plane. In particular, the increase E. Rotational masses
in the large β region is remarkable.
Figure 6 shows how the ratios of the LQRPA vibra- In Fig. 7, the LQRPA rotational masses D (β,γ) cal-
k
tional masses to the IB vibrational masses vary on the culated for 68Se are displayed. Similarly to the vibra-
(β,γ) plane. It is clearly seen that the LQRPA vibra- tional masses discussed above, the LQRPA rotational
tional masses are considerably larger than the IB vibra- massesalsoexhibitaremarkablevariationoverthe(β,γ)
tional masses and their ratios change depending on β plane, indicating a significant deviation from the irrota-
and γ. In this calculation, the IB vibrational masses are tional property.
10
FIG.8: (Coloronline)RatiosoftheLQRPArotationalmasses
FIG. 7: (Color online) Rotational masses D (β,γ) in unit of
MeV−1,calculatedfor68Se. SeeEq.(39)forktherelationwith to the IB rotational masses, Dk(β,γ)/Dk(IB)(β,γ), calculated
therotational momentsof inertia J (β,γ). for 68Se.
k
Figure8showshowtheratiosoftheLQRPArotational shown that the ratio of the Migdal term to the crank-
masses Dk(β,γ) to the IB cranking masses Dk(IB)(β,γ) ing term in the rotational moment of inertia (about the
vary on the (β,γ) plane. The rotational masses calcu- 1st axis) increases as β decreases. Needless to say, the
lated for 70,72Se exhibit behaviors similar to those for Migdal term (also called the Thouless-Valation correc-
68Se. tion) corresponds to the time-odd mean-field contribu-
As we have seen in Figs. 5–8, not only the vibra- tiontakenintoaccountintheLQRPArotationalmasses,
tional and rotational masses but also their ratios to the so that the result of Ref. [54] implies that the ratio
IB cranking masses exhibit an intricate dependence on D (β,γ)/D(IB)(β,γ), increases as β decreases, in agree-
1 1
β and γ. For instance, it is clearly seen that the ra- ment with our result. To understand this behavior, it is
tios, D (β,γ)/D(IB)(β,γ), gradually increase as β de- importanttonotethat,inthepresentcalculation,thedy-
k k
creases. This result is consistent with the calcula- namical effect of the time-odd mean-field on D (β,γ) is
1
tion by Hamamoto and Nazarewicz [54], where it is associatedwiththeK =1componentofthequadrupole-