Spin-flip M1 giant resonance as a challenge for Skyrme forces

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by  V. O. Nesterenko

Spin-flip M1 giant resonance as a challenge for Skyrme forces
V.O. Nesterenko1, J. Kvasil2, P. Vesely2,3, W. Kleinig1,4, P.-G. Reinhard5, and V.Yu. Ponomarev 6
1 BLTP, Joint Institute for Nuclear Research, 141980,
Dubna, Moscow region, Russia; [email protected]
2 Institute of Particle and Nuclear Physics, Charles University, CZ-18000, Praha, Czech Republic;
[email protected]ff.cuni.cz; [email protected]ff.cuni.cz
3 Department of Physics, P.O. Box 35 (YFL), University of Jyvaskyla, 40014, Jyvaskyla, Finland
4 Technische Universita¨t Dresden, Inst. fu¨r Analysis, D-01062, Dresden, Germany
5 Institut fu¨r Theoretische Physik II, Universita¨t Erlangen, D-91058, Erlangen, Germany and
6 Institut fu¨r Kernphysik, Technische Universita¨t Darmstadt, D-64289, Darmstadt, Germany
(Dated: January 22, 2010)
Despite a great success of the Skyrme mean-field approach in exploration of nuclear dynamics,
0 it seems to fail in description of the spin-flip M1 giant resonance. The results for different Skyrme
1 parameterizations are contradictory and poorly agree with experiment. In particular, there is no
0 parameterization which simultaneously describes the one-peak gross structure of M1 strength in
2 doublymagicnucleiandtwo-peakstructureinheavydeformednuclei. Thereasonofthismismatch
could lie in an unsatisfactory treatment of spin correlations and spin-orbit interaction. We discuss
n
the present status of the problem and possible ways of its solution. In particular, we inspect i)
a
J the interplay of the collective shift and spin-orbit splitting, ii) the isovector M1 response versus
isospin-mixed responses, and iii) the role of tensor and isovector spin-orbit interaction.
2
2

I. INTRODUCTION of the spin-flip M1GR within the Skyrme-Hartree-Fock
]
h (SHF) approach [16–18]. To the best of our knowledge,
t there are only few early SHF studies of this resonance
- The spin-flip M1 giant resonance (M1GR) was a sub-
l [19, 20] but even they are not fully consistent. Indeed,
c jectofintensivetheoreticalandexperimentalstudiesdur-
u the study [19] exploits a hybrid model with a partial
ing the past decades [1–3]. The resonance is known to
n implementation of SHF in the Landau-Migdal formula-
be amajorsourceofknowledgeonspincorrelations. Be-
[ tion while the work [20] uses early Skyrme forces and,
sides, it strongly depends on the spin-orbit splitting and
what is crucial, omits the important spin-density corre-
1 socanserveasarobusttestofthespin-orbitinteraction.
v lations. Only recently the first fully self-consistent sys-
The M1GR was widely explored within various empiri-
5 tematicSHFinvestigationofthespin-flipM1GRwasper-
cal microscopic models, see e.g. [4–7], which allowed to
6 formed[15]. Thecalculationsweredonewithinthesepa-
clarify its main features. Meanwhile, the nuclear density
9 rableRandom-Phase-Approximation(SRPA)model[22–
3 functionaltheory(DFT)hasbeendeveloped. Itprovided
24] extended to magnetic excitations [15, 25]. The reso-
. elaborate self-consistent methods (Skyrme, Gogny, rela-
1 nance was explored in spherical nuclei 48Ca and 208Pb
tivistic) with high descriptive power [8–10]. Hence, it is
0 and deformed nuclei 158Gd and 238U. Eight different
0 now desirable to study the M1GR in this context. Un- Skyrme parameterizations were considered and it was
1 til recently, most of the DFT applications to nuclear dy-
shown that none of them is able to describe simultane-
: namicswereconcentratedonelectricmodesandGamow-
v ously the one-peak structure of the resonance in doubly-
Teller (GT) resonance [8–10], while much less work was
i magic nuclei together with the two-peak structure in de-
X done for magnetic excitations. At the same time, the
formed nuclei. The main reason of the failure seems to
r exploration of magnetic modes in general and spin-flip
a beinapoordescriptionoftheinterplaybetweenthe col-
M1GR in particular could be extremely useful to clarify
lective shifts (caused by spin-density correlations) and
the spin and spin-orbit correlations in the nuclear den-
spin-orbit splitting in the static mean field. Obviously,
sityfunctionals. ThisholdsespeciallyfortheSkyrmeand
this failure of Skyrme forces is also an alarming message
Gogny functionals where, unlike the relativistic models,
for SHF investigations of the GT resonance.
the spin-orbit interaction is an independent part of the
modeling. Further, magnetic modes allow to explore the It is also worth mentioning the very recentSHF study
spintermsinspin-saturatedeven-evennuclei,wherethey [14]where,inaccordancewithresults[15],aconsiderable
cannot be fitted by the ground state properties. The influenceoftensorforcesonthespin-flipM1GRinspher-
M1GR can help to clarify still vague role of the ten- ical nuclei was found. Hence, the tensor forces become
sor forces [11–15]. And, last but not least, the spin-flip indeed an important, though still not well understood,
M1GR is a counterpart of the GT resonance which is of factor in the exploration of this resonance.
great current interest in connection with astrophysical
Altogether, one may state that the M1GR is still a
problems [9, 10]. So, a satisfactory treatment of M1GR
challengeforSHFandleavesveryseriousopenproblems.
is relevant for the DFT description of the GT resonance
A discussion of these problems is the main scope of the
as well.
present paper. We will also discuss the possibility to
In this paper, we will concentrate on the exploration use the M1GR for testing the spin, spin-orbit and ten-
2
sor terms in the Skyrme functional. The interplay of where b , b′, ˜b , ˜b′ are the force parameters. The func-
i i i i
thesetermsisratherinvolvedandmakestheproblemin- tional involves time-even (nucleon ρ , kinetic-energy τ ,
q q
deed demanding. We will discuss the present status of spin-orbit J ) and time-odd (current j , spin s , vec-
q q q
the studies, scrutinize some particular important points tor kinetic-energy T ) densities where q denotes protons
q
(isovectorcharacterofM1GRanditsmanifestationinex- and neutrons. The total densities, like ρ = ρ + ρ ,
p n
periment, contributions of the tensor and isovector spin- are without the index. The contributions with b and
i
orbitterms, etc) andsketchthe possible waysof the fur- b′ (i=0,1,2,3,4)representthe standardterms responsible
i
ther progress. for the ground state properties and electric excitations
Theexplorationisperformedwithintheself-consistent of even-even nuclei [8, 10]. In traditional SHF function-
separable Random-Phase-Approximation(SRPA) model als, the isovector spin-orbit interaction is linked to the
[15, 21–23, 25] based on the Skyrme functional [16–18]. isoscalar one by b′4 = b4. The tensor spin-orbit terms
The model was shown to be an effective and accurate ∝ ˜b ,˜b′ are often omitted. In Eq. (1) they can be
1 1
tool for systematic study of multipole electric giant res- switchedbytheparameterγ . Thespintermswith˜b ,˜b′
T i i
onances [22–24]. Recently, it was extended and applied becomerelevantonlyforoddnucleiandmagneticmodes
to magnetic excitations [15, 21, 25]. in even-even nuclei. Though ˜b ,˜b′ may be uniquely de-
i i
The paper is outlined as follows. In Sec. 2, the SRPA termined as functions of b ,b′ [10], their values were not
i i
model is sketched. In Sec. 3, the present status of the yet well tested by nuclear data and so are usually con-
SHF description of M1GR and related difficulties are sidered as free parameters. Just these spin terms are of
summarized. In Sec. 4,the isospincharacterofthe mea- paramount importance for the spin-flip M1GR.
sured and computed M1GR responses is discussed as a
SRPAis afully self-consistentmodelasits residualin-
possible sourceofthe discrepancies. InSec. 5, the possi-
teractionincludes allthe terms followingfromthe initial
ble important role of the tensor and isovector spin-orbit
Skyrmefunctional. Formagneticmodes,these termsare
termsisconsidered. InSec. 6,theconclusionsaredrawn.
determined through the second functional derivatives
II. MODEL AND CALCULATION SCHEME δ2E δ2E δ2E δ2E
, , , . (2)
δjq′δsq δsq′δsq δJq′δJq δTq′δsq
SRPAis afully self-consistentDFT modelwhereboth
the static meanfieldandresidualinteractionarederived
from the Skyrme functional [16–18]. The SRPA residual
The pairing comes through the functional V =
interaction includes all contributions from the Skyrme pair
1/2 G χ χ∗ whereχ isthepairingdensityandG is
functional as well as the Coulomb (direct and exchange) q q q q q q
the Ppairingstrength[10]. In the presentstudy, pairingis
andpairing(atBCSlevel)terms. Theself-consistentfac-
includedattheBCSlevelthroughthequasiparticleener-
torization of the residual interaction in SRPA consider-
gies and Bogoliubov’scoefficients. Unlike the case of the
ably reduces the computational expense while maintain-
scissorsmode,apossibleviolationoftheparticlenumber
ing a high accuracy. This makes the model very suitable
conservation is not critical for spin-flip M1GR with its
forsystematicstudies. Themodelwasfirstlyderivedand
ratherhigh energy. Anyway,a better pairing description
widely used for electric excitations [22–24]. Recently it
within SRPA is in progress.
was extended to magnetic modes [15, 21, 25].
StartingpointistheSkyrmeenergy-densityfunctional Thespectraldistributionofthespin-flipM1modewith
[8, 10] Kπ =1+ is given by the strength function
b b′
H = 0ρ2− 0 ρ2+b (ρτ −j2)−b′ (ρ τ −j2)
Sk 2 2 Xq q 1 1Xq q q q S(M1;ω)= |hΨν|Mˆ|Ψ0i|2ζ(ω−ων) (3)
νX6=0
b b′ b b′
− 2ρ∆ρ+ 2 ρ ∆ρ + 3ρα+2− 3ρα ρ2
2 2 q q 3 3 q
Xq Xq
where Ψ is the ground state, ν runs over the RPA
0
−b4(ρ∇J+(∇×j)·s)−b′4 (ρq∇Jq+(∇×jq)·sq) Kπ = 1+ states with energies ων and wave functions
Xq Ψ . Further, ζ(ω −ω ) = ∆/[2π[(ω −ω )2 + ∆2]] is a
ν ν ν 4
˜b ˜b′ ˜b ˜b′ Lorentzweightwiththeaveragingparameter∆=1MeV.
+ 0s2− 0 s2− 2s·∆s+ 2 s ·∆s
2 2 q 2 2 q q Such averaging serves to simulate broadening effects be-
Xq Xq yond SRPA (escape widths, coupling with complex con-
˜b ˜b′ figurations) and the width ∆ is chosen to be optimal for
+ 3ραs2− 3ρα s2
3 3 q the comparison with experiment. The strength function
Xq (3)is computeddirectly, i.e. without calculationofRPA
+γ (˜b (s·T−J2)+˜b′ (s ·T −J2)) (1) states ν, which reduces the computation expense even
T 1 1 q q q
Xq more.
3
The operatorof spin-flipM1 transitionin (3)reads [3] 6 a) 6 b) 6 c)
V] SkO SG2 SV-bas
3 Z N Me 158Gd
Mˆ = µBr8π[gsp sˆi+gsn sˆi] (4) 2/N4 4 4
AXi=1 Xi=1 m1) [
= µBr83π [21gs0−τ3gs1]sˆi (5) B(M 2 2 2
Xi=1
where sˆ is the spin operator, gp = 5.58ς and gn =
i s p s
−3.82ςnareprotonandneutronsping-factors,gs0 =gsp+ 0 4 6 8 10 4 6 8 10 4 6 8 10 12
gasnnd=is1o.v3e5ctaonrd(Tgs=1 1=)gspspi−n gg-snfa=cto6r.s2,4thaereisiosospscinalaτr i(sT-=1/02) V]10 d) exper. 10 e) exper. 10 f) exper.
3 e
for protons and 1/2 for neutrons. All the g-factors are M 8 8 8 208Pb
quenched by ς =0.68 and ς =0.64. Note that g1 >> 2/N
p n s 6 6 6
g0 which shows the predominantly isovectorcharacterof m) [
s 1
the spin-flip M1 resonance. As we are interested in the M 4 4 4
B(
spin-flip transitions, the orbital part in (4) is omitted.
2 2 2
Note that in the experimental data [26, 27] used later
for the comparison, the orbital contribution is strongly 0
2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 12
suppressed.
w [MeV]
SRPA calculations employ a coordinate-space grid
withameshsizeof0.7fm. Fordeformednuclei,cylindri-
FIG.1: Theunperturbed(short-dashcurve)andSRPA(solid
cal coordinates are used and the equilibrium quadrupole curve) M1 strength in 158Gd and 208Pb for the forces SkO,
deformationisfoundbyminimizationofthetotalenergy
SG2, and SV-bas. The experimentaldata are given byboxes
[23,24]. Thesingle-particlestatesaretakenintoaccount with bars for 158Gd [26] and vertical arrows for 208Pb [27].
fromthebottomofthepotentialwellupto+20MeV.In The strength is smoothed by the Lorentz weight with ∆=1
theheaviestnucleusunderconsideration,238U,thisgives MeV.
∼17000 two-quasiparticle (2qp) Kπ = 1+ pairs with the
excitation energiesup to 50-70MeV. More details of the
SRPAformalismandcalculationscheme canbe foundin
[15, 21, 25]. data which indicate a two-peak structure of the M1GR
in 158Gd and one (isovector) peak in 208Pb.
The figure illustrates a typical situation already
III. PRESENT STATUS OF THE PROBLEM pointed out in the study [15], namely, that none of the
considered Skyrme forces is able to describe the M1
In the present study, the M1 strength is considered strength simultaneously in both deformed and doubly-
in spherical (208Pb) and deformed (158Gd and 238U) nu- magic nuclei. Indeed, we see that SkO well reproduces
clei. ArepresentativesetofeightSHFparameterizations theone-peakstructureoftheM1GRin208Pbbutfailsin
is used: SkT6 [28], SkO [29], SkO’ [29], SG2 [30], SkM* offering two peaks in 158Gd. Vise versa, SG2 and espe-
[31],SLy6[32],SkI4[33],andSV-bas[34]. Theyexhibita ciallySV-bassucceedinthe two-peakstructure in158Gd
variety ofeffective masses(from m∗/m=1in SkT6 down but deliver a wrong resonance shape in 208Pb.
to0.65inSkI4)andothernuclearmattercharacteristics. These results may be understood in terms of two key
Some of the forces (SLy6) were found best in the de- factors: i) the proton and neutrons spin-orbit splittings,
scription of E1(T=1) GR [23, 24, 35]. Others were used Ep and En, which set the proton and neutronbranches
so so
instudiesofGamow-Tellerstrength(SG2,SkO’)[30,36– of the unperturbed resonance, and ii) the residual inter-
38]orpeculiaritiesofspin-orbitsplitting(SkI4)[33]. The action which produces a collective shift E (defined as
coll
forces SkT6, SG2 and SkO’ involve the tensor spin-orbit a difference between SRPA and unperturbed resonance
term added with (SkO’) and without (SkT6, SG2) refit- centroids). Fig. 1showsthat,fortheforcesSG2andSV-
ting the Skyrme parameters. SV-bas is one of the latest bas, the proton and neutron unperturbed branches ap-
SHF parameterizations [34] where the spin-orbit isovec- pear as separated peaks in 208Pb and as one single peak
tor interaction is varied freely by setting b′ 6=b . with a right shoulder in 158Gd. In both cases the proton
4 4
InFig. 1thespin-flipM1strength(3),calculatedwith low-energypeakishighersincegp >gn. Theresidualin-
s s
g-factors gp = 5.58ς and gn = −3.82ς , is presented in teractionupshifts thestrengthby1-2MeV,redistributes
s p s n
thedeformed158Gdandspherical208Pb. BothSRPAand itinfavorofthe upper peak,andsomewhatenlargesthe
unperturbedstrengthsareshowntodemonstratethecol- splitting. As a result, a distinctive two-peak structure is
lectiveshiftcausedbytheresidualinteraction. Theforces formed. Insteadfor SkO,the relative spin-orbitsplitting
SkO, SG2 and SV-bas are used as representative exam- E =En−Ep isverysmallandtheprotonandneutron
so so so
ples. The results are compared with the experimental branches actually form one peak which is then upshifted
4
3 158Gd Eso IV. ISOVECTOR SPIN-FLIP M1(T=1)
E RESPONSE
2 coll
1 In the above discussion and Ref. [15], we analyzed
thespin-flipM1strengthincluding bothisovector(T=1)
V] 0
e and isoscalar (T=0) contributions. This strength was
M
E [so -13 SkM* SLy6 SkT6 SG2 SV-bas SkI4 SkO SkO' cAaslcwulaastemdewnittihonge-dfaicntoSrsecg.sp =II,5t.5h8eςpisaonvedctgosnr=g-−fa3c.t8o2rςnis.
Ecoll, 2 208Pb mMu1cshtrleanrggetrhtshhaonultdhebeisporsecdaloamrionnaen,tglys1i>so>vegcs0t,ora.nIdnsooththeer
1 words, the M1 and purely isovector M1(T=1) responses
are to be about the same. However, these arguments
0
consider the M1GR as one entity and do not take into
-1 account possible local differences (i.e. features at par-
SkM* SLy6 SkT6 SG2 SV-bas SkI4 SkO SkO'
ticular energies) in M1 and M1(T=1) strengths. As is
FIG. 2: The relative spin-orbit splittings Eso = Esno−Espo shownbelow, these local differences can be essentialand
(full squares) and M1 collective shifts Ecoll (full circles) in considerably change the appearance of the M1GR.
158Gd and 208Pb for 8 Skyrme forces. For better view, the In this connection, it is worth to compare with exper-
symbols are connected by lines. The horizontal line E=0 is iment the spin-flip M1(T=1) response calculated with
drawn for convenience of thecomparison. g0 = 0 (similar calculations were recently performed for
s
208Pbin[14]). Thisdiffersfrommostofthepreviouscal-
culations [4–7, 19] where the common M1 strength was
considered. However, the isovector separation is reason-
ablebecausetheexperiment[26,27]treatsthe M1GRas
by the residual interaction.
the isovector mode.
This analysis illustrates the well known fact [1–3, 15, In Fig. 3, the isovector M1(T=1) strength in 208Pb
19] that the quality of the description of the M1GR is
computedwitheightdifferentSkyrmeforcesispresented.
mainly determined by the ratio E /E between the
coll so Todiscriminatethedetails,asmallwidthof∆=0.2MeV
collective shift and relative spin-orbit splitting. If the
is used in the Lorentz smoothing. In this doubly-magic
initialE islarge,thenastrongresidualinteractionwith
so nucleus the RPA spectrum is dilute and so the small
E > E is necessary to mix the proton and neutron
coll so smoothing does not cause an excessive complication of
branches, redistribute the strength to a higher energy,
the strength. Fig. 3 shows that the results depend
and thus produce a one-peak resonance. Otherwise, a
strongly on the force. However, unlike the M1 case, the
two-peak structure persists. If instead E is small, then
so M1(T=1) strength already exhibits mainly a one-peak
the unperturbed resonance already has one peak which
structure provided by the dominant right peak. This
is then merely upshifted by the residual interaction.
structureisobviousevenfortheforcesSG2,SkM*,SkT6,
InFig. 2,thekeyingredientsoftheM1GRdescription, andSV-bas,whichshowtwopeaksinM1strength. Only
E andE ,arecomparedforeightSkyrmeforces. One SLy6 maintains the structure of the M1 result.
coll so
seesthatE <E formostoftheforces(SkM*,SLy6, SuchadifferencebetweenM1andM1(T=1)responses
coll so
SkT6,SG2,SV-bas)whichshouldresultinadouble-peak may be explained in terms of spin g-factors. The M1
M1GR.Andindeed,Fig. 1demonstratethisforSG2and transitions deal with gsp = 5.58ςp and gsn = −3.82ςn
SV-bas in 158Gd and208Pb. Instead, for the forces SkI4, and so, as was mentioned above, the unperturbed M1
SkO, and SkO’, we have E > E and hence the one- strength exhibits the left proton peak ∝ (gp)2 about
coll so s
peak M1GR. The results of Fig. 2 for 158Gd and 208Pb twice higher than the right neutron peak ∝ (gn)2. The
s
remind those for 238U [15]. So the similar results may residual interaction recasts the M1 strength in favor of
be expected for other medium and heavy nuclei as well. the right peak, which finally yields the two-peak struc-
This means that the M1GR structure is mainly deter- turewithcomparablepeakheights. Instead,inM1(T=1)
mined by the Skyrme force rather than by the particu- transitions we use gp = −gn = g1/2 = 3.12 and so, un-
s s s
lar nucleus. In other words, the forces of the first (sec- like the M1 case, the unperturbed proton and neutron
ond) group should always yield a two-peak (one-peak) peaks already have about the same heights (with a bit
structure. Hence a failure in simultaneous description of higherneutronpeak). Thenthefurthercollectiveupshift
M1 strength in nuclei like 158Gd and 208Pb by one and of the strength results in a strict dominance (more than
the same force. This is a very serious drawback of the in M1 case) of the right peak, hence mainly a one-peak
present-day Skyrme parameterizations. Besides, this is structure.
an alarming message for the SHF description of the GT Asaresult,someforcesprovideanacceptabledescrip-
resonancewhich,beingacounterpartofM1GR,isdeter- tionoftheexperimentaldatafortheM1(T=1)case. The
mined by the same factors. Possible ways to cure this forces SG2, SkO, and SV-bas give a dominant peak at
problem will be discussed in the next sections. the energies 6.8, 7.2, and 8 MeV, i.e. close to the ex-
5
40 exper. 7.3 MeV 40 exper. 7.3 MeV
SG2 SkI4
30 30
208
Pb
20 20
10 10
0
40 4 6 8 10 40 4 6 8 10
V] 30 SkM* 30 SkO
e
M 20 20
2/N10 10
4 6 8 10
m) [ 400 4 6 8 10 40
=1 30 SkO' 30 SkT6
T
1, 20 20
M
10 10
(
B
0
40 4 6 8 10 40 4 6 8 10
30 SLy6 30 SV-bas
20 20
10 10
0
4 6 8 10 4 6 8 10
w [MeV] w [MeV]
FIG. 3: Isovector M1(T=1) strength in 208Pb, calculated with 8 Skyrme forces as indicated. The 2qp (short-dash curve) and
SRPA (solid curve) results are presented. The vertical dash line marks the average experimental resonance energy 7.3 MeV.
The strength is smoothed bythe Lorentzweight with ∆=0.2 MeV.
5 5 5
a) 158Gd b) c)
SkO SG2 SV-bas
4 4 4
3 3 3
2 2 2
V] 1 1 1
e
M
2/N 02 4 6 8 10 02 4 6 8 10 02 4 6 8 10
m=1) [10 d) 238U 10 e) 10 f)
T
1, 8 8 8
M
B(
6 6 6
4 4 4
2 2 2
0 0 0
2 4 6 8 10 2 4 6 8 10 2 4 6 8 10
www [[[MMMeeeVVV]]]
FIG.4: TheisovectorM1(T=1)strengthin158Gdand208PbfortheforcesSkO(left),SG2(middle),andSV-bas(right). The
experimentaldata[26]aregivenbythegreyboxesandbars. ThestrengthissmoothedbytheLorentzweightwith ∆=1MeV.
perimental average value 7.3 MeV. For other forces the 158Gd and 238U shows that SkO does not reproduce the
disagreement is larger than 1 MeV. two-peak structure at all while SG2 and SV-bas suggest
it, but with strongly attenuated left peak. Though in
However, as already observed in case of mere M1
generaltheM1(T=1)responsebetteragreeswiththeex-
strength, an acceptable agreement for spherical nuclei
perimentfor208PbthantheM1response,asimultaneous
does not mean the same for deformed ones. Fig. 4 for
6
10 exper. 10 exper.
208Pb V] 208Pb
e
V] 8 M1 M 8 M1(T=1)
e 2/
M N
2/N 6 m1) [ 6
m) [ 4 T= 4
M1 M1,
B( 2 B( 2
0 0
4 6 8 10 12 V] 4 6 8 10 12 14
V] 4 158Gd Me 4 158Gd
2/Me 3 M1 2m) [/N3 M1(T=1)
N 1
m1) [ 2 1,T= 2
M M
B( 1 B( 1
0 0
2 4 6 8 10 12 2 4 6 8 10 12 14
w [MeV] w [MeV]
FIG.5: Thespin-flipM1(left)andM1(T=1)(right)strengthsin208Pband158GdfortheforceSV-basinthestandardversion
(solid curve), with tensor contribution (bold curve) and with b′4 =b4 (short-dash curve). The experimental data are given by
boxes with bars for 158Gd [26] and vertical arrows for 208Pb [27]. The strength is smoothed by the Lorentz weight with ∆=1
MeV.
descriptionofthe experimentaldata insphericalandde- E . Instead the T=1 spin-orbit interaction changes Ep
so so
formed nuclei still fails. and En on scale and so affects E .
so so
The comparison of Figs. 1 and 4 shows that for SV- Here we take into account both static mean-filed and
bas the computed M1 strength is closer to the experi- collective impacts of the tensor and T=1 spin-orbit in-
mentthan the M1(T=1)one. Then the naturalquestion teractions. The results are demonstrated in Fig. 5. It
arises, to which extent the experimental data [26] from is worth reminding that the spin-orbit term in Skyrme
(p,p′)reactionand[27]from(γ,γ′)reactionwithtagged functional (1) reads
photons give just the isovectorM1GR? Both experimen-
−b (ρ∇J+(∇×j)·s)−b′ (ρ ∇J +(∇×j )·s ). (6)
tal studies claim this. However this claim, being based 4 4 q q q q
on the general reaction conditions, is actually not sup- Xq
ported by thorough estimations and checks. Moreover, The standard SHF calculations use b′ = b , i.e. only
4 4
these reactions should actually involve both T=0 and the isoscalar (T=0) contribution. At the same time, the
T=1 channels. Thus the computation of the reaction relativistic models employ b′=0 [8–10]. So, it is worth
4
cross-sections are called for more adequate comparison todecouplethecoefficientsb′,andb andthusintroduce
4 4
with the experimental. This uncertainty also could be the isovector (T=1) spin-orbit interaction, as was done
one ofthe reasonsofthe disagreementbetweenSHF and e.g. in [33]. Such decoupling is natural since a similar
experimental results for M1GR. separation is already used for other Skyrme coefficients,
b and b′, with i = 0,1,2,3. Actually, the force SV-
i i
bas already follows this track and uses b = 34.117 and
4
b′ = 0.547b . The effect is demonstrated in Fig. 5 for
V. TENSOR AND ISOVECTOR SPIN-ORBITAL 4 4
FORCES M1 and M1(T=1) responses, where the SV-bas result
is compared with b′ = b variant (after refitting). It
4 4
is seen that the effect is not large. However, any final
Another point to be discussed in connection with the
conclusions on its scale can be done only after thorough
M1GR problems is the influence of the tensor and T=1
checks involving various Skyrme forces and nuclei.
spin-orbit interactions. Both interactions come to the
Wenowconsidertheimpactoftensorinteraction. Itis
Skyrmefunctionalthroughthetermswithspin-orbitden-
oftenomittedinthe standardeffectivetwo-bodySkyrme
sities [8]. As shown in our recent study [15], these in-
interaction. If included, it adds to the functional (1) the
teractions can affect the M1GR through the spin-orbit
term
splittings Ep and En. The tensor interaction changes
so so
Espo andEsno likewise,thus producing atotalM1GRshift γT(˜b1(s·T−J2)+˜b′1 (sq·Tq−J2q)) (7)
without a noticeable variation of the relative splitting Xq
7
where the squared spin-orbit densities J2 and J2 repre- VI. CONCLUSIONS
q
sent the tensor contribution while s·T and s ·T terms
q q
serve to restore in (7) the Galilean invariance. The ex-
The open problem of the description of the spin-flip
change part of the zero-range Skyrme interaction also
M1giantresonance(M1GR)withintheSkyrme-Hartree-
leads to similar spin-orbit terms, see e.g. [12]. To be
Fock (SHF) approach is analyzed. It is shown that
accurate, the tensor and central exchange contributions
presently available Skyrme parameterizations poorly re-
should be treated separately and their parameters are
producetheexperimentaldataand,inparticular,cannot
to be determined from the initial effective two-body in-
provideasimultaneousdescriptionofM1GRgrossstruc-
teraction [12, 14]. However, from the point of view of
ture in deformed and spherical (doubly magic) nuclei.
a zero-range Skyrme interaction, it is reasonable not to
The two main factors responsible for the M1GR prop-
distinguish the tensor and central exchange terms and
erties, spin-orbit splittings and spin correlations, are in-
use for both of them the same fitting parameters˜b and
1 spected for eight different Skyrme parameterizations.
˜b′. We use here just suchcommonpractice. For simplic-
1 Some criticalaspects areworkedout. One pointis the
ity,thetensorandcentralexchangecontributionswillbe
essential difference between M1 and M1(T=1) responses
furthercalledtensorterms. Notethatthesetensorterms
which leads to the open question: how much is the ob-
influence both groundstate properties anddynamics. In
served strength of the isovector nature and which of the
theresultsshowninFigs. 1-4,theforcesSkOandSV-bas
responsesshouldbecomparedwithit? Furthermore,the
havenotensorterms. However,thesetermsareaddedin
essential influence of the tensor force was demonstrated,
SG2astheynoticeablyimprovedescriptionoftheground
which can have really dramatic effects. So the tensor in-
state for this particular parameterization.
teractioncanbeakeyelementinthefurtherdevelopment
Asanexample,wewillnowcomparetheSV-basresults
ofabetterM1GRdescription. AnappropriateT=1part
with and without tensor terms. They are fully switched
of the spin-orbit interaction could also be an important
on by γ = 1. As shown in [15], the refitting of other
T ingredient.
Skyrmeparametersmayconsiderablydecreasethetensor
Altogether,theSHFdescriptionoftheM1GRremains
effect. So,weuseforγ =1the refittedSV-basparame-
T yet open as a quite complicated problem where many
ters. The tensor contributions to both ground state and
contributionsareentangled. Theproblemmayhavegen-
SRPA residual interaction are taken into account. The
eral consequences for the SHF description of nuclear dy-
results of the calculations are shown in Fig. 5. It is seen
namics in the spin-isospinchannel. More developmentis
that the tensor effect is indeed dramatic (a large ten-
needed to establish SHF as a reliable model also for spin
sor impact on M1GR was also found in [14]). Moreover,
properties.
it considerably improves agreement with the experimen-
tal data in this particular case. Hence the tensor forces
can indeed be an important factor in the description of
M1GR. Acknowledgments
It is also worth noting that tensor forces significantly
influencetheLandau-Migdalparametersg andg′ inthe The work was partly supported by the DFG RE-
0 0
spin and spin-isospin channels and affect the estimation 322/12-1,Heisenberg-Landau (Germany - BLTP JINR),
of spin instability of nuclear matter for Skyrme forces and Votruba - Blokhintsev (Czech Republic - BLTP
[13, 36]. Eight Skyrme parameterizations used in the JINR) grants. W.K. and P.-G.R. are grateful for the
present study have g′ > −1 and so are spin-stable at BMBF support under contracts 06 DD 9052D and 06
0
the equilibrium density in the spin-isospin channel. Just ER 9063. Being a part of the research plan MSM
this channel determines the isovector spin-flip M1 and 0021620859(Ministry of Education of the Czech Repub-
Gamow-Teller GR. Anyway, our knowledge on the inter- lic) this work was also funded by Czech grant agency
play between tensor forces and spin correlations is still (grant No. 202/06/0363). P.V. is grateful for the
rather poor and M1GR could be used here as a robust FIDIPRO support. V.Yu.P. thanks the DFG support
and important test in clarification of this interplay. under the Contract SFB 634.
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