Continuum quasiparticle linear response theory using the Skyrme functional for multipole responses of exotic nuclei

0.32 MB·English

by  Kazuhito Mizuyama

Continuum quasiparticle linear response theory using the Skyrme functional for
multipole responses of exotic nuclei
Kazuhito Mizuyama,1,∗ Masayuki Matsuo,2 and Yasuyoshi Serizawa1
1Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan
2Department of Physics, Faculty of Science, Niigata University, Niigata 950-2181, Japan
(Dated: January 12, 2009)
We develop a new formulation of the continuum quasiparticle random phase approximation
(QRPA) in which the velocity dependent terms of the Skyrme effective interaction are explicitly
∗
treated except the spin dependent and the Coulomb terms. Numerical analysis using the SkM
parametersetisperformedfortheisovectordipoleandtheisovector/isoscalar quadrupoleresponses
in 20Oand 54Ca. It isshown that theenergy-weighted sum ruleincludingtheenhancementfactors
for the isovector responses is satisfied with good accuracy. We investigate also how the velocity
9
dependent terms influence the strength distribution and the transition densities of the low-lying
0
surface modes and thegiant resonances.
0
2
PACSnumbers: 21.10.Pc,21.10.Re,21.60.Jz,24.30.Cz

n
a
J I. INTRODUCTION since most of excitation modes including even the low-

2 lying excitations are located near or above the nucleon
1 separationenergy. This canbe achievedby means ofthe
Nucleineartheneutrondrip-lineprovideuswithmany
continuum QRPA methods [10, 11, 12, 13, 14, 15]. Sec-
] new physics issues which arise from the presence of
h ondly, the QRPA description should be consistent with
weaklyboundneutronsandthecouplingtounboundneu-
t the HFB description of the ground state in the sense
- tron states. The ground state and the excitation modes
l that the same effective interaction or the same density
c of a near-drip-linenucleus are indeed very different from
functional should be used for both descriptions. If this
u those of stable nuclei as is testified by the observations
n is achieved, one can calculate the ground and excited
of the neutron halo[1], the neutron skin[2] and the soft
[ states solely fromthe effective interaction(or the energy
dipole excitation[3]. In addition the nucleon correlations
density functional) without relying on phenomenologi-
2 such as the pairing may also be influenced in the new
cal parameterization of the mean-fields. This is often
v circumstances[4,5]. Consequentlytherehasbeenconsid-
called the requirement of the self-consistency. The two
5
erable efforts in the last two decades to develop nuclear
1 requirements, however, have been in a trade-off relation
many-body theories toward this direction.
1 in the actual implementations. Namely in the contin-
1 Focusing on near-drip-line nuclei in the medium uum QRPA methods which fulfill the first requirement
6. mass region, theoretical approaches based on the self- the self-consistency has been left behind since the resid-
0 consistent mean-field methods or the density functional ualinteractionfortheQRPAdescriptionisoftenapprox-
7 theories are of great promise. The Hartree-Fock- imated to a tractable simple contact force[10, 11, 12] or
0 Bogoliubov(HFB) theory[6],especiallythoseemploying the Landau-Migdal forces [13, 14, 15]. On the other
v: the coordinate-space representation [4, 5, 7], has been hand, recently developed fully self-consistent QRPA’s
i playing a central role to describe the ground state and using the Skyrme functional[23, 24] and the relativis-
X
the pair correlation. The HFB theory provides us also tic mean-field functional[18, 19] treat approximately the
r withthebasisforfurthertheoreticaldevelopmentstode- continuum states by employing the finite-box discretiza-
a
scribe the dynamics, e.g. the excitation modes built on tion or the discrete oscillator basis.
the ground state. Indeed new schemes of the quasiparti- It is therefore importantto developa new formulation
clerandomphaseapproximations(QRPA)formulatedon of the continuum QRPA which is based on the nuclear
thebasisofthecoordinate-spaceHFBhavebeenrecently density functional and thus satisfies the self-consistency
proposed and applied extensively to studies of multipole as precisely as possible. In the present paper we try to
responses of unstable nuclei [8, 9, 10, 11, 12, 13, 14, 15, make a one step progress in this direction.
16, 17, 18, 19, 20, 21, 22, 23, 24, 25] (see also references To this end we shall proceed in the following way.
in Ref.[26]). We start with the Skyrme’s Hartree-Fock energy func-
Therearetwoimportantrequirementstobeconsidered tional combined with the pair correlation energy. We
whenthe HFB+QRPAtheoriesareappliedtonear-drip- then use this functional not only for the static Hartree-
line nuclei. First of all, the coupling of excitation modes Fock-Bogoliubovmean-fieldsbutalsotoderivetheresid-
to the continuum states have to be taken into account ual interaction to be used in the continuum QRPA. In
formulating the new continuum QRPA, we pay special
attention to the energy weighted sum rule, which is not
satisfied in the previous continuum QRPA’s [10, 11, 12,
∗Electronicaddress: [email protected] 13,14,15]. Tosatisfythiswetakeintoaccountexplicitly
2
the velocity dependent central terms of the Skyrme ef- Fock energy functional E [ρ,∇ρ,∆ρ,τ,j,s,J] is
Skyrme
fective interaction to derive the residualinteraction, and completelyspecified. Concerningthepaircorrelationen-
then implement the residualinteractioninto the Green’s ergy E , we use the one evaluated for the the density-
pair
function formulation of the continuum QRPA proposed dependent delta interaction (DDDI) [34, 35]
in Ref.[10]. In the present paper, however,we do not in-
1 ρ(r) γ
clude the spin dependent densities and the Coulomb in- V (1,2)= V (1−P ) 1−η δ(r−r′).(1)
pair 0 σ
teraction in deriving the residual interaction, and hence 2 (cid:20) (cid:18) ρc (cid:19) (cid:21)
the goal of the full self-consistency is not achieved yet.
E is a functional of the local density ρ (r,t) and the
pair q
Our formulation of the Skyrme QRPA is similar to that
local pair densities
of Ref.[20, 21] apart from the treatments of the contin-
uum quasiparticle states, on which we impose the out- ρ˜ (r,t)=hΦ(t)|ψ†(r ↓)ψ†(r ↑)±ψ (r ↑)ψ (r ↓)|Φ(t)i.
±q q q q q
going wave boundary condition instead of the finite-box (2)
discretization. Application of the static variational principle to the
Byperformingnumericalcalculations,weshalldemon- total energy functional E + E leads to the
Skyrme pair
strate that the Skyrme continuum QRPA in the present Hartree-Fock-Bogoliubovequation
formulation indeed satisfies the sum rule as far as the
H φ (rσ)=E φ (rσ) (3)
dipole and quadrupole responses with natural parities 0q q q q
are concerned. We shall also show that the inclusion for the quasiparticle wave function
of the velocity dependent terms gives better description
of the strength function and the transition densities of φ (rσ)= ϕq,1(rσ) . (4)
the multipole responses in comparison with the previ- q ϕq,2(rσ)
(cid:18) (cid:19)
ous continuum QRPA that utilizes residual interactions
Here
of the simple contact forces.
The paper is organized as follows. The formulation is h −λ ˜h
H = q q q (5)
given in the next section. In Section III we present re- 0q h˜∗ −h∗+λ
(cid:18) q q q (cid:19)
sultsofnumericalcalculationperformedfortheisovector
isthe2×2matrixrepresentationoftheHFBmean-field
dipole and the isoscalar/isovector quadrupole responses
Hamiltonian
in neutron-rich O and Ca isotopes. We discuss both
the low-lying excitations and the giant resonances. We
hˆ = drdr′ h (rσ,r′σ′)ψ†(rσ)ψ (r′σ′)
shallillustrateindetailtheimportanceoftakingaccount q q q
of the velocity dependent terms by comparing with the Xq Z Xσ,σ′
Landau-Migdal approximation of the Skyrme effective +1 drdr′ h˜ (rσ,r′σ˜′)ψ†(rσ)ψ†(r′σ˜′)
interaction[27, 28]. The conclusions are drawn in Sec- 2 q q q
Z σ,σ′
tion IV. X
+h.c. (6)
The Hartree-FockHamiltonianh andthe pairpotential
q
II. CONTINUUM QRPA USING THE SKYRME h˜ are defined through the functional derivative of the
q
FUNCTIONAL
energy functionals E and E , respectively.
Skyrme pair
We considermultipole responseofthe nucleus under a
Inthissection,wegiveaformulationofthecontinuum small time-dependent external perturbation
QRPA which is based on the Skyrme functional.
We start with the energy functional of the system de- Vˆ (t) = e−iωt drf (r) ψ†(rσ)ψ (rσ)
ext q q q
fined for a determinantal many-body state vector |Φ(t)i q Z σ
X X
of the generalized form in which the pair correlation is +h.c. (7)
taken into account by means of the Bogoliubov’s quasi-
particle method[6]. The time-dependence is explicitly expressed in terms of a one-body spin-independent local
written here since we consider dynamical multipole re- field fq(r), for which we take a multipole field ∝rLYLM
sponses of the system under a time-dependent perturba- such as the electric dipole and the isoscalar/isovector
tion. TheenergyfunctionalE =E +E consists quadrupole fields.
Skyrme pair
oftheSkyrmeHartree-FockenergyE andthepair The external perturbation causes the induced fields in
Skyrme
correlation energy E . The Skyrme Hartree-Fock en- theHartree-Fockmean-fieldandthepairpotential,which
pair
ergy ESkyrme is expressed in terms of the local density we denote δhq and δh˜q, respectively. δhq and δh˜q are
ρ (r,t), its spatial derivatives ∇ρ (r,t) and ∆ρ (r,t), expressedintermsoffluctuationsinthevariousone-body
q q q
the current density j (r,t), the kinetic energy density densities
q
τ (r,t), the spin density s (r,t) and the spin-orbit ten-
q q δρ (r,t),δ∇ρ (r,t),δ∆ρ (r,t),δτ (r,t),δj (r,t),
sor J (r,t) where q = n,p stands for the neutron or q q q q q
q
proton components[29, 30]. Given a parameter set such δJq(r,t),δsq(r,t),
as SIII[31], SkM*[32] andSLy4[33], the Skyrme Hartree- δρ˜ (r,t) (8)
±q
3
←− −→ ←− −→
and the second derivatives of the energy functional. A + ∇+∇ ·cqq′δ∇ρq′ + ∇−∇ ·bqq′δ2ijq′ (9)
fully self-consistent QRPA based on the Skyrme HFB
h i h i
functional can be constructed if one considers all the
kinds of density fluctuations in Eq.(8). In the previous and
continuum QRPA approaches, however, only the fluctu-
ations in the local densities δρq(r,t) and δρ˜±q(r,t), and δh˜ =a˜ δρ˜ −a˜ δρ˜ . (10)
the induced fields associated with these density fluctua- q q +q q −q
tionshavebeentakenintoaccount[10,11,12,13,14,15].
Althoughthisapproximationhasalargenumberofprac- Thefunctionsaqq′,bqq′,cqq′ anda˜q areexpressedinterms
tical usefulness, it is not sufficient in some respects: it of the local densities and the effective interaction pa-
violates the energy weighted sum rule when the Skyrme rameters. The definition of aqq′,bqq′ and cqq′ follows
HF mean-field with the effective mass is adopted. This Ref. [43], and their detailed expressions are given in Ap-
is because the current conservation law is not satisfied pendix. Note here that we use δ∆ ρ (r,t) ≡ (∆ +
+ q
when the velocity dependent parts (the terms propor- ∆′)δρq(rr′,t)|r′=r = δ∆ρq(r,t)−2δτq(r,t) in place of
tional to the t1 and t2 terms) of the Skyrme interac- δ∆ρq(r,t), where ρq(rr′,t) is the density matrix. We
tionandthecurrentfluctuationsδj areneglectedinthe alsoattachedafactor2itothecurrentfluctuationδj so
q q
RtthhPiasAtp[a3ur6er,pr3oe7ss,ep3ow8ne,s3isb9hl]ae.lflWoirnectahluiemdeeanatelrilgmtyhpweroedvieginnhgstiettdyhisflsuupmcotiurnuatl.teiFofnoorsr tahllaetl wδ2itijhqδ=∇ρ(∇q =−(∇∇′)+δρ∇q(′r)rδρ′,qt()r|rr′′=,tr)|rb′e=corm. eTshienflpuacr--
tuation in the kinetic energy density δτ (r,t) does not
the responses caused by the spin independent local mul- q
appear here since it can be eliminated from the induced
tipole fields. They arethe fluctuations δj , δ∇ρ ,δ∆ρ ,
q q q fieldby apartialintegration. Neverthelesswe eventually
andδτ inthe current,the spatialderivativesofthe den-
q include δτ as explained later.
sity and the kinetic energy density. It is more preferable q
to take into account also the fluctuations in the spin- The induced fields are also represented in the 2 × 2
dependent densities s and J , but we neglect them in matrix form as
q q
the present work. This is one approximation which re-
mains in the present approach. We neglect the residual δh δh˜
Coulombinteraction. Thecrossderivativesamongρq and δh˜∗q −δhq∗ = BβOˆβ κβγδργ (11)
ρ˜±q arealsoneglected. These arethe secondapproxima- (cid:18) q q (cid:19) Xβ Xγ
tion we introduce. Consequently the full self-consistency
is not fulfilled, but the treatment of the residual interac-
where δρ is a collective notation for
γ
tion is significantly improved compared to the previous
continuum QRPA approaches [10, 11, 12, 13, 14, 15] in
thesensethatthepresentformalismallowsustodescribe δργ ∈δρq,δ∆+ρq,δ∇ρq,δ2ijq,δρ˜±q. (12)
the correct energy weighted sum rule for the multipole
responses. Note also that the approximate treatment ←− −→ ←− −→
andOˆβ denotesthederivativeoperators1,∆+∆,∇−∇
of the particle-hole residual interaction is comparable to ←− −→
and ∇+∇ while Bβ stands for one of the 2×2 matrices
that adopted in the currently available continuum RPA
1 0 1 0 0 1 0 1
approaches which utilize the Skyrme-Hartree-Fock func- , , and . κ rep-
0 −1 0 1 1 0 −1 0 βα
tional without taking into the pairing [40, 41, 42, 43]. (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
On the other hand, we should keep in mind that the ap- resentsthe functions aqq′,bqq′,cqq′ anda˜q inEqs.(9) and
proximationneglectingthespindependentdensitiesmay (10). The correspondenceamong Oˆα, Bα κβγ and δργ is
not be justified for multipole responses with unnatural shown in Table I.
parities involving spin excitations. Theexternalperturbationandtheinducedfieldscause
Theinducedfieldsundertheaboveapproximationsare quasi-particle excitations, which in turn bring about
expressed as fluctuations in the densities δρ ,δρ˜ ,δ∆ ρ ,∇δρ and
q ±q + q q
δ2ij . This relationis givenbythe linearresponseequa-
←− −→ q
δhq = aqq′δρq′ +bqq′δ∆+ρq′ + ∆ + ∆ bqq′δρq′ tion, which is written as
Xq′ h i
δρ (r,ω)= dr′Rαβ(rr′,ω) κ (r′)δρ (r′,ω)+δ f (r) (13)
α 0q βγ γ β,0q q
" #
β Z γ
X X
in the frequency domain.
Here Rαβ(rr′,ω) is the unperturbed response function forthedensityδρ andthefieldBβOˆβ. UsingtheGreen’s
0q α
4
function formalism of the continuum QRPA[10], the un- perturbed response function is expressed as
1
Rαβ(rr′,ω) = dETr AαOˆα(r)G (rr′,E+h¯ω+iǫ)BβOˆβ(r′)G (r′r,E)
0q 4πi 0q 0q
ZC h i
1
+ dETr AαOˆα(r)G (rr′,E)BβOˆβ(r′)G (r′r,E−¯hω−iǫ) (14)
0q 0q
4πi
ZC h i
in terms of the quasi-particle Green’s function G (E)= quantities δρ = δ∇ρ and δ2ij , they are expanded
0q α q q
(E −H )−1 and a contour integral in the complex en- as
0q
ergyplane. The complexenergyintegralisperformedon
a rectangular contour C enclosing the negative energy δρα(r,ω)= YLλM(rˆ)[δρα]λL/r2, (16)
partofthe realE axis withthe twosides locatedat±iǫ λ=L±1
2 X
[10]. Here ǫ is a small parameter which plays a role of
the smoothing energy width. The matrices Aα and Bβ in terms of the vector spherical harmonics YLλM
and the operators Oˆα and Oˆβ follow Table I, but we re- and the radial functions [δρα]λL = [δ∇ρq]Lλ=L±1 and
markthatthematrixAα takesaformAα = 2 0 for [δ2ijq]Lλ=L±1. Note that only the terms λ = L±1 re-
0 0 main here since we consider the multipole excitations
the particle-hole densities δρ ,δ∆ ρ ,δ∇ρ(cid:18),δ2ij (cid:19)and with the natural parity. Then the linear response equa-
q + q q q
δτ while the matrix Bβ has the following definitions: tion (13) is rewritten as an equation for the relevant
q radial functions [δρ ] ,[δρ˜ ] ,[δ∆ ρ ] ,[δ∇ρ ]λ=L±1
1 0 q L ±q L + q L q L
Bβ = 0 −1 forthe’time-even’quantitiesδρq,δ∆+ρq and [δ2ijq]Lλ=L±1. Denoting collectively these density
(cid:18) (cid:19) fluctuations δρ , the linear response equation for these
1 0 αL
and δ∇ρq, and Bβ = 0 1 for the ’time-odd’ δ2ijq variables is given by
(cid:18) (cid:19)
(See Table I).
δρ (r,ω)
αL
Let us assume the spherical symmetry of the ground
state, and we apply the multipole decompositions. Let = dr′Rαβ (rr′ω)
0,qL
L be the multipolarity of the excitation modes under
β Z
X
consideration. The fluctuation in the scalar quantities
× κ δρ (r′,ω)/r′2+δ f (r′)
δρα =δρq,δρ˜±q and δ∆+ρq are expanded as βγ γL β,0q qL
" #
γ
X
δρ (r,ω)=Y (rˆ)[δρ ] /r2, (15) (17)
α LM α L
and we now consider only the radial functions [δρ ] = using the unperturbed response function for the fixed
α L
[δρ ] ,[δρ˜ ] and [δ∆ ρ ] . Concerning the vector multipolarity L
q L ±q L + q L
Rαβ (rr′,ω)
0,qL
1 |hl′j′||Y ||lji|2
= 4πi dE 2L+L 1 Tr AαOˆlαjl′j′(r)G0,ql′j′(rr′,E+h¯ω+iǫ)BβOˆlβ′j′lj(r′)G0,qlj(r′r,E)
ZC lXjl′j′ h i
+Tr AαOˆlα′j′lj(r)G0,qlj(rr′,E)BβOˆlβjl′j′(r′)G0,ql′j′(r′r,E−¯hω−iǫ) .
h i (18)
Here G (r′r,E) is the 2×2 radial HFB Green’s func- responding to the previously defined Oˆβ. Their explicit
0,qlj
tion for specified orbital and total angular momenta l forms are given in Table I. We adopt the exact form for
and j, and Oˆβ is the radial derivative operator cor- the radial HFB Green’s function[44] constructed as
l′j′lj
5
G (rr′,E)= css′(E) θ(r−r′)φ(+s)(r,E)φ(rs′)T(r′,E)+θ(r′−r)φ(rs′)(r,E)φ(+s)T(r′,E) (19)
0,qlj qlj qlj qlj qlj qlj
s,sX′=1,2 (cid:16) (cid:17)
intermsoftwoindependentsolutionsφ(rs)(r,E)(s=1,2) makes sense in Eq.(18).
qlj
regular at the origin r = 0 of the radial HFB equation To obtain a numerical solution of the linear response
(+s)
andtwoindependentsolutionsφ (r,E)(s=1,2)satis- equation, we need to rewrite further Eq.(17). When the
qlj
fying the out-going boundary condition. (The construc- radial derivative operators−→∂∂r and ∂∂r′ act←−on the radial
tion (19) is the same as that used in Refs.[10, 44] except HFBGreen’sfunctionlike ∂ G (rr′,E) ∂ ,asingular
that the effective mass should be taken into account in ∂r 0,qlj ∂r′
the definitions of the Wronskian and the coefficients csqslj′ term proportional to 2mh¯∗q2(r)δ(r−r′) emerges. We need
while in Refs.[10, 44] the bare mass is assumed.). In this to treat these singular terms separately in the numerical
waytheexacttreatmentofthecontinuumsingle-particle calculation. Forthispurposewerewritethederivativeof
statessatisfyingthe properboundaryconditionisimple- the Green’s function into singular and regular parts
mented in the QRPA formalism.
−→ ←−
Note that in Table I we use the following convention ∂ ∂ 2m∗(r)
G (rr′,E) = − q δ(r−r′)
for the derivative opera←−tors m−→arked with the right/left- ∂r 0,qlj ∂r′ ¯h2
sided arrows such as ∂ ± ∂ . When this derivative −→ ←−
∂r ∂r ∂ ∂
is inserted in Oˆlβ′j′lj(r′) in the first term of r.h.s. of +∂rG0,qlj(rr′,E)∂r′ (20)
−→
∂ f g
Eq.(18), the derivative symbol with the right-sided
∂r′
arrow indicates that it acts on the coordinate r′ in where the regular part (the second term in r.h.s de-
the Green’s function G (r′r,E) while the other one noted with the tildered derivatives) is defined as a part
0,qlj −→ ←−
←∂− of ∂ G (rr′,E) ∂ that arises from the action of
∂r′ with the left-sided arrow acts on the Green’s func- ∂r 0,qlj ∂r′
tion G0,ql′j′(rr′,E + h¯ω + iǫ). The same−→rule is ap- the derivatives on the wave functions φl(jrs) and φl(j+s)
plied also to the operator Oˆα (r), i.e., ∂ acts on r in Eq.(19), but not on the Heaviside theta function
ljl′j←′− ∂r θ(r−r′). Inserting this decomposition into the response
inG0,ql′j(rr′,E+h¯ω+iǫ)while ∂∂r onr in G0,qlj(r′r,E). function(Eq.(18)), the r.h.s. of the linear response equa-
The ordering of the operators and the Green’s functions tion (17) is decomposed into two parts:
δρ (rω) = dr′R˜αβ (rr′ω) κ (r′)δρ (r′ω)/r′2+δ f (r′)
αL 0,qL βγ γL β,0q qL
" #
β Z γ
X X
+2 Sαβ(r) κ˜ (r)δρ (rω)/r2+δ f (r) (21)
q βγ γL β,0q qL
" #
β γ
X X
where
δρ ∈[δρ ] ,[δ∆ ρ ] ,[δ∇ρ ]λ=L±1,[δτ ] ,[δ2ij ]λ=L±1,[δρ˜ ] . (22)
αL q L + q L q L q L q L ±q L
Here R˜αβ denotes a part of the response function tributionfromthesingulartermssuchas 2m∗q(r)δ(r−r′).
0,qL h¯2
whichcontainsonlytheregularpartsofthederivativesof The integral dr′ disappearsin this term because of the
eG−→x0c,qelpj.tItthsaetxpthrees←sd−ieorniviasttihveessaomfethaestGhraeteonf’sRf0αu,βqnLc(tEioqn.,(1e8.)g). dpeelntdaixfu.nNctoitoenR.thTahteSeqαxβprisesasioonnse-opfoiSnqαtβfuanrcetgioivneinndinepAepn--
∂ G (rr′,E) ∂ in Eq.(20) is replaced by the corre- dentofthefrequencyω,expressedintermsoflocalquan-
∂r 0,qlj ∂r′
−→ ←− tities such as ρ (r), τ (r), m∗(r) and their derivatives.
sponding regular part ∂ G (rr′,E) ∂ . On the other q q q
∂r 0,qlj ∂r′ It is noted that the linear response equation (21) in-
hand,thesecondtermofr.h.s. ofEq.(21)representscon-
f f cludesthefluctuation[δτ ] inthekineticenergydensity
q L
6
δρα Aα, Bα Oˆα(r) Oˆlαjl′j′(r) καβ
2 0 1 0
δρq 0 0 , 0 1 1 1 aqq′
(cid:18) (cid:19) (cid:18) − (cid:19)
δ∆+ρq ←∆−+→−∆ ∂←∂−r22 + ∂−∂→r22 − l′(lr′2+1) − l(lr+21) bqq′
2LL+1 ←∂∂−r + →−∂∂r + L−r1 (for[δ∇ρq]Lλ=L−1)
δ∇ρq ←∇−+→−∇  −q 2LL++1(cid:18)1 ←∂∂−r + →−∂∂r − L+r(cid:19)2 (for [δ∇ρq]λL=L+1) cqq′
q (cid:18) (cid:19)
 ←∂−→−∂ 1→−∂ ←∂−1
δτq ←∇−·→−∇ ∂r ∂+rl(−l+r1)+∂rl′(−l′+∂1r)−r(L+2)(L−1) 1 –
2 r2
(cid:0) (cid:1)
L ←∂− →−∂ + l(l+1)−l′(l′+1)1 (for [δ2ij ]λ=L−1)
2 0 1 0 − 2L+1 ∂r − ∂r L r q L
δ2ijq (cid:18)0 0(cid:19) , (cid:18)0 1 (cid:19) −←∇−+→−∇  q2LL++11 (cid:18)←∂∂−r − →−∂∂r − l(l+1)L−+l′1(l′+1)1r (cid:19)(for[δ2ijq]λL=L+1) −bqq′
q (cid:18) (cid:19)

0 1
δρ˜+q 1 0 1 1 a˜qδqq′
(cid:18) (cid:19)
0 1
δρ˜−q 1 0 1 1 −a˜qδqq′
(cid:18)− (cid:19)
TABLEI:ThecorrespondenceandtheexpressionsforthematricesAαandBβ,theoperatorsOˆα andOˆlαjl′j′ andκαβ appearing
in Eqs. (11), (14) and (18). See also thetext and Appendix.
τ as a dynamical variable to be considered. This is be- Green’sfunctions (insteadofoneGreen’s function inthe
q
cause [δτ ] emerges from the singular terms associated case of the continuum RPA[40, 43]). Looking at the ex-
q L
with the linear response equation for [δ∆ ρ ] . Finally pression of Eq.(18), it may appear that products of two
+ q L
we make a little remark on the structure of the singu- delta functions 2m∗q(r)δ(r−r′) emerge from the singular
lar terms. The presence of the singular terms has been h¯2
terms of two HFB Green’s functions in Eq.(18). Such a
notifiedintheformulationoftheSkyrme-HFpluscontin-
term however does not contribute to the response func-
uum RPA[40, 43] where the pairing is neglected. In the
tion since it has no energy dependence and hence it van-
present Skyrme-HFB plus continuum QRPA approach,
ishes when the contour integral in the complex energy
thestructureofthesingulartermsismoreinvolvedsince
plane is performed.
the response function contains two single-particle HFB
III. NUMERICAL ANALYSIS A. Numerical procedure
Let us first describe the detailed procedure of the nu-
merical calculation.
In this section, we shall demonstrate the Skyrme con- We adopt the SkM∗ parameter set of the Skyrme
tinuum QRPA by performing numerical calculations for interaction and the mixed-type parametrization of the
the dipole and quadrupole responses in 20O and 54Ca. DDDI pairing interaction (η = 0.5, γ = 1, ρ = 0.16
0
7
fm−3)[35]formostofthecalculations. Theforcestrength
0.8
V of the DDDI is chosen so that the average neutron Full
0
0.7
pairing gap h∆ i reproduces the overall magnitudes of LM
n
] t +t
the experimental odd-even mass differences for the iso- V 0.6 0 3
e
topic chain, obtained with the three-point formula[45]. M 20
0.5 O, E1
Here we use the average pairing gap defined by h∆ i = 2/
n m
drρ˜n(r)∆n(r)/ drρ˜n(r). The adopted value is V0 = 2 f 0.4
−280 and −285 MeVfm−3 for 20O and 54Ca producing [e 0.3
Rh∆ni= 1.91 MeVRand 1.29 MeV, respectively. E) x 0.2
(
Since we use the contact interaction for the effective S
0.1
pairinginteraction,weneedacut-offofthequasi-particle
statesintheHFBcalculation. Wedefinethecut-offwith 0
Full
respect to the quasi-particle energy E < E = 60
α max 2 LM
MeV. Concerning the angular momentum quantum
]
V
numbers lj we sum up the quasi-particle states up e
M
to l = 7h¯ and 8h¯ for 20O and 54Ca, respectively. 1.5
In pmearxforming the HFB and the continuum QRPA 2m/ 54Ca, E1
f
calculations,wediscretizethe radialcoordinatespaceup 2e 1
to rmax = 15 fm with an equidistant interval ∆r = 0.2 ) [x
fm. InthecontinuumQRPAcalculations,thedynamical E
S( 0.5
quantities to be obtained are the eighteen functions
[δρ ] ,[δρ˜ ] ,[δ∆ ρ ] ,[δ∇ρ ]λ=L±1,[δ2ij ]λ=L±1
q L ±q L + q L q L q L
and [δτ ] which obey the linear response equation 0
q L
0 10 20 30 40 50
(21). Using the same radial mesh, these functions
are represented as a grand vector while the linear Ex [MeV]
response equation is represented as a linear algebraic
equation where the response function R˜αβ (and Sαβ) FIG. 1: The B(E1) strength function of isovector dipole re-
0,qL q sponse in 20O (upper panel) and in 54Ca (lower panel) cal-
corresponds to a matrix. Since the number of the ∗
culated with the parameter set SkM . The solid curve is the
functions to be solved is larger (18 vs. 6) than in the
resultobtainedinthefullcalculationwhilethedashedcurveis
previous continuum QRPA that handles only the local
that in theLandau-Migdal (LM) approximation. The dotted
densities [δρq]L and [δρ˜±q]L, the number of the matrix curvein theupperpanel istheone in the t0+t3 approxima-
elements of the response functions is therefore about tion. Seealso thetext.
ten times larger than in the previous continuum QRPA
calculations. Toreducetheincreasedcomputationalcost
thus caused,we havechosenthe values of l andr
max max
smaller than those used in our previous calculations
[10, 11, 12, 13]. For the same reason we have used here
a relatively large smearing parameter ǫ = 1.0 MeV in the continuum QRPA in the Green’s function formalism
most of the following calculations. We evaluate the fulfillstheself-consistencyintheparticle-particlechannel
strength function at discretized excitation energies with with high accuracy [10, 11]. The renormalization factor
an interval of 0.5MeV. is f =1.0470 and 1.0142 for 20O and 54Ca, respectively.
It is noted here that the self-consistency is not com-
pletely satisfied in the present formulation since a few In the following analysis, we would like to demon-
approximations are introduced in deriving the residual strate how the description of the multipole response
interaction from the Skyrme HFB functional. Conse- is improved in comparison with the previous contin-
quently the spuriousmodes ofmotionwhichshouldhave uum QRPA where the residual interaction is simpli-
exact zero excitation energy according to the Thouless’s fied to a contact force. For this purpose, we per-
theorem[6]donotemergeattheexpectedenergy. Acom- form calculations where the Landau-Migdal (LM) ap-
monly adopted procedure to circumvent this problem is proximation to the residual interaction is introduced
to renormalize the residual interaction κ in Eq.(21) [13, 14, 15, 16, 17, 20, 21, 22]. This is an approxima-
αβ
by an overall factor f as κ → f × κ so that the tion which replaces the residual interaction by a contact
αβ αβ
excitation energy of the spurious mode is forced at the force ∝δ(r−r′) whose strength is given by the density-
zero energy[10, 11, 12, 14, 15, 20, 21, 22, 46]. We ap- dependent Landau-Migdal parameters F and F′ evalu-
0 0
ply this renormalization procedure to the particle-hole ated for the Skyrme functional[9, 27, 28] using the local
residual interactions that are derived from the Skyrme densityapproximation. Itshouldbenoted,however,that
HF functional E . The residual interaction in the the Landau-MigdalparametersF andF′ containa part
Skyrme 0 0
particle-particle channel, derived from the pair correla- ofthe t andt terms,andthis approximationshouldbe
1 2
tion energy E , is kept in the original strength since distinguished from dropping all the t and t terms.
pair 1 2
8
2L+1
120 = − Im drf (r)[δρ ] (r,ω)(23)
qL q L
Full π
100 LM Xq Z
eV] 80 20 + for the operator FˆLM with the multipolarity L can be
M O, IS 2
evaluated in terms of the solution [δρ ] (r,ω) of the
4/ q L
m 60 linear response equation (21) obtained for the external
E) [fx 40 field FˆL0. We evaluate the B(E1), B(IS2) and B(IV2)
S( strengthfunctions associatedwiththe electric dipole op-
20 erator
0 eN Z eZ N
Full FˆIV = r Y (Ω )− r Y (Ω ) (24)
700 LM 1M A i 1M i A i 1M i
i=1 i=1
V] 600 X X
Me 500 and the isoscalar/isovectorquadrupole operators
4m/ 400 54Ca, IS 2+
E) [fx 300 Fˆ2IMS = A ri2Y2M(Ωi), Fˆ2IMV = A τzri2Y2M(Ωi). (25)
S( 200 Xi=1 Xi=1
100
The B(E1) strength functions calculated for 20O and
0 54Ca are shownwith the solid curve in Fig.1. The broad
0 10 20 30 40 50
peaks around E = 20 MeV in 20O and E = 16 MeV
x x
Ex [MeV] in54Cacorrespondtothegiantdipoleresonance(GDR).
There is s small bump around E = 8 MeV in 54Ca,
FIG. 2: The same as Fig.1 but for the B(IS2) isoscalar x
which corresponds to the soft dipole excitation or the
quadrupolestrength function in 20O and 54Ca.
pygmydipole resonance. We findhoweverthatthe small
peakE =13MeVin20OisneithertheGDRnorthesoft
x
40 dipole excitation, but rather a non-collective two quasi-
35 FLuMll particle excitation (cf. Section IIID). The strength at
V] 30 E ≈ 0 is due to the spurious mode, and it is caused by
Me 25 20O the incomplete self-consistency.
4m/ 20 + For the sake of comparison, the B(E1) strength func-
E) [fx 15 IV 2 toifotnhoebrteasiindeudalininttheeraLcatinodnaius-aMlsiogdpalolt(tLeMd)waitphprtohxeimdaasthioedn
S( 10
curveinFig.1. Notethattherenormalizationfactorused
5
in the Landau-Migdal approximation (f = 0.6686 and
0
Full 0.7515 for 20O and 54Ca, respectively) deviate signifi-
200 LM cantlyfromoneincontrasttothoseinthefullcalculation
V] (f =1.0470and1.0142). This factsuggeststhatthere is
Me 150
4m/ 54Ca significantimprovementintheself-consistencycompared
E) [fx 100 IV 2+ winitahStkhyerLmMe-QapRpPrAoxcimalactuiloant.ioTnhuissinfegatduirsecriestipzoeidntceodntoiunt-
S( 50 uum quasiparticle states[21].
Itis seenin Fig.1that the profile ofthe strengthfunc-
0 tion obtained in the LM approximation differs signifi-
0 10 20 30 40 50
cantly from that in the full calculation. The peak po-
E [MeV]
x
sitions of the giant dipole resonance are apparently dif-
FIG. 3: The same as Fig.1 but for the B(IV2) isovector ferent. Estimating the centroid energy of the GDR by
quadrupolestrength function. E(GDR) = m /m using the energy weighted sum m
1 0 1
andthe non-weightedsum m , we find E(GDR)=20.66
0
MeV(20O)and15.80MeV(54Ca)forthefullcalculation
while E(GDR) = 17.67 and 14.20 MeV in the LM ap-
B. Strength function
proximation,exhibitingaratherlargedifferencebyabout
2-3 MeV. It is clear that the LM approximation is not
The strength function
very appropriate to give precise quantitative description
of the GDR.
S(h¯ω) ≡ |hν|Fˆ |0i|2δ(h¯ω−E ) If we evaluate the energy weighted sum integrated up
LM ν
toE =15MeVfor20O,thefullcalculationgives9.7%of
ν,M
X
9
theclassicalThomas-Reiche-Kuhn(TRK)sumrulevalue
0.8
while it is 20.1% in the LM approximation. Comparing Full
0.7
withtheexperimentalvalue12%[47],wefindthatthefull LM
] t +t
calculation is in better agreement with the experiment. V 0.6 0 3
e
The B(E1) strength function in 20O is calculated in a M 20
0.5 O, E1
fully self-consistent Skyrme-QRPA calculation[24] using 2/
m
the same SkM∗. We find only small difference between 2 f 0.4
ourcalculationandthatinRef.[24]. Itmaybeattributed [e 0.3 SLy4
)
to the neglect ofthe Coulomband spin-dependent terms Ex 0.2
in our calculations. The observed effect of the velocity S(
dependent terms on the GDR centroid energy is essen- 0.1
tially the same as that discussed in Ref.[21]. 0
Figure 2 displays the B(IS2) isoscalar strength func- 0 10 20 30 40 50
tion for the quadrupole responses in 20O and 54Ca. In E [MeV]
x
both nuclei there are two significant peaks, one around
Ex =2−3MeVcorrespondingtothelow-lying2+ collec- FIG.4: ThesameastheupperpanelofFig.1,buttheSkyrme
tivevibrationalmodeandtheotheraroundEx =15−20 parameter set SLy4is used.
MeVcorrespondingtotheisoscalargiantquadrupoleres-
onance (ISGQR). (The experimental 2+ energy in 20O
1
is 1670 keV[48].) The calculated isoscalar quadrupole of the velocity dependent terms depends on the adopted
strength function for 54Ca is quite similar to that ob- Skyrme parameter set. To demonstrate this we show in
tained in the fully self-consistent Skyrme QRPA using Fig.4 the B(E1) strength function in 20O obtained with
the same SkM∗[24] apart from features associated with SLy4[33] instead of SkM∗. It is seen that there is no big
different choices of the smoothing width. differenceintheGDRpeakpositionbetweenthefullcal-
Concerning the effect ofthe velocity dependent terms, culation and in the Landau-Migdal approximation, and
itisseeninFig.2thatthedifferencebetweenthefullcal- hence the effect of the velocity dependent terms in the
culation and the LM approximation is less significant in case of SLy4 appears smaller than in the case of SkM*.
comparison with the isovector dipole response: the peak Note however that even in this case there is significant
positions of the giant isoscalar quadrupole resonance differencebetweentheLMandt0+t3 approximations. If
(E = 19.0 MeV) and of the low-lying state (E = 3.0 welookatthedifferencebetweenthefullcalculationand
MeV) in20O is affectedonly little by inclusionofthe ve- thet0+t3approximation,theeffectofthevelocitydepen-
locity dependent terms. The same is seen also in 54Ca. denttermsisnotnegligible. Notealsothatthedifference
Note however that the influence of the velocity depen- between the full calculation and the LM approximation
denttermsontheB(IV2)isovectordistributionisclearly is not negligible in the sum rule (cf. next subsection).
larger than in the case of the B(IS2) isoscalar strength
distribution as is seen in Fig.3. Combining Figs. 1, 2
and 3, we see an apparent trend that the influence of C. Energy weighted sum rule
the velocity-dependent terms is more significant in the
isovector responses than in the isoscalar responses. Let us analyze whether the energy weighted sum rule
Beforemovingtothenextsubsection,wewouldliketo is satisfied in the present calculations. For this purpose
makea few additionalremarksonthe effectofthe veloc- we evaluate the running energy weightedsum definedby
itydependentterms. Wefirstremarkthatitispossibleto
consideranotherwaytoevaluatetheeffectofthevelocity Ex
W(E )= dE E S(E), (26)
x
dependent terms, e.g. by comparing with a calculation
Z0
where all the velocity dependent terms containing the t
1
which integrates the sum up to an excitation energy E .
and t parameters are completely neglected. (In other x
2
The limiting value of W(E ) for a sufficiently large E
words,itisthecalculationwhereonlythesimplecontact x x
is to be compared with the energy-weighted sum rule
interactionassociatedwith the t andt terms aretaken
0 3
(EWSR).
into account. It is different from the Landau-Migdal
TheEWSRfortheB(ISL)isoscalarmultipolestrength
(LM) approximation since in the latter a part of the t
1 function is identical to the classical sum rule mcl ≡
aranmdett2etresrFmsiasnrdenFo′r.m)aTlizheedBin(tEo1t)hestLreanngdtahu-fMunigctdiaolnpain- 1 h¯2 L(2L+1)2Ahr2L−2i which is expressedin term1s of
0 0 4π2m
this t +t approximation calculated for 20O is plotted the expectation value of the radial moment r2L−2 with
0 3
in the upper panel of Fig.1 together with the other two respecttothe groundstate[6,38]. Forthe isovectormul-
curves representing the full calculation and the LM ap- tipole strength functions, however, the EWSR contains
proximation. Wefindherethattheresultobtainedinthe theenhancementfactorwhicharisesfromtheresidualin-
t +t approximation is almost identical to that in the teraction, in particular the velocity-dependent terms in
0 3
LM approximation. Secondly, we remark that the effect the case of the Skyrme effective force[36, 38, 39, 49, 50].
10
The EWSR for the B(E1) electric dipole strength func- 120
tion is givenby mEWSR(E1)= NZmcl(1+κ)where κ is Full
tchaeseeonfhtahnecSemkyernmtef1afoctrocre[2w3h,i5ch1].iAsT2ehaes1ivlyaleuveaolufaκtefodriSnktMh∗e eV] 100 EWSR (κ=0.32) LM
M 80
is κ=0.32 and 0.36 in 20O and 54Ca, respectively. 2 TRK
m
The upper panel of the Fig.5 displays the running en- 2 f 60 20O
ergy weighted sum W(E ) for the B(E1) strength func- e
tion of the dipole respoxnse in 20O (cf. Fig.1). The E)[x 40 E1
running sum evaluated at the highest calculated energy W(
20
E = 55 MeV reaches 96% of the EWSR. This suggests
x
that the EWSR is satisfied in the present calculation. It 0
isnotedthatW(E )approachestheEWSRvalueinclud- 80 Full
x
LM
ing the enhancement factor, but not the classical TRK V] 70 EWSR(classical)
value plotted with the dotted horizontal line in Fig. 5. Me 60
Namely the effect of the velocity dependent terms in the 4 50
m
residualinteractionis indeedincludedin the presentcal- 3f 40 54Ca
0
culation. In the same figure, we also show the result ob- 1
tained in the LM approximation. In this case, however, E)[x 30 IS 2+
the running sum reaches only 86 % of the EWSR, and W( 20
hencetheapproximationfailstodescribetheEWSRand 10
the enhancement factor. The fluctuations δ∆+ρ, δ∇ρ, 0
δτ and, in particular, δ2ij play the essential role to re- 0 10 20 30 40 50
store the EWSR since these are the fluctuations associ- E [MeV]
x
ated with the velocity-dependent terms (∝ t and t ).
1 2
Note that the LM approximation neglects these fluctu- FIG. 5: The running energy weighted sum for the B(E1)
ations although a part of the velocity-dependent t1+t2 electric dipole strength function in 20O(upper panel) and
terms is taken into account via the Landau-Migdal pa- thatfor theB(IS2)isoscalar quadrupolestrengthfunction in
rameters F and F′. 54Ca(lower panel) obtained using the Skyrme parameter set
0 0 ∗
SkM . Thesolidcurverepresentstheresultofthefullcalcula-
The lower panel in Fig.5 displays the running en-
tionofthepresentSkyrmecontinuumQRPAwhilethedashed
ergy weighted sum for the B(IS2) isoscalar quadrupole
curve is the result obtained in the Landau-Migdal (LM) ap-
strength function in 54Ca (cf. Fig.2). The energy
proximation to the residual interaction. The horizontal solid
weighted sum amounts to 98% of the EWSR, and we
lineindicatesthevalueoftheenergy-weightedsumruleinthe
confirm more clearly than in the analysis of the elec- presentcalculation,whichincludestheenhancementfactorin
tric dipole strength that the EWSR is satisfied in the the isovector response. The dotted horizontal line in the up-
present calculation. When we adopt the LM approxi- perpaneldenotesthevalueoftheThomas-Reiche-Kuhnsum
mation where the fluctuations δ∆ ρ, δ∇ρ, δτ and δ2ij rule.
+
are neglected, the running sum W(E ) at the maximum
x
energy E = 55 MeV overshoots the EWSR by about
x
ten percent. Again the consistent inclusion of the veloc- inclusion of the dynamical pairing effect is important to
itydependenttermsintheQRPAdescriptionisessential guarantee the energy weighted sum-rule because other-
to guarantee the EWSR. It is noted also that the differ- wise the self-consistency in the pairing channelwould be
encebetweenthefullcalculationandtheLandau-Migdal violated. This means, in the present context, that we
approximation is significant mainly in the energy region needtoinclude boththe velocitydependenttermsofthe
(E >18 MeV) higher than the giant resonance peak. Skyrme effective interaction and the residual pairing in-
x
We confirmed that the EWSR is satisfied also in the teraction.
case of the other Skyrme parameter set SLy4. For the Figures 7 and 8 show the B(E1) and B(IS2) strength
B(E1)andB(IS2)strengthfunctionsin20O,therunning functions in 20O, respectively, calculated with use of a
sum W(E ) at the highest energy E =55 MeV reaches small smearing width ǫ = 0.2 MeV. They are compared
x x
95% and 97% of the EWSR, respectively. In the LM with those obtained with ǫ = 1.0 MeV (cf. Figs.1 and
approximation, on the contrary, the sum overshoots the 2). The running energy weighted sum of the strength
EWSRby6%and12%intheB(E1)andB(IS2)strength function is also shown. Finer structures of the strength
functions, respectively. functions are visible here: we can distinguish the first
InFig.6,wedemonstrateinfluenceoftheresidualpair- excited2+,whichisabounddiscretestatelocatedbelow
ing interaction on the energy-weighted sum. When the the separation energy.
residual pairing interaction is dropped in the continuum It is seen from the bottom panels of Figs. 7 and 8
QRPA calculation (i.e., the dynamical pairing effect is thatthe agreementwith the energyweightedsumrule is
neglected), the energy-weighted sum overestimates the improved with use of the smaller smearing width. The
EWSR value by 5%. As pointed out already[10, 11, 18], running sums at the highest energy E = 55(50) MeV
x