Spin-isospin Response in Finite Nuclei from an Extended Skyrme Interaction

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by  Peiwei Wen

Spin-isospin Response in Finite Nuclei from an Extended Skyrme Interaction
Peiwei Wen1,2, Li-Gang Cao1,3,4,5, J. Margueron6, H. Sagawa7,8
1Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
2 University of Chinese Academy of Sciences, Beijing 000049, China
3 State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China
4 Kavli Institute for Theoretical Physics China, CAS, Beijing 100190, China
5 Center of Theoretical Nuclear Physics, National Laboratory of
Heavy Ion Accelerator of Lanzhou, Lanzhou 730000, P.R. China
6 Universit´e de Lyon, Universit´e Lyon 1 and CNRS/IN2P3,
Institut de Physique Nucl´eaire de Lyon, 4 rue Enrico Fermi, 69622 Villeurbanne, France
7Center for Mathematics and Physics, University of Aizu,
Aizu-Wakamatsu, Fukushima 965-8580, Japan and
8RIKEN, Nishina Center, Wako, 351-0198 ,Japan
4
1 Themagneticdipole(M1)andtheGamow-Teller(GT)excitationsoffinitenucleihavebeenstud-
0
ied in a fully self-consistent Hartree-Fock (HF) plus random phase approximation (RPA) approach
2
by using a Skyrme energy density functional with spin and spin-isospin densities. To this end, we

n adopt the extended SLy5st interaction which includes spin-density dependent terms and stabilize
a nuclear matter with respect to spin instabilities. The effect of the spin-density dependent terms is
J examined in both themean field and thespin-flip excited state calculations. The numerical results

9 showthatthosetermsgiveappreciablerepulsivecontributionstotheM1andGTresponsefunctions
of finitenuclei.

]
h PACSnumbers: 21.60.Jz,24.30.Cz,24.30.Gd,25.40.Kv
t
-
l
c I. INTRODUCTION 1981, the Skyrme SGII interaction was design to give,
u
for the first time, a detailed description of the GT
n
[ The properties of spin asymmetric matter are still data [13]. Some other Skyrme interactions, such as
very difficult to access experimentally since the ground SLy230a & SLy230b [14], SLy4 & 5 [15], SkO [16], and
1
more recently SAMi [17], have been determined with a
v states of nuclei have a weak or an almost zero spin-
specialcareofthespinandspin-isospinpropertiesofnu-
1 polarization: even-even spherical nuclei are not spin-
8 polarized, while their closest odd nuclei can be weakly clear matter and nuclei. Calculations of GT within the
9 spin-polarized by the last unpaired nucleon, but with relativistic frameworkweredone morerecently [18, 19].
1 The relation between the spin or the spin-isospin exci-
a moderate impact on the ground-state energy [1]. In
. tations and the central part of the nuclear interaction
1 well deformed nuclei, ground-state spin and parity as-
0 signments are still difficult to predict globally [2]. It is is however not a one-to-one relation and other effects
4 therefore difficult to probe the nuclear interaction in should be considered such as the spin-orbit splitting of
1 the single-particle states and the residual spin-orbit in-
spin and spin-isospin channels from the ground-state
: teraction in the RPA calculations [20].
v properties of nuclei. However, in the excitation spectra
i of nuclei, some collective modes can provide a unique Recently an extension of the Skyrme interaction, in-
X
opportunity to explore the nuclear interactions in spin cluding spin-density and spin-isospin density depen-
r
a and spin-isospin channels [3–5]. The M1 and GT exci- dent terms, was proposed by some of the present au-
tations are the most common collective modes of spin thors [1, 21]. At variance with predictions in nuclear
andspin-isospintypesinnuclei. Thesemodeshavebeen matter of ab initio methods based on realistic bare in-
extensivelystudiedduringthelastdecadeandmuchin- teractions [22–25], most of the standard Skyrme inter-
formation on spin and spin-isospin excitations becomes actions predict spin or spin-isospin instabilities beyond
now available [5, 6]. They are of interest not only in the saturation density of nuclear matter [26]. The ad-
nuclear physics but also in astrophysics. They play, for ditionalparametersofthe extendedSkyrme interaction
instance, an important role in predicting β decay half- were therefore adjusted to reproduce the results given
lives of neutron rich nuclei involved in the r-process of by microscopic G-matrix calculations better. The ex-
the nucleosynthesis [7]. In core-collapse supernova, the tension of the Skyrme interaction was designed to keep
GT transitions ofpf-shell nuclei give animportant con- the simplicity of the standard Skyrme interaction and
tribution to the weak interaction decay rates that play to remove the ferromagnetic instability or to shift it to
an essential role in the core-collapse dynamics of mas- larger density.
sive stars [8–10]. The neutrino-induced nucleosynthesis The extended spin-density dependent terms can im-
may take place via GT processes in neutron-rich envi- provethepropertiesofthe Skyrmeenergydensityfunc-
ronment[11]. For neutrino physics and double β decay, tional in spin and spin-isospin channels by adding the
accurate GT matrix elements are necessary to under- weak repulsive effect. For example, the dimensionless
stand the nature of neutrinos [12]. Landau parameter G′ is increased by about 0.3 for
0
In the beginning of the 1980s,GT experiments made three interactions SLy5 [15], LNS [27] and BSk16[28].
great progress when the (p,n) facility at the Indiana In Refs. [1, 21], the authors explored the effect of spin-
University Cyclotron Facility became operational. In density dependent terms on the response functions and
2
the mean free path of neutrinos in nuclear matter as where ρ = ρ − ρ is the spin density and ρ =
s ↑ ↓ st
well as the ground state properties of finite odd nuclei. ρ −ρ −ρ +ρ is the spin-isospin density. The
n↑ n↓ p↑ p↓
The model proposed in Ref. [1, 21] was constrained by spin symmetry is satisfied if the power of the density
microscopicG-matrix predictions in uniform matter. It dependent terms γ and γ are both even integers.
s st
will be quite interesting to investigate the effect of the In the following study, the spin-density dependent
proposed extension of the Skyrme interaction for the terms (2) are added to the original Hamiltonian. Then
spin and spin-isospin excitations of finite nuclei. In the the density dependent part of the Skyrme energy den-
presentwork,we study the contributionof spin-density sity functional,
dependenttermstothe M1andGTexcitationsinfinite
nuclei 90Zr and 208Pb with a fully self-consistent HF H = t3ρα 3ρ2+(2x −1)ρ2−(2x +1)ρ2−ρ2 ,(3)
plusRPAframework[29]. TheSLy5Skyrmeparameter 3 48 3 s 3 t st
(cid:2) (cid:3)
set is employed in our calculations by adding the spin-
has the extra density dependent terms Hs and Hst
density dependent terms. The new parametrization is 3 3
which read,
calledasSLy5st,whichisthesameusedinRefs.[1,21].
In present study we switch on and off the spin-density ts
dependenttermsingroundstatesandexcitedstatescal- Hs = 3ργs 3ρ2+(2xs−1)ρ2−(2xs+1)ρ2−ρ2 ,(4)
3 48 s 3 s 3 t st
culationstoseehowmuchtheyaffectthespinandspin- (cid:2) (cid:3)
tst
isospin response functions in finite nuclei. Hst = 3 ργst 3ρ2+(2xst−1)ρ2−(2xst+1)ρ2−ρ2 ,(5)
3 48 st 3 s 3 t st
This paper is organized as follows. In Sec. II we (cid:2) (cid:3)
willbrieflyreportthetheoreticalframeworkoftheRPA
whereρ =ρ −ρ . The meanfieldpotentialU , where
t n p q
basedonthe Skyrmeinteractionandits extension. The
q =n,p, gets additional terms
results and discussion are presented in Sec. III. Sec-
tion IV is devoted to the summary and perspective for ts
Uadd. = 3ργs (2+xs)ρ−(1+2xs)ρ
future. q 12 s 3 3 q
(cid:2) (cid:3)
tst
+ 3 ργst (2+xst)ρ−(1+2xst)ρ . (6)
12 st 3 3 q
II. FORMULA (cid:2) (cid:3)
In symmetric nuclear matter the Landau parame-
We adopt the standard form of Skyrme interaction ters [30] are also modified by the following additional
withthenotationsofRef.[15]. Thetwonucleonsarein- terms
teracting through a zero-range,velocity-dependent and
Fadd. ts tst
density-dependent Skyrme interaction with space, spin 0 = 3ργs + 3 ργst , (7)
and isospin variables r , σ and τ which reads [15]: N0 8 s 8 st
i i i
F′add. ts tst
V(r1,r2) = t0(1+x0Pσ)δ(r) 0N0 = −234(2xs3+1)ργss − 234(2xs3t+1)ργstst ,(8)
+ 21t1(1+x1Pσ)[P′2δ(r)+δ(r)P2] GNa0dd. = 4ts38γs(γs−1)[3ρ2−(2xs3+1)ρ2t
0
+ t2(1+x2Pσ)P′·δ(r)P tst 1
+ 1t (1+x P )ρα(R)δ(r) − ρ2st]ρsγs−2+ 132(xs3t− 2)ργstst
3 3 σ
6 ts 1
+ iW0(σ1+σ2)·[P′×δ(r)P] , (1) + 234(xs3− 2)(γs+1)(γs+2)ργss , (9)
G′add. tst
wPh′eirsethre=herr1m−itria2n, Rcon=ju12g(art1e+ofrP2),(Pact=ing21io(∇n t1h−e l∇ef2t)),, N00 = 438γst(γst−1)[3ρ2+(cid:0)2xs3t−1(cid:1)ρ2s
ρPσ==ρ21+(1ρ+isσt1h·eσto2)tailsntuhcelesopnind-eenxscihtay.ngWeiotphienratthoer,staannd- − (2xs3t+1)ρ2t]ργstst−2− 2ts34ργss
n p
dard formalism, the total binding energy of a nucleus tst
− 3 (γ +2)(γ +1)ργst . (10)
can be expressed as the integral of a Skyrme density 48 st st st
functional [15], which includes the kinetic-energy term
K, a zero-range term H , the density-dependent term It was mentioned in [1, 21] that the spin-density de-
0
H ,aneffective-masstermH ,afinite-rangemomen- pendent terms may lead very important effects on the
3 eff
tum dependent term H , a spin-orbit term H , a spin and the spin-isospin properties of finite nuclei and
fin so
spin-gradient term H , and a Coulomb term H . nuclear matter. That is, the dimensionless Landau pa-
sg Coul
rameter G′ is increased by about 0.3 for three inter-
TheSkyrmeinteractionhasbeenextendedtoinclude 0
actions SLy5, LNS and BSk16. The improved Skyrme
spin-density dependent terms which can improve the
energydensity functionalin spinandspin-isospinchan-
properties of the energy density functional in the spin
nels will give also substantial contributions to the spin
and spin-isospin channels [21]. That is
and spin-isospin excitations in finite nuclei, such as M1
1 and GT excitations. In present work, we will study
Vadd.(r ,r ) = ts(1+xsP )[ρ (R)]γsδ(r)
1 2 6 3 3 σ s the effect of spin-density dependent terms on the spin-
1 dependent M1 and GT excitations in finite nuclei 90Zr
+ 6ts3t(1+xs3tPσ)[ρst(R)]γstδ(r),(2) and 208Pb.
3
ThecalculationsaredonewithintheSkyrmeHFplus coordinater andtheir detailedexpressionsaregivenby
RPA. The well known RPA method [31, 32] in matrix
ts tst
form is given by vqq(r) = − 3(xs−1)ργs − 3 (xst−1)ργst
0 12 3 s 12 3 st
vqq′(r) = ts3(xs +2)ργs + ts3t(xst+2)ργst
0 12 3 s 12 3 st
A B Xν 1 0 Xν ts
=E , (11) vqq(r) = 3 γ (γ −1)ργs−2 3ρ2−(2xs+1)ρ2−ρ2
(cid:18)B∗ A∗ (cid:19)(cid:18) Yν (cid:19) ν(cid:18)0 −1(cid:19)(cid:18) Yν (cid:19) σ 48(cid:20) s s s (cid:18) 3 t st(cid:19)
+ργs (γ +1)(γ +2)(2xs−1)−2
s (cid:18) s s 3 (cid:19)(cid:21)
tst
wYhνearreeEthνeiscotrhreesepnoenrdgyingoffothrweaνr-dthanRdPbAacsktwataerdanadmXplνi-, +438(cid:20)γst(γst−1)ργstst−2(cid:18)3ρ2+(2xs3t−1)ρ2s
tudes, respectively. The matrix elements A and B are
−(2xst+1)ρ2
expressed as 3 t(cid:19)
+ργst −(γ +1)(γ +2)+2(2xst−1) ,
st (cid:18) st st 3 (cid:19)(cid:21)
Ami,nj =(ǫm−ǫn)δmnδij +hmj|Vres|ini, (12) vσqq′(r) = 4ts38(cid:20)γs(γs−1)ρsγs−2(cid:18)3ρ2−(2xs3+1)ρ2t −ρ2st(cid:19)
+ργs (γ +1)(γ +2)(2xs−1)+2
s (cid:18) s s 3 (cid:19)(cid:21)
tst
+ 3 −γ (γ −1)ργst−2 3ρ2+(2xst−1)ρ2
48(cid:20) st st st (cid:18) 3 s
B =hmn|V |iji. (13) −(2xst+1)ρ2
mi,nj res 3 t(cid:19)
+ργst (γ +1)(γ +2)+2(xst−1) (16)
st (cid:18) st st 3 (cid:19)(cid:21)
We will use the following operator for M1 excitation,
The p-h matrix elements are obtained from the
SinkyErqms.ee(n3e)∼rg(y5d).enTsihteyfeuxnpclitcioitnafolrimncsluodfinthgealmltahteritceersmAs FˆM1 = A {gis−→si +gil−→l }, (17)
andB aregiveninRef.[29]inthe caseofSkyrmeforce. Xi=1
In general, the expression of the residual interaction is where the spin g−factors are gs = 5.586 for protons
derivedfromthesecondderivativeoftheenergydensity and gs = −3.826 for neutrons, respectively, and the
withrespecttothedensityρst withthespinabdispspin orbitalg−factorsaregl =1.0forprotonsandgl=0.0for
indices,
neutrons, respectively, in unit of the nuclear magneton
µ = e~/2mc. We will also study the charge-exchange
N
GT excitations. The GT external operator reads
δ2H A
Vres =sXts′t′ δρstδρs′t′, (14) FˆGT± =Xi=1−→σ(i)t±(i). (18)
III. RESULTS AND DISCUSSIONS
whereH istheHFenergydensityfunctional. According
to Eq. (14), the antisymmetrized particle-hole interac- In Table I we show the parametersused inthis study
tion induced by the spin-density dependent terms (2) and the Landau parameters G and G′ calculated with
0 0
are expressed as, the corresponding Skyrme interactions. To keep the
spinsymmetry,wesetγ andγ equalto2. Thevalues
s st
for the other parameters ts, tst, xs and xst are fixed
3 3 3 3
by an optimal fit of the BHF results in spin and spin-
Vqq = vqqδ(~r −~r )+vqqδ(~r −~r )σ ·σ , isospinchannelsinahigherdensityregionthanthenor-
res 0 1 2 σ 1 2 1 2 mal density. The ground state properties of nuclei 90Zr
Vrqeqs′ = v0qq′δ(~r1−~r2)+vσqq′δ(~r1−~r2)σ1·σ2,(15) and208Pbarecalculatedinthe coordinatespacewith a
boxapproximation. Theradiusoftheboxistakentobe
20fm in whichthe continuumis discretizedin the large
box. The 8 oscillator shell is included as the particle
wherethefunctionsv andv dependonlyontheradial states to build the RPA model space. All calculations
0 σ
4
(208Pb). Including the spin-density dependent terms,
TABLE I: Parameters of the spin-density dependent terms
ts3 (in MeV.fm3γs−2), ts3t (in MeV.fm3γst−2), xs3 and xs3t for the centroid energies of the low-lying and high-lying
strengthsbecome5.53MeV(10.92MeV)and16.68MeV
the interactions SLy5st. We also show the dimensionless
Landau parameters G0 and G′0 deduced from the original (19.07 MeV) for 90Zr (208Pb). The energy shift is 0.3
MeV (1.05 MeV) for the low-lying and 0.42 MeV (0.94
SLy5and SLy5st,respectively.
MeV) for the high-lying states in 90Zr (208Pb). The
ts3 ts3t xs3 xs3t γs γst G0 G′0 energy shift given by the Skyrme HF plus RPA calcu-
SLy5 - - - - - - 1.12 −0.14 lations is qualitatively the same as those estimated by
SLy5st 0.6×104 2×104 -3 0 2 2 1.19 0.15 the semi-classicalSteinwedel-Jensenmodel for 208Pb in
Ref. [21]. The upwardshift of the centroidenergies can
be understood as follows: the spin-density dependent
Exp. terms give an strong repulsive contribution to the ma-
SLy5 trix elements of RPA for the GT calculations because
10
1) SLy5st the residual interactions or the Landau parameter G′0
-V 90Zr changes to be more positive from −0.14 to 0.15 when
e
M the spin-density dependent terms are included. It also
(T- 5 can been seen that the RPA collective state located at
G 19.07 MeV with the spin-density dependent terms in
R
208Pb is very close to the experimental GT excitation
energy of 19.2 ± 0.2 MeV [33, 34]. For 90Zr, the cal-
culated values with or without the contribution of the
0
spin-density dependent terms both are larger than the
0 5 10 15 20 25
E (MeV) experimental value of 15.60 MeV [35].

50
Exp. 10
40 SLy5
SLy5
) SLy5st
1 8 SLy5st
-eV 30 208Pb -1V) 90Zr
M e
R (GT-20 2( MµN46 2 ()µN1
10 M1 BM
R
2
0
0 5 10 15 20 25
0
E (MeV)
6 8 10 12 14
FIG.1: (Coloronline)RPAResponsefunctionsof90Zr(up- E (M eV)
per panel) and 208Pb (lower panel) for GT excitations cal- 25
SLy5
culatedbytheSkyrmeHFplusRPAapproachbasedonthe
SLy5st
SLy5 interaction. Solid (Dotted) line is the result given by 20
including (excluding) the spin-density dependent terms. A 208Pb
)
1
Lorentzian smearing parameter equals 1 MeV. The experi- -V 15 )
mental responses from Ref. [34, 35] are shown by thedots. e 2µN
M (
2µN10 M1
( B
1
areperformedwithintheSLy5parametersetbyinclud- M
R 5
ing or excluding the spin-density dependent terms.
InFig.1wedisplaythe responsefunctions for GTex-
citation in 90Zr and 208Pb calculated with and with- 0
0 5 10 15
out the contribution of the spin-density dependent E (MeV)
terms (2). The solid(dotted) line representsthe results
FIG.2: (Coloronline)RPAResponsefunctionsof90Zr(up-
including (excluding) the spin-density dependent terms per panel) and 208Pb (lower panel) for M1 excitations cal-
both at the HF and RPA calculations. The dots show
culatedbytheSkyrmeHFplusRPAapproachbasedonthe
the corresponding experimental GT response. As one
SLy5 interaction. Solid (Dotted) line is the result given by
cansee fromFig.1, the inclusion of the spin-density de- including ( excluding) the spin-density dependent terms. A
pendent terms tends to slightly increase the high-lying Lorentzian smearing parameter equals 1 MeV. The exper-
strength on the one hand and to decrease the low-lying imental B(M1) values from Ref. [36–41] are shown by the
strength on the other hand. The excitation energies bars.
are shifted up in energy both the low-lying and high-
lying strengths by the spin-dependent terms. Without We alsoinvestigatetheeffectofthe extendedSkyrme
the spin-density dependent terms, the centroid ener- interaction on the M1 excitation by the RPA calcula-
gies of the low-lying and high-lying strengths are 5.23 tions for 90Zr and 208Pb. The results are shown in Fig.
MeV (9.87 MeV) and 16.26 MeV (18.13 MeV) for 90Zr 2. The spin-density dependent terms are included or
5
excluded in the calculations to clarify their influence. tions. The calculations are carried out with the SLy5
For the RPA results of 208Pb, the proton 1h → Skyrme interaction and the extended Skyrme interac-
11/2
1h configuration contributes mainly to the lower en- tion SLy5st,in which the spin-density dependent terms
9/2
ergy peak of the response function, and the neutron areaddedtotheSLy5parametersettomimic the BHF
1i → 1i configuration plays the main role in the results in spin and spin-isospin channels. Those terms
13/2 11/2
higher energy peak. For 90Zr, the M1 response func- are switched on and off in both the HF and the RPA
tion mainly comes from the neutron configuration of calculations in this study. The inclusion of spin-density
1g → 1g . The results show that the inclusion of dependent terms is known to give no contribution to
9/2 7/2
the spin-density dependent terms increasesthe energies the ground state of even-even nuclei, while the residual
ofM1statesbothin90Zrand208Pb. Thepeakenergies interactionsfromthespin-densitydependenttermsgive
of M1 excitation in 208Pb are 7.6 MeV (7.4 MeV) for substantial repulsive effect and shifts the M1 and GT
the lower one and 9.3 MeV (9.1 MeV) for the higher response function of finite nuclei to higher energy.
one with including (excluding) the spin-density depen- The main conclusion we can draw from the present
dent terms. The peak energy of the M1 excitation in study is that the spin and spin-isospin response func-
90Zr are 9.4 MeV (9.1 MeV) by including (excluding) tions can be changed without altering the ground-state
the spin-density dependent terms. The energy shift is properties. Since the parameters related to the spin-
lessthan0.3MeVforboth90Zrand208Pb. Theeffectof density dependent terms are introduced to the existing
the spin-dependent terms is predicted to be smaller on Skyrme interaction, it is marginal whether the new in-
the distribution of the M1 response function compared teractionimprovestheagreementwiththeexperimental
withthatontheGTexcitation. Thiscanbeunderstood data of spin dependent excitations or not. The better
bythechangeoftheLandauparameterswhenthespin- strategy could be to perform a global fitting of the pa-
density dependent terms are included. The difference rameters of the spin-independent and spin-dependent
of G which contributes to the M1 excitation is about density terms on the same foot. Recently, the effect of
0
0.07 (from 1.12 to 1.19), while the change of G′ which tensor force on the various response of nuclear systems
0
plays the dominant role in GT excitation is about 0.3 has been studied extensively [42–44] and the important
(from -0.14 to 0.15). In Fig. 2 we show also the exper- contributions to the spin and spin-isospin response in
imental data of the M1 excitations for 90Zr and 208Pb. nuclear matter and finite nuclei are pointed out. It is
The experimental data for the M1 excitations in 208Pb a future challenge to include both the spin dependent
arefoundatEx=5.85MeVforlow-lyingcomponentand terms and the tensor force in the parameter fit proce-
between Ex=7.1 and 8.7 MeV for the high-lying com- dure. This study will be discussed in the forthcoming
ponent[36–38],whilein90ZrtheM1strengthsexistbe- paper.
tween Ex= 9.0 and 9.53 MeV [39–41]. We can see that
thepresenttheoreticalresults,takingornottakinginto
account the contribution of the spin-density dependent ACKNOWLEDGEMENTS
terms, slightly overestimate the experimental data in
energy. This work is supported by the National Natural Sci-
ence Foundation of China under Grant Nos 10875150
and 11175216, and is supported in part by the Project
IV. SUMMARY AND PERSPECTIVE
of Knowledge Innovation Program (PKIP) of Chinese
Academy of Sciences, Grant No. KJCX2-EW-N01.
In summary, we have studied the effect of the spin- This work is partially supported by the Japanese Min-
density dependent terms of the Skyrme energy den- istry of Education, Culture, Sports, Science and Tech-
sity functional on the M1 and GT giant excitations in nologybyGrant-in-AidforScientificResearchunderthe
90Zr and 208Pb by the Skyrme HF plus RPA calcula- program number (C(2))20540277.
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