Deformation properties of the BCP energy density functional

0.22 MB·

by  L. M. Robledo

Deformation properties of the BCP energy density fun
tional
∗
L.M. Robledo
Dep. Físi
a Teóri
a C-XI, Universidad Autónoma de Madrid, 28049 Madrid, Spain
M. Baldo
Instituto Nazionale di Fisi
a Nu
leare, Sezione di Catania, Via Santa So(cid:28)a 64, I-95123 Catania, Italy
P. S
hu
k
Institut de Physique Nu
lèaire, IN2P3-CNRS, UMR8608, F-91406 Orsay, Fran
e and
Université Paris-Sud, F-91406 Orsay, Fran
e
8 X. Viñas
0 Departament d'Estru
tura i Constituents de la Matèria and Institut de Cièn
ies del Cosmos,
0 Universitat de Bar
elona, Diagonal 647, 08028 Bar
elona, Spain
2
We explore the deformation properties of the newly postulated BCP energy density fun
tional
n
(EDF).Theresultsobtainedforthreeisotope
hainsofMg,DyandRaare
omparedtotheavailable
a
exp24e0rimental data as well as to the results of the Gogny-D1S for
e. Results for the (cid:28)ssion barrier
J
of Puare also dis
ussed.
4
2
I. INTRODUCTION the spin-orbit strength
an be found in [1℄. They were
]
h adjusted to reprodu
e the ground-state energy of some
t In a re
ent work [1℄ we have shown that a fully mi- spheri
al nu
lei. For the pairing part of the intera
tion
-
l
ros
opi
input from nu
lear and neutron Equation of we simply take the density dependent delta for
e stud-
c
u State(EOS)
al
ulations[2℄
omplementedbyadditional ied in [3℄ for e(cid:27)e
tive mass equal to the bare one. With
n terms a

ounting for (cid:28)nite size e(cid:27)e
ts, to nu
lear Den- this so-
alled Bar
elona-Catania-Paris(BCP) fun
tional
[ sity Fun
tional Theory (DFT)
an be very su

essfull. ex
ellent results for 161 even-even spheri
al nu
lei with
The density fun
tional for the ground state energy E of rms values for ground state energies and
harge radii,
1
a nu
leus, we
onsidered in [1℄, is of the form
omparablewiththemostperformantfun
tionalson the
v
65 E =T0+Es.o.+Ei∞nt+EiFnRt +EC (1) maIrnkethti[s4,p5a,pe6r, 7w℄e, w
oenretionbuteaiinnveeds.tigating the properties
7 T0,Es.o.,EC oftheBCPfun
tionalandexplorehowdeformationprop-
3 where are the standard expressions for ki- erties of nu
lei are des
ribed. We will (cid:28)nd that the per-
1. neti
energy, spin-orbit term, and Coulomb energEy.∞The forman
e is again ex
ellent, as well in
omparison with
0 dependen
e on proton and neutron densities in int is experiment asin
omparisonwith the resultsof the very
8 the one obtained from the mi
ros
opi

al
ulation [2℄ su

essfull Gogny D1S for
e [4℄. We have
omputed the
0 and treated in Lo
alDensity Approximation(LDA). We PotentialEnergy Surfa
es(PES) asafun
tion oftheax-
v: added aFinite Range (FR) term to a

ountforaproper ially symmetri
quadrupole deformation of several iso-
i des
ription of the nu
lear surfa
e: topesoftheMagnesium,DysprosiumandRadiumspe
ies
rX EiFnRt[ρn,ρp] = 21 Z Z d3rd3r′ρt(r)vt,t′(r−r′)ρt′(r′) wfeirtehntthneui
dleeaarins
meinnadrioofs
oofvetrhiengNdui
(cid:27)leidreenCtrheagrito.nsTaon(cid:28)dndisifh-
a Xt,t′
these exploratory
al
ulations we have
2o40nsidered also
1
− 2Xt,t′ γt,t′Z d3rρt(r)ρt′(r) (2) vtheery(cid:28)ssesniosnitivbeartroieqruoafdrtuhpeohleeadveyfonrmu
alteiuosn proPpuerwtihesi
.h is
t γt,t′
with =proton/neutron and the volume integral of
vt,t′(r). The se
ond term in the r.h.s. of Eq (2) is in- II. METHODS
trodu
edtopreservethenu
learmatterpropertiesofthe
mi
ros
opi

al
ulation in the bulk. For the (cid:28)nite range Asitis
ustomaryinthiskindof
al
ulationstheHFB
vt,t′(r)
tfohramt isfav
tt,ot′r=Vt,t′e−wr2e/rm02.adWeea
hsiomsepalemGinauimssuiamnoafntsharteze, ethqeuaetnioerngyhadsenbseiteynfuren

atsitonaasl wahmerineitmhiezaHtFioBnwparov
eefsusno
-f
V = V = V V = V = V
open parameters: p,p n,n L, n,p p,n U, tion of the Bogoliubov transformation [8℄ is
hosen to
r0
and . The values of these parameters together with minimize the energy. The variational set of HFB wave
fun
tions is given by means of the standard Thouless
parametrization [8℄. The minimization pro
ess is per-
∗ formedbyusingthegradientmethodasitallowsaneasy
Ele
troni
address: luis.robledouam.es and e(cid:30)
ient implementation of
onstraints by using the
2
26Mg 32Mg 38Mg III. RESULTS
V)V)1100..00
ee
MM
E (E ( 55..00 Wehaveperformed
al
ulationsofthepotentialenergy
surfa
e(PES), asafun
tion ofthe massquadrupolemo-
Q20
ment , of several isotopes of the nu
lei Magnesium,
00..00 Dysprosium and Radium. The idea is to explore some
representative nu
lei distributed all over the Nu
lide
1155..00 144Dy 154Dy 164Dy Chart. We have
omparedthe resultsof the
al
ulations
V)V)1100..00 with the BCP energy density fun
tional (parametriza-
ee tionsBCP1andBCP2)withthoseobtainedbyusingthe
MM
E (E ( 55..00 mD1aSrkG. oFgonrythfoerM
egwishoi
thopiess
woenshidaevreed
ohmepreutaesdathbeeenv
ehn-
A = 20 A = 40
mass ones between and (
overing the
00..00 N=8 and N=20neutron shell
losures). For the Dyspro-
A=140
sium isotopes we have
omputed the ones from
1155..00 218Ra 228Ra 236Ra A=170
(N=74)upto (N=104). Finally,fortheRadium
A = 216
V)V)1100..00 isotopes we havAe
=on2s3id6ered isotopes between
MeMe (N=128) up to (N=148). Inβ2Fig. (1) we have
E (E ( 55..00 rpalomtteetder,thdee(cid:28)PnEeSd(aassβa2fu=n
qtio4n5πohQrf22t0ih)eof sodmefeorrmeparteiosnenptaa--
00..00 tivenu
lei. AstheresultsobtainedwithBCP1andBCP2
--00..55 00..00 00..55 --00..55 00..00 00..55 --00..55 00..00 00..55
parametrizationsarealmostidenti
al, only the (cid:28)rst ones
bb bb bb
22 22 22 are shown. The (cid:28)rst noti
eable fa
t is that in all the
nu
leistudied thetwo
urveslookrathersimilarshowing
β2
minima,maximaandsaddlepointsalmostatthesame
Figure1: ThePotentialEnergySurfa
es(PES)ofsomerepre-
sentativenu
leiasafun
tionoftheβ2deformationparameter values. However, the relative energy of those
on(cid:28)gura-
(fullline,BCP1fun
tional;dashedoneGogny-D1Sfor
e). In tions obtained with BCP1 is di(cid:27)erent from the one with
allthe
asesthezeroofthe
urvesisreferredtotheirabsolute the Gogny for
e. A

ording to the number of neutrons
minimum. we
andistinguishtwodi(cid:27)erentregions: the(cid:28)rstone
or-
responds to neutron numbers greater than the mid-shell
value, where there are di(cid:27)eren
es in the prolate side be-
β2
tweenGognyandBCPwhi
hextenduptoratherhigh
values2o6f around 0.5. 3E2xamples of this be1h44avior are the
nu
lei Mg (N=14), Mg (N=20) and Dy (N=78).
te
hnique of Lagrange multipliers. For the present ex-
These
ondregion
orrespondstoneutronnumberslower
ploratory
al
ulations we have restri
ted the
al
ulation
than the mid-shell value where the spheri
al
on(cid:28)gura-
to
on(cid:28)gurations preserving axial symmetry (but re(cid:29)e
-
tion lies, in the Gogny
ase, at an energy higher than
tion symmetry breaking is allowed) in order to redu
e
in the BCP results and the di(cid:27)eren
e in
reases as N in-
the
omputational e(cid:27)ort. The quasiparti
le operators Edef

reases. This means that the deformation energy ,
are expanded in a harmoni
os
illator basis written as
z de(cid:28)ned as the energy di(cid:27)eren
e between the spheri
al
the tensor produ
t of one os
illator in the dire
tion
ω
on(cid:28)guration and the deformed ground state, is larger
z
with os
illator frequen
y and another in the perpen-
ω for the Gogny for
e than for the BCP fun
tional. As
⊥
di
ularoneand
hara
terizedby . Largeenoughbasis
z a
onsequen
e of this behavior the ex
itation energy of
(up to 26 shells in the dire
tion in the
ase of (cid:28)ssion)
the oblate minimum is higher with the Gogny intera
-
and optimizationof the harmoni
os
illatorlengths have
tion than with the BCP fun
tional, rea
hing the di(cid:27)er-
been used to guarantee good
onvergen
e for the values
en
2e28roughly a fa
tor of two, as in the example shown
of all physi
al quantities.
of 154Ra. E16x4ampl2e2s8 of this 2b3e6havior shown in Fig. (1)
are Dy, Dy, Ra and R2a18. Finally, forsomenu-
In the
al
ulations with the Gogny for
e the stan-
lei
lose or at shell
losure like Ra shown in Fig. (1)
dard approximations have been
onsidered, namely the the two PES are rather similar. These systemati
di(cid:27)er-
Coulomb ex
hange
ontribution to the energy has been en
es observed between Gogny and BCP1 are probably
repla
ed by the Slater approximation and the Coulomb a
onsequen
e of the di(cid:27)erent surfa
e energy
oe(cid:30)
ient
a a = 17.74
S S
pairing (cid:28)eld has been negle
ted. The two body kineti
(the values are MeV and 17.84 MeV for
a =18.2
S
energy
orre
tion has been fully taken into a

ount. For BCP1 and BCP2 respe
tively [9℄ and MeV for
detailspertaining theimplementationofthe HFB pro
e- Gogny D1S [5℄). Another possible explanation for the
dure and
enter of mass
orre
tion with the BCP fun
- di(cid:27)eren
es in the ex
itation energy of prolateand oblate
tional see Ref. [1℄ minima
ould be the lower level density obtained with
3
20
theGognyfor
e(e(cid:27)e
tivemassratioof0.7)as
ompared V)15
e
twoitthhethpehoynsei
aolftohnee)BtChPatfumna
kteiosnsahlEe(ldele(cid:27)fgea
ptsivsetrmonasgsereq(useael E (Mdef1005
below). The deformation energy also depends on 1.4
the amountof pairing
orrelations(see, forexample Ref. 1.2
h 1
[10℄) in su
h a way that the stronger the pairing
orrela-
Edef 0.8
tions are the smaller the value of . It turns out that 0.2
E = Tr(∆κ)
pp
the parti
le-parti
le
orrelation energy ,
r0.1
whi
hisameasureofpairing
orrelations,istypi
ally30 d
to 40 % sEtrdoenfger in the Gogny-D1S
ase. However, its 0
e(cid:27)e
t on is not strong enough as to over
ome the 24
other e(cid:27)e
ts mentioned above that lead to an in
rease
22
of the deformationenergy of Gogny D1S with respe
t to
BCP. 20
V)
Con
erning the phy26si
s, we observe in Fig. (1) how Me18
ashpappeear
soienxbisotethn
teheinBCMPganwditDh1iSts
aolb
lualtaetigornosu.nWdestaalstoe S (2N16 BGCOPG2
observe the shoulder pre
ursor of the defor3m2ed ground 14 BCP1
Exp
stateafterangularmomentumproje
tionin Mgaswell
12
a32stheprolategroundstatethatdevelopsin heavierthan
Mfgpisotopes as38a
onsequen
e of neutrons populating 10 140 144 148 152 156 160 164 168 172
the shell(see Mg). Inthe
aseofDywehavetwoex-
A
a1m44ples of shape
oexisten
e1w54ith an oblate ground state
( Dy) and a prolat1e64one ( Dy) as well as a well de-
formed sy14s2tem like Dy. Shape
oexisten
e also ap- Figure 2: Mean (cid:28)eld results for the self
onsistent minimum
pears in Dy (not shown) but in this
ase BCP pre- of all the Dy isotopes
onsidered. In the lower panel the
S2N
di
ts a prolate g.s. whereas Gogny predi
ts an oblate twoneutronseparationenergy isshownagainstthemass
one. Finally in the lowerrow of Fig. (12)18we haveseveral numberof theisotopes alongδrw=ithhrt2hi1e/2ex−peri3m/e5n×ta1l.2da×taA.1I/n3
Raisotopesrangingfromthespheri
al Rashowingan thenextpanelthequantity p η
is depi
ted. In the following panel the axis ratio (see text
ex
ited su2p2e8rdeforme2d36minimum, to the reasonably well
for details) is represented. Finally, in the upper panel the
deformed Ra and Ra where the ex
itation energies Edef
deformation energy is plotted.
ofthe oblateminimaarequiteabit higherin the
aseof
the Gogny for
e than with the BCP fun
tional.
0.6
In Fig (2) we have displayed the magnitude of several r0.4
physi
al quantities
orresponding to the ground state of d 0.2
35
the
omputed Dy isotopes, using the two BCP fun
tion-
30
als and the D1S for
e. In the lower panel of this (cid:28)gure
GOG-D1S
ttyhpei
tawlodnise
uotnrtoinnusietypaarattitohne esneemrig-imesagair
enpulo
ltetueds.148TDhye eV)2205 EBxCpP1
M
(N=82,Z=66) is
learly observed. As usual, our predi
- (N15
tions do not
ompare well with experiment around that S210
point be
ause of the impa
t of missing
orrelations in 5
the binding energies of transitional (not well deformed)
0
systems. In well deformed systems the agreement with
20 24 28 32 36 40
experiment is mu
h better for all three
ases
onsid- A
ered here. The absolute values of the binding energies
also agree well in all the
ases and, as an example, we

an mention that in a
al
ul1a6t0ion with a basis of 15 Figure 3: Same as Fig. (2) butfor the Mg isotopes and only
thetwo lower panels.
shells the binding energies of Dy are -1305.894 MeV
for BCP1, -1305.607 MeV for BCP2, -1304.288 MeV for
D1S whereas the experimental value is -1309.457 MeV.
The theoreti
al results do not in
lude any kind of
or- pareratherwellwith experimentaldataand surprisingly
relation energy beyond mean-(cid:28)eld like the rotationalen- theradiiare
losertoexperimentinwelldeformednu
lei
ergy
orre
tion that
an be estimated to be 2.18 MeV whi
hwerenot
onsidere1d42in theoriginal(cid:28)t [1℄. The dif-
for BCP and 3.16 MeV for D1S. In the next panel of ferentresultobtainedin DyforBCP1andBCP2(and
Fig (2) δarq=uanhrti2tiy1/r2e−lated3/to5×the1.2m×eaAn1s/q3uare radius, D1S) is due to the almost degenerate oblate and prolate
namely , is plotted. minima in this nu
leus: with BCP1 the ground state is
p
We observe that the three theoreti
al predi
tions
om- prolate whereas it is oblate with BCP2 and Gogny-D1S.
4
0.2 150
d r-00..110 GOG-D1S 3/22Q (b), Q (b)341102257055050 QQQQQQ343434
12 BCP1 -1790
V) Exp
e
(M2N10 -1795 240Pu
S
8
Gog-D1S
V) BCP2
216 220 224 A 228 232 236 E (MeHFB-1800 BCP1
Figure 4: Sameas Fig. (3)butfor theRaisotopes. -1805
η = hz2i/hx2i 1/2
Iisnathmeenaesxutrepaonfedletfhoermaxaitsiornat(iβo2 =q(cid:0) 45πηη22−+21)(cid:1)(η >w1hif
ohr -1810-10 0 10 20 30 40 50 60 7Q02 (b8)0 90 100 110 120 130 140 150 160
η < 1 240
prolate deformation, for oblate deformations and
η = 1 Figure 5: Fission properties for the nu
leus Pu
omputed
for spheri
al states) is plotted. As in previous
with the two parametrizations of the BCP fun
tional as well

ases, the agreement between the three theoreti
al re- aswiththeGogny-D1Sfor
e. Inthelowerpanelthepotential
sultsisverygood
on(cid:28)rmingwhatwassaidindis
ussing energy surfa
es are depi
ted as a fun
tion of the quadrupole
Figure (1) about the
oin
iden
e of the position of max- moment. In the upper panel the o
tupole and hexade
apole
ima and minima of the PES. Finally in the upper panel momentsare presented.
Edef
the deformation energy is shown. The Gogny-D1S
values for this magnitude are systemati
ally larger by a
240
few MeV than the BCP ones as dis
ussed previously.
spondingtothe(cid:28)ssionpro
essofthenu
leus Pu
om-
The results for the self
onsistent ground state in the
puted using the two BCP fun
tionals and the Gogny
Mg isotopes is depi
ted in Fig. (3) where we have only
S2N δr for
e. As it is
ustomary in this kind of
al
ulations
plotted the
urves for and . The two additional
[11℄ the rotational energy
orre
tion
omputed with the
quantities shown in the
ase of the Dy isotopes have a
Yo

oz moment of inertiahas been in
luded in the PES.
similarbehaviorforthe Mg nu
lei and arenot presented
S2N It
an be
on
luded from this (cid:28)gure (lower panel) that
here. Again, we observe for the a reasonably good
the PES obtained with the BCP1 and BCP2 fun
tion-
agreement with experiment whi
h is of the same qual-
als are very similar and both
losely follow the shape
ity for the three s
hemes. The theoreti
al radii alsolook
of the Gogny D1S PES up to the se
ond minimum. The
rather similar in all the theoreti
al approa
hesand
om-
heightsofthe(cid:28)ssionbarriersprovidedbyboththeBCP1
pare well with the s
ar
e experimental values.
and BCP2 fun
tionals are lower than the one
al
ulated
S2INn Fig. (4) we show the self
onsistent results for the withD1S,asitisexpe
tedfromthelowersurfa
eenergy
220−22o6f the Ra isotopes. The results for the isotopes valuesofBCP reportedabove. The smallervaluesofthe
Ra iβn3
lude o
tupole deformation in their ground (cid:28)ssion barriers for the BCP fun
tional results go in the
state with values of the order of 0.15 for all the
ases right dire
tion as
ompared to the experimental estima-

onsidered. It is not the s
ope of the present work to tions [12, 13℄ but the e(cid:27)e
t of triaxiality in the (cid:28)rst bar-
dis
uss o
tupole deformation in detail but it is worth rierinthe
aseoftheBCPfun
tionalremainstobestud-
pointing out that also this deformation multipole
omes 2ie4d0. Theimpa
tinthespontaneous(cid:28)ssionhalflifetSF of
out quiteS2tNhe same independent of the for
e. The the- Puisalsoun
ertainasthis quantitynotonlydepends
oreti
al 218 results look rather similar ex
ept fo21r4the on the topology of the PES but also on the
olle
tive
nu
leus Ra whi
h is
lose to the semi-magi
Ra. inertia that has not been
onsidered here. Preliminary
Atiosnitenweragsiedsish
auvseseadsobmefeowreh,attheerratwtio
bneehuatvroionr saerpoaurnad- i
nagl
tuoloathioignh1s.vn1ao×ltu1ein0s
2f7olurdtihneghtarlifalxiifae)lliytiye1l.(d5th×tSeFr1e0=f2o6r1e.2p×ro1d0u28
s-
semi-magi

on(cid:28)gurations. Most surprising is the re- forBCP1, s forBCP2 and s forD1S.
gionaroundA=226-228wherethetheoreti
alpredi
tions Therefore it seems that the larger values of the
olle
-
move away from experimental values. Con
erning the tive inertia obtained forthe BCP fun
tionals (
onsistent
radiiagoodagreementbetweentheresultsobtainedwith with their lower pairing
orrelations) somehow
ounter-
BCP, Gogny and the experiment is observed. a
ttheirlower(cid:28)ssionbarriersprovidingalongerhalflife.
In Figure (5) we display the potential energy surfa
e Although the shapes involved in (cid:28)ssion are mainly
har-
(PES) as a fun
tion of the quadrupole moment
orre- a
terizedbyitsquadrupolemoment,highermultipolede-
5
5 hip1913///222 pf53/2/2 h9/2 fi173/2/2 5 ddj1535///222 sgi1171///222 dd53//22 gs19//22 dthhii(cid:27)gehenreeurn
ltleevlueesvle1dl6e0dnDesnyitsyitthiineesstwihneeglfheoarpvmeaerprtlio

ltaetseelde.viTenlosFialilgtuussrtperha(et6re)i
tfihotyer
0 f7/2 0 g9/2 (Q20 = 0) for BCP1 and Gogny-D1S both for protons
and neutrons. The higher level density of BCP's single
eV) -5 dsh151//122/ 2 gd73//22 gdh731//122/2 ds15//22 eV) -5 ppf531/2//22 fhi1793/2//22 hfp591/2//22 fpi1733/2//22 nBpoaCtrPtii1n
l.
eluledveedl,ssisinI

Vlee.atrhlCeyyOoabNrseeCrvLveeUrdyS.IsROimeNsiulSaltrstfoorthBeCoPn2esaroef
M-10 M-10
e (s.p.-15 pp31//22 fg59/2/2 g9/2 e (s.p.-15 dsh151//122/ 2 gd73//22 dh31/12/2 s1/2 t
hereWnpseeqhrfuaoavrdemraaunnpa
oleylezoedfdethfboeyrmnmeawetaionBnsC.oPfTafhufenew
tteissotensleap
lteienrdfowerhxmaaetmd
poalneres-
quite demanding as they in
lude (cid:28)ssion barriers for Pu
f7/2 p1/2 p3/2 g9/2 g7/2 d5/2 ovarrqiouuasdrruegpioolnesporfoptheertNieus
olvideeraChwaidrte.rWanegheaovfeis
ootmoppeasreind
-20 -20
s1/2 d3/2 f5/2 pp31//22 f5/2 tmheentraelsudltastaofwohuernemveeranpo(cid:28)seslidble
aal
nudlaftoiounnsdwgiotohdeaxgpreerei--
BCP GOG g9/2 ment
omparabletotheoneobtainedforspheri
almagi
-25 d5/2 f7/2 -25 BCP GOG or semimagi
nu
lei. Other quantities not dire
tly re-
lated to experiment like the topology of the potential
energy surfa
es have been
ompared to the results of a
Figure 6: Spheri
al single parti
le energies for pr1o60tons (left
wellperformantfor
eandwithlongtraditioninthe(cid:28)eld,
panel) and neutrons(rightpanel) for thenu
leus Dy. The
results for the BCP1 fun
tional and the Gogny for
e (GOG) namely the Gogny-D1S for
e and the results are
ompa-
are given. To help identify the levels their labels alternate rable and of the same quality. Minor di(cid:27)eren
es
an be
their position with respe
t to theline assigned to them attributed to a slightly lower surfa
e tension for BCP

ompared to D1S and to the di(cid:27)erent pairing intera
-
tions used. From these exploratory
al
ulations we
an
formations
anbeimportantfordes
ribingsome(cid:28)ne de-
on
lude that the BCP fun
tional with the BCP1 and
tails of the nu
lear dynami
s. In the top panel of Figure BCP2 parametrization
an be used with
on(cid:28)den
e in
(5) the expe
tation values of the o
tupole and hexade- the study of nu
lear properties related to deformation.
Q20

apole moments are displayed as a fun
tion of and Obviously amore thorough study of the BCP fun
tional
striking similarity between the three results is observed. with respe
t to deformationhas to be
arriedout as, e.g
The similarity between the results obtained with the the study of o
tupole deformation, triaxiallity,
olle
tive
BCPfun
tionalsandtheGognyfor
e
allsfora
ompari- inertia, in
luding the moment of inertia, et
. Work in
sonoftheunderlyingsingleparti
lestru
turethat,asitis this dire
tion is underway.
wellknown,isessentialindeterminingtheresponseofthe
system to deformation. The most thorough
omparison
would involve the analysis of the single parti
le energy
A
knowledgments
plots as a fun
tion of quadrupole deformation (Nilsson
diagrams) but this is a demanding task that is deferred
to a longer publi
ation. We just want to mention here ThisworkhasbeenpartiallysupportedbytheCICyT-
that the most relevant feature of the Nilsson diagrams, IN2P3 and CICyT-INFN
ollaborations. L.M.R. a
-
namely the shell gaps determining the lo
ation of min- knowledges (cid:28)nan
ial support from the DGI of the MEC
ima are rather similar for both the BCP fun
tionals and (Spain) under Proje
t FIS2004-06697. X.V. a
knowl-
the Gogny for
e. Tmhi∗s/ims so in spite of the di(cid:27)erent ef- edges support from grants FIS2005-03142 of the MEC
fe
tive mass ratios in the two
ases, one for the (Spain) and FEDER and 2005SGR-00343of Generalitat
BCP fun
tionals and 0.7 for Gogny-D1S, that imply a de Catalunya.
[1℄ M. Baldo, P. S
hu
kand X. Viñas, nu
l-th/0706.0658v2 [5℄ J.F. Berger, M. Girod and D.Gogny, Nu
l. Phys. A502,
[2℄ M. Baldo, C. Maieron, P. S
hu
k and X. Viñas, Nu
l. 85
(1989).
Phys.A736 (2004) 241 and referen
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