Table Of ContentMagnetic dipole excitations of 50Cr
H. Pai,1,∗ T. Beck,1 J. Beller,1 R. Beyer,2 M. Bhike,3,4 V. Derya,5 U. Gayer,1 J. Isaak,6,7
Krishichayan,3,4 J. Kvasil,8 B. Lo¨her,1,6 V. O. Nesterenko,9 N. Pietralla,1 G. Mart´ınez-Pinedo,1,10
L. Mertes,1 V. Yu. Ponomarev,1 P.-G. Reinhard,11 A. Repko,8 P. C. Ries,1 C. Romig,1 D. Savran,6,7
R. Schwengner,2 W. Tornow,3,4 V. Werner,1 J. Wilhelmy,5 A. Zilges,5 and M. Zweidinger1
1Institut fu¨r Kernphysik, Technische Universita¨t Darmstadt, D-64289 Darmstadt, Germany
2Institut fu¨r Strahlenphysik, Helmholtz-Zentrum Dresden-Rossendorf, D-01328 Dresden, Germany
3Department of Physics, Duke University, Durham, North Carolina 27708, USA
4Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708, USA
5Institut fu¨r Kernphysik, Universita¨t zu K¨oln, D-50937 K¨oln, Germany
6ExtreMe Matter Institute EMMI and Research Division,
GSI Helmholtzzentrum fu¨r Schwerionenforschung GmbH, D-64291 Darmstadt, Germany
6 7Frankfurt Institute for Advanced Studies FIAS, D-60438 Frankfurt am Main, Germany
1
8Institute of Particle and Nuclear Physics, Charles University, CZ-18000, Prague 8, Czech Republic
0
9Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow region, 141980, Russia
2
10GSI Helmholtzzentrum fu¨r Schwerionenforschung, Planckstraße 1, D-64291 Darmstadt, Germany
n 11 Institut fu¨r Theoretische Physik II, Universita¨t Erlangen, D-91058, Erlangen, Germany
a (Dated: January 11, 2016)
J
8 The low-lying M1-strength of the open-shell nucleus 50Cr has been studied with the method
of nuclear resonance fluorescence up to 9.7 MeV, using bremsstrahlung at the superconducting
] DarmstadtlinearelectronacceleratorS-DALINACandComptonbackscatteredphotonsattheHigh
x Intensityγ-raySource(HIγS)facilitybetween6and9.7MeVoftheinitialphotonenergy. Fifteen1+
e stateshavebeenobservedbetween3.6and9.7MeV.Followingouranalysis,thelowest1+stateat3.6
-
l MeVcanbeconsideredasanisovectororbitalmodewithsomespinadmixture. Theobtainedresults
c generally match theestimations andtrendstypicalfor thescissors-like mode. Detailed calculations
u
within the SkyrmeQuasiparticle Random-Phase-Approximation method and the Large-Scale Shell
n
Model justify our conclusions. The calculated distributions of the orbital current for the lowest
[
1+-state suggest the schematic view of Lipparini and Stringari (isovector rotation-like oscillations
1 inside the rigid surface) rather than the scissors-like picture of Lo Iudice and Palumbo. The spin
v M1 resonance is shown to be mainly generated by spin-flip transitions between the orbitals of the
5 fp-shell.
0
9 PACSnumbers: 21.10.Re;23.20.Lv;25.20.Dc;21.60.Jz;27.50.+e
1
0
. I. INTRODUCTION complicatedbecauseheredeformationandpairingeffects
1
come to play. At the same time, these effects make the
0
6 During the last decades, investigation of strong mag- physics of fp-shell nuclei much richer. In particular, de-
1 neticdipole(M1)transitionsinmedium-massnucleiwas formationresultsinappearanceofthelow-energyorbital
: scissors-like M1 mode (1+) [13–16].
v of a keen interest [1–6]. First of all, this was caused by sc
Xi a diversity of various physicaleffects related to these ex- So far the experimental data on the M1 1+sc strength
citations: quenching of spin strength and impact of non- in the fp-region were limited to a few axially deformed
r nucleonic degrees of freedom, relation to Gamow-Teller nuclei: 46,48Ti (Z=22)[1] and 56Fe (Z=26) [9]. It was
a
transitions and relevant astrophysical problems, isospin found that the 1+sc in these nuclei is mainly represented
degrees of freedom of valence-shell excitations, peculiar- byonelow-energy(intheregion3.4-4.4MeV)statewith
ities of the orbital scissors-like mode, cross-shell and l- Kπ = 1+. As expected [13–16], this state is character-
forbidden transitions, impact of complex configurations, ized by a strong M1 transition to the ground state. The
etc. Furthermore, these nuclei are accessible to modern theoretical analysis [1, 9, 12] indicates that the 1+sc in
large-scale shell model calculations [7, 8] that have the this mass regionis ofa mixed(orbital+ spin)character,
capability to describe the effects mentioned above. The- although with a dominant orbital contribution. Shell-
oreticaleffortsandadvancedexperimentshavefinallyled modelcalculations[9]demonstrateaconsiderabledepen-
to a significantprogressin understanding the features of dence of the description on a subtle balance of different
M1 excitations in this mass region [6]. effects such as equilibrium deformation, pairing,interac-
tion parameters, etc.
Open-shellfp-shellnucleiareparticularlyimportantin
this activity [9–11]. Description of these nuclei is rather For a better understanding of the 1+ features in fp-
sc
shell nuclei, we certainly need experimental data for de-
formed Cr (Z=24) isotopes placed between the Ti and
Fe chains. Jπ = 1+ states of 50Cr are known from pre-
∗Electronicaddress: [email protected],[email protected] vious (p,p′) experiments [17] and for three of them γ-
2
ray transitions have been measured in nuclear resonance number of events and energy-integrated cross-sections,
fluorescence (NRF) [18, 19]. B(M1) values are known transition widths as well as M1 transition strengths,
only for two 1+ states of 50Cr. A comprehensive M1 we refer the reader to the reviews by Metzger [22] and
strength distribution is missing. It is the purpose of the Kneissl and coworkers [20, 21]. Spin quantum numbers,
present study to provide data for a more comprehensive cross-sectionsI (energy-integratedscatteringcrosssec-
i,0
M1 strengthdistribution for the deformed nucleus 50Cr. tionsforexcitingastateianddeexcitingthisstatetothe
This nucleus was studied in photon scattering ex- ground state 0), ground-state transition widths Γ , and
0
periments using NRF [20–22]. The experiment was transitionstrengthsB(M1)weremeasuredattheDHIPS
performed at the superconducting Darmstadt electron setup.
linear accelerator S-DALINAC [23] using unpolarized A series of NRF experiments was performed at TU
bremsstrahlung with a continuous spectral distribu- Darmstadt and at Duke University. The photon-
tion from the Darmstadt High Intensity Photon Setup scattering experiment with unpolarized bremsstrahlung
(DHIPS) [24]. Parity quantum numbers of spin-1 from the Darmstadt linear electron accelerator S-
states were determined using the High Intensity γ-Ray DALINAC[23]wasperformedattheDarmstadtHighIn-
Source[25]operatedbytheTriangleUniversitiesNuclear tensity Photon Setup (DHIPS) [24]. Two measurements
LaboratoryandtheDukeFreeElectronLaserLaboratory on50Crwereperformedwithbremsstrahlungatendpoint
(DFELL) at Duke University in Durham, NC, USA. energiesof7.5(1)MeVand9.7(1)MeV,respectively. The
In the presentexperiment, the spectrum of 50Cr up to measurement with 7.5(1) MeV bremsstrahlung has been
9.7 MeV and the corresponding M1 strength were deter- carriedouttoidentifytransitionsviaintermediatestates.
mined. The M1 strength for the lowest Kπ = 1+-state Bremsstrahlung has been produced by completely stop-
with excitation energy of 3628.2 keV was obtained and ping the intense electron beam from the S-DALINAC’s
the B(M1) values for 12 strongly excited 1+ states were injectorina thickcopper radiatortarget. Thegenerated
measured for the first time. photon beam passes a massive copper collimator result-
Two theoretical models were used for further in- inginabeamspotwithasizeofabout2.5cmdiameterat
terpretation: Skyrme Quasiparticle Random-Phase- theNRFtargetposition. Targetnucleiareexcitedbythe
Approximation (QRPA) [26, 27] and Large-Scale Shell resonant absorption of photons and subsequently decay
Model(LSSM) [8]. Theself-consistentQRPAandLSSM either directly or via intermediate states to the ground
rather well reproduce the experimental axialquadrupole state.
deformation βexp=0.2897(44) [28] of 50Cr and provided Scattered γ-rays were detected by three HPGe detec-
qualitatively close results. The calculations agree upon tors with 100% efficiency relative to a standard 3"×3"
the feature that the 3628.2-keV state has predominan- NaI detector at a γ-ray energy of 1.3 MeV. They are
tely orbitalcharacterwith aminor spinadmixture. This placed at polar angles of 90◦ and 130◦ with respect to
means that the present experiment provides observation the incident beam. The detectors were surrounded by
ofthe1+scinCrisotopes. Thenewdataessentiallysupple- lead and BGO Compton suppression shields. The first
mentthe1+sc systematicsinfp-shellnucleiand,asshown measurement was performed with 2.0 g of an isotopi-
below, lead to a better understanding of 1+sc features in cally enriched 50Cr target (96.416% enriched) using an
this mass region. Besides our analysis of the orbitalflow end-point energy of E = 9.7(1) MeV and average elec-
0
implies a closer correspondence to the schematic view of tron beam currents of about 20 µA. In this measure-
Lipparini and Stringari [29] corresponding to isovector ment data were taken for about 131 hours. For energy
rotation-like oscillations of the nucleons inside the rigid and photon-flux calibrations 399.3 mg of enriched 11B
nuclear surface than to the scissors-likepicture of Lo Iu- (99.52% enriched) were used that were irradiated simul-
dice and Palumbo [13]. taneously. The efficiency ofthe two HPGe detectors was
The paper is organized as follows. In Sec. II, the determinedwitha56Cosourceuptoabout3500keVen-
experimental method and data analysis are outlined. In ergy. For higher photon energies the gamma-ray detec-
Sec. III, the theoretical analysis of the results is given. tion efficiencies were extracted from a simulation using
Sec. IV provides a summary. the Geant4 toolkit [30]. The second measurement was
performedwith the sametargetmassusing anend-point
energy at E = 7.5(1) MeV and average electron beam
0
II. EXPERIMENTAL METHOD AND DATA currentsofabout33µA.Inthismeasurement,datawere
ANALYSIS taken for about 105 hours.
Photon scattering spectra of the 50Cr(γ,γ′) reaction
In NRF measurements, the excitation mechanism is from DHIPS between 3000 and 10000 keV for the detec-
purely electromagnetic. Therefore, intrinsic properties tor at 130◦, measured at E = 9.7(1) MeV are shown in
0
like spin, parity, and transition probabilities can be de- Fig. 1. We have observed 33 excited states of 50Cr with
terminedfromthemeasuredquantities(angulardistribu- spin quantum numbers J = 1. In this work, we focus
tion, polarization asymmetries, γ-ray energy, and inten- on Jπ=1+ states. M1 transitions to the ground state of
sity [20–22]) in a model-independent way. For details of 50Cr are indicated in Fig. 1 by arrows. In the following,
the NRF method, basic relations between the detected transitions corresponding to direct decays to the ground
3
123000000000000 5Eθ0C 0= r=( 1γ 9,3γ.07’) 0MeV 11B (a) 11B w9100e0◦r%ewpirtlehalcaretedisvpeienecffitatccoiertnohscsey-liainkncedidgetehnoetmboetetharemyr,attwtwopooholaafvrteha6enm0g%lehsraevole-f
ative efficiency, respectively. In this setup, two detectors
V 0
3000 3500 4000 4500 5000 were placed in the (horizontal) polarization plane of the
e2000
4 k1600 11B (b) incident γ-ray beam at azimuthal angles ϕ = 0◦ and
1.1200 ϕ = 180◦ and the other two were placed perpendicular
nts/ 480000 to it at ϕ=90◦ and ϕ=270◦, respectively.
Cou10000 5500 6000 6500 7000 7500 8000 tioTnsheobesxeprevreidmiennttahleamsyemasmueretrmieesntεs faotrHthIγeSMar1etsrhaonwsin-
800 11B (c) in Fig. 2. The asymmetry ε is defined by the ratio of
600
400 themeasuredandefficiency-correctedpeakintensitiesAk
200 and A , within and perpendicular to the polarization
⊥
80000 8500 9000 9500 10000 plane, respectively:
E (keV)
γ
A −A
k ⊥
ε= =q·Σ. (1)
FIG. 1: Photon scattering spectra of the 50Cr(γ,γ′) reaction Ak+A⊥
from DHIPS between 3000 keV and 10000 keV for the 130◦
detector,measuredatE =9.7MeV.Ground-stateM1tran- Here,Σis the analyzingpowerandq is the experimental
0
sitions of 50Cr are indicated by arrows. sensitivity which accounts for the finite opening angles
of the detectors and the finite size of the target. The
analyzingpowerΣ ofNRFamountsto+1foraJπ =1+
state and −1 for Jπ = 1− state, respectively. For this
statearecalledelastictransitionsandthosedecayingvia setup, the experimentalsensitivity q amounts to0.86(1).
intermediate states will be referredto as inelastic transi- Intotal13M1excitationsof50Crhavebeenobservedin
tions. this experiment.
50
1 Cr 50
1 Cr
0.8
)N0.8
0.6 2
µ
ε)y ( 0.4 M1) ( 0.6
etr 0.2 B( 0.4
m 0
m 0.2
y-0.2
s 0
A-0.4 3 4 5 6 7 8 9 10
Energy (MeV)
-0.6
-0.8
FIG.3: TheexperimentalB(M1)valuesforthegroundstate
-1 transitions.
5000 6000 7000 8000 9000 10000
E(keV)
γ
FIG. 2: Experimental asymmetry for theM1 transitions ob-
served at HIγS. III. RESULTS AND DISCUSSION
A. Experimental results
Parity quantum numbers of spin-1 states were deter-
mined at the High Intensity γ-Ray Source [25] operated The experimental results are summarized in Table I
by the Triangle Universities Nuclear Laboratoryand the and Figure 3. In the presentwork,we have deduced M1
Duke Free Electron Laser Laboratory(DFELL) at Duke strengths in 50Cr for the first time through the (γ,γ′)
UniversityinDurham,NC,USA.Thisfacilityprovidesa reaction in addition to the 3628.0-keV transition, whose
quasi-monoenergetic beam of nearly completely linearly M1strengthhaspreviouslybeenmeasuredvia(γ,γ′)[18]
polarized photons from Laser Compton Backscattering and (e,e′) [1, 17, 33] experiments albeit with larger un-
(LCB) in the entrance channel. For details of the analy- certainties. In the present experiment, the uncertainty
sis procedure, we refer the reader to Refs. [31, 32]. The of the cross section has been reduced for the 3628.0-keV
measurement was performed with 2.0 g of an isotopi- transition. Nine of the observed excited states of 50Cr
cally enriched 50Cr target (96.416% enriched). The tar- may be alreadyknownfrom a (p,p′) experiment[17] and
get was centered between four HPGe detectors which threestates(3628.2,7645.7and8885.6keV)wereknown
4
TABLE I: Transitions observed in the 50Cr(γ,γ′) reaction with an endpoint energy of 9.7 MeV. Experimental uncertainties of
theexcitation energies are less then 0.5 keV.
Exa Eγa WW((19300◦◦)) Jπ Ii,0 Γ0 ΓΓ0ib B(M1)↑ B(M1)↑ Γ0red
(keV) (keV) (eVb) (eV) (µ2 ) (µ2 ) (meV/MeV3)
N N
d
3628.2 3628.0 0.74(2) 1+c 120(5) 0.205(9) 1.113(49) 0.99(12) 4.29(19)
2845.0 0.49(1)
4997.0 4996.7 0.78(6) 1(+)e 16.2(12) 0.070(7) 0.145(15) - 0.56(6)
4213.8 1.0(1)
5931.2 5930.8 0.92(9) 1+ 23.9(20) 0.073(6) - 0.091(7) - 0.35(3)
7600.8 7600.2 0.80(10) 1+ 66.4(73) 0.334(37) - 0.197(22) - 0.76(8)
7645.7 7645.1 0.69(17) 1+ 23.2(28) 0.118(14) - 0.068(8) - 0.26(3)
7948.1 7947.4 0.78(3) 1+ 197.7(96) 1.382(79) 0.714(41) - 2.75(16)
7164.5 0.27(2)
8045.8 8045.1 0.77(10) 1+ 42.2(47) 0.238(26) - 0.118(13) - 0.46(5)
8121.5 8120.8 0.76(20) 1+ 16.4(20) 0.094(11) - 0.045(5) - 0.18(2)
8528.1 8527.4 0.70(8) 1+ 96(11) 0.85(11) 0.353(48) - 1.36(18)
7743.1 0.39(6)
8885.6 8884.8 0.82(8) 1+ 77.7(68) 0.534(47) - 0.197(17) 0.277(74)e 0.76(7)
9007.9 9007.0 0.88(13) 1+ 40.5(48) 0.286(34) - 0.101(12) - 0.39(5)
9208.3 9207.4 0.77(20) 1+ 50(12) 0.369(89) - 0.123(30) - 0.47(11)
9409.5 9408.5 0.78(14) 1+ 105(17) 0.81(13) - 0.252(41) - 0.97(16)
9579.1 9578.1 0.97(25) 1+ 37.7(78) 0.301(62) - 0.089(18) - 0.34(7)
9719.1 9718.1 0.85(11) 1+ 173(21) 1.42(17) - 0.402(49) - 1.55(19)
aThedifferencebetweenthetheexcitationenergy(Ex)andtran-
sitionenergy(gamma-rayenergy(Eγ))isduetothenuclearrecoil.
bBranchingratio.
cFromRefs.[1,17,33].
dFromRef.[18].
eFromRef.[34].
from (γ,γ′) [18, 19] experiments. For twelve 1+ states of by A. Richter et al. [15] in heavy deformed nuclei. As
50Cr, ourdata provide firstinformationontheir γ-decay mentioned above, the 1+ state was also reported in the
sc
transitions. open fp shell nuclei 46Ti, 48Ti [1] and 56Fe (Z=26) [9].
Using linearly polarized quasi-monoenergetic photons
Theexperimentalvalueofthequadrupoledeformation
atHIγS,positiveparityquantumnumberswereassigned parameter β for 50Cr is 0.2897(44) [28], which indi-
exp
to 13 states apart from the 3628.2-keV state. The defi-
cates that this nucleus is well deformed and hence can
nite positive parity quantum number assignment for the support the formation of a 1+ state. The ratio of the
3628.2-keV state has been adopted from earlier (p,p′ ) M1 transition rates of the 1+slcevel at 3628.2 keV to the
and (e,e′) [1, 17, 33] experiments. Three new M1 tran- ground state and the first-excited 2+ state (at 783.32
sitions (5930.8, 8045.1 and 8120.8 keV) have been iden- B(M1;1+→2+)
tified in the present work and for thirteen 1+ states M1 keV excitation energy) is found to be B(M1;11+→01+) =
1 1
excitationstrengthshavebeenobtainedforthefirsttime. Γ1·(Eγ0)3 = 1.02(2) with E being the γ-ray tran-
Γ0 Eγ1 γ0,(1)
sition energy to the ground (first-excited) state. This
The experimental B(M1)values into the groundstate
value is larger by a factor of 2 compared to the value of
are shown in Figure 3. The M1 strength distribution
0.5 obtained by employing the Alaga rule [36]. This dis-
exhibits anintriguing pattern: anisolated,rather strong crepancyhintsatsomedegreeofKmixing,perhaps50Cr
M1 excitation at low energy at 3.6 MeV is separated
exhibits some degree of gamma-softness [37].
from an accumulation of fragmented M1 strength above
7 MeV. A strongly excited Jπ = 1+ state at an excita- Since in 50Cr neutron and proton Fermi levels are in
the same open subshell, spin magnetism is also expected
tion energy of 3628.2keVwith a strengthof B(M1)↑ =
1.113(49) µ2 is observed. The rest of the observed M1 in the lowest 1+ state. Therefore, the state at 3628.2
N
excitations in 50Cr is expected [17] to be generated by keV should be of a mixed orbital/spin nature. It is also
excitedin(p,p′)reactions[17],whichpointstoasufficient
spin excitations, mainly from f → f orbitals like
7/2 5/2
in 52Cr [35]. We will first concentrate on the 1+ state contribution of spin-M1 strength.
at 3628.2 keV. This state could correspond to a moder- The available experimental results for the 1+ in fp-
sc
ately collective scissors-like mode. This mode has been shell nuclei, including the present data, are exhibited in
predicted [13, 14] and then discovered in (e,e′) reactions Fig.4. Itisinstructivetoseehowmuchthesedatamatch
5
1.5 βexp = 0.2897(44) 50Cr 10
1 (a) exp
) est (2)
0.5 V
e est (4)
M 5
) 3 3.5 4 4.5 5 (
N E 46
2µ( 1.5 βexp = 0.3175(42) 46Ti 56Fe 48Ti 50Cr Ti
) 1 0
1
0.24 0.26 0.28 0.30 0.32 0.34 0.36
M 0.5
exp
(
B 2.0
3 3.5 4 4.5 5
(b)
1.5 β = 0.2575(56) 48Ti 1.5 exp
1 exp ) est (3)
1
0.5 M1.0 est (5)
3 3.5 4 4.5 5 B(0.5 50Cr 46Ti
56 48
1.5 β = 0.239(2) 56Fe 0.0 Fe Ti
1 exp 0.06 0.07 0.08 0.09 0.10 0.11 0.12
2
0.5
0 exp
3 3.5 4 4.5 5
FIG. 5: Experimental 1+ energies (top) and B(M1)-values
Energy (MeV) sc
(bottom) as compared to estimations (2)-(3) and (4)-(5) in
46,48Ti[1],50Cr (presentwork) and56Fe[9]. Theexperimen-
FIG.4: B(M1)↑valuesfor1+sc statesof46,48Ti[1]and56Fe[9] tal deformation parameters βexp are taken from [28].
along with the 3.628 MeV state of 50Cr. The experimental
deformation parametersβ [28]aregivenforeachnucleus.
exp
components and the orbital motion is supplemented by
a significant spin admixture (see discussion below). Al-
early [2, 6, 16, 38]
together one may state that our results for the state at
3.628MeVratherwellfitthe1+ systematics,whichjusti-
E = 66δA−1/3MeV, (2) sc
fies the assignment of predominantly scissors-like nature
B(M1) = 0.0042EA5/3δ2g2 µ2 , (3) tothisstate. Inthenextsection,ourtheoreticalanalysis
r N
presentsadditionalargumentsinfavorofthisconclusion.
and modified [39]
E = 13.4p1+(3δ)2A−1/3MeV, (4) B. Theoretical analysis
δ3
B(M1) = 0.66 A4/3g2 µ2 , (5)
1+(3δ)2 eff N The M1 strength in 50Cr was analyzed within the
Skyrme Quasiparticle Random-Phase-Approximation
empirical estimations for 1+ energies and B(M1) val- (QRPA) [26, 27] and a Large-Scale Shell Model (LSSM)
sc
ues. Here, δ = 0.946β is the deformation parameter; [8].
g = 2Z/A is the orbital factor, g = c g with c =0.8
r eff g r g
[39]. The deformation parameter β in Figs. 4-7 is re-
lated to the value B(E2) ↑= B(E2,0+ → 2+) as [40] 1. Skyrme QRPA
1 1
βexp = 3Z4πR2pB(E2)↑/e2 with R0 = 1.2 A1/3fm. As
0
seen below from Fig. 5, estimations (2)-(3) and (4)-(5) Results of the self-consistent Skyrme-QRPA calcula-
give for fp-shell nuclei rather close results. tions are presented in Figs. 6-9 and Tables II and III.
The data are compared to the estimates according to The calculations are performed with a QRPA restricted
Eqs. (2)-(3) and (4)-(5) in Fig. 5. We see in panel (a) to axial symmetry [27] for axially deformed nuclei. The
that the experimental values lie somewhat below the es- Skyrme QRPA method is fully self-consistent since i)
timation but closely follow the linear dependence on the both the mean field and the residual interaction are de-
deformation parameter, which is a strong fingerprint of rivedfrom the same Skyrme functional and ii) the resid-
a scissors-like excitation. Our data on 50Cr well fit this ual interaction includes all the functional contributions
trend. Followingthepanel(b),thecorrespondenceofthe as well as the Coulomb direct and exchange terms. The
experimental and estimated B(M1)-values is also quite Skyrme parameterization SGII [41] is used. The volume
satisfactory. Some deviations from the phenomenologi- δ-force pairing is treated at the BCS level [44]. The pa-
cal estimates can be explained by that in fp-nuclei the rameterofthe equilibriumaxialquadrupoledeformation
states are determined by a few two quasiparticle (2qp) β=0.30isdeterminedbyminimizationofthetotalenergy
6
ofthesystem. Thecalculatedβ isinasatisfactoryagree-
50
ment with the experimental value β =0.2897(44) [28].
exp
The spuriousstrengthis concentratedbelow 2 MeV(not (a) 50 Skyrme QRPA
showninFigs.6,7 and 9) anddoes not noticeablyaffect 25 Cr K=1 proton
the results. Note that applications of the self-consistent
)
Skyrme QRPA to magnetic orbital and spin excitations 2 m 0
3 4 5 6 7 8 9
in deformed nuclei are still very limited and mainly de- 2 f (b) x x
voted to heavy deformed nuclei, see e.g. [42, 43]. To our e 25 T=0
(
knowledge,the presentSkyrme QRPA explorationis the )
2
first one for medium deformed nuclei. E 0
B( (c) 3 4 5 6 7 8 9
25 T=1
2 50 Skyrme QRPA
(a)
Cr K=1 orb+spin
1 0
2 3 4 5 6 7 8 9 10
2 )N 0 E (MeV )
2 3 4 5 6 7 8 9
FIG. 7: Proton (a), isoscalar (b), and isovector (c) E2
) orb
M1 1 (b) strengthsK=1calculated withintheSkyrmeQRPAwiththe
( force SGII. In panel (b), high peaks at 3.9 and 5.5 MeV
B
0 (with B(E2)=116 and 101 e2fm4, respectively) are marked
3 4 5 6 7 8 9 bycrosses.
2
spin
(c)
1
∆K=1 B(E2;Iπ = 0+ → 2+ ) values of the associ-
K 0,gs 1,ν
0 ated2+ statebelongingtothesameband. Thesevalues
2 3 4 5 6 7 8 9 10 1,ν
E (Me V) ewpere=obetnain=ed1w; iatnhdtheepeff=ec−tiveench=ar1g,esreespepffe=cti1v,eelyneff. =Th0e;
FIG.6: Total(a),orbital(b),andspin(c)M1strengthsK=1 eff eff eff eff
B(E2) strengths are exhibited in Fig. 7. For the conve-
calculated within SkyrmeQRPAwith the force SGII.
nience of comparisonwith Fig. 6, the strengths are plot-
ted as a function of the excitation energies E for the
1+
1,ν
QRPA states with K=1 (The energies of (0+ → 2+ )
Figure 6 presents calculated B(M1) values obtained 0,gs 1,ν
transitions can be roughly estimated in the approxima-
for the total(orbital + spin), orbital,and spin M1 tran-
sitions Iπ = 0+ → 1+ from the ground state to the tionthatthemomentofinertiaofK=1bandsisthesame
K 0,gs 1,ν as in the ground-state band. Then the rotational in-
ν-th QRPA state with Kπ = 1+. In the spin part, the
crement (E -E ) to E is ∼522 keV.). Fig. 7
familiar quenching factor of 0.7 is used. The calcula- 2+ 1+ 1+
1,ν 1,ν 1,ν
tions result in a 1+ state at 2.71 MeV with a large M1 shows that, for the band based on the calculated 2.71-
excitation strength of B(M1)↑=1.52 µ2 . By compar- MeV state, we get a negligible B(E2, ∆T=0) but size-
N
ing B(M1) values in the plots a), b) and c), it is easy able B(E2, ∆T=1) value. This clearly emphasizes the
to see that the magnetic strength of this state is pro- isovectorexcitationcharacterofthestate. Moreover,the
duced by a constructive interference of the orbital and B(E2,∆T=1)valuecalculatedforthisstateisthelargest
spin parts. The orbital strength dominates which indi- forallstatesbelow6MeV.Thisobservationpointstothe
catesapredominantlyscissors-likecharacterofthestate. close relation of this state to the underlying quadrupole
The mainly scissors-like nature of this state is also sup- deformation. Bothaspects matchthe isovectornatureof
ported by a satisfactory agreement of its excitation en- thescissors-likemodewhichisusuallyviewedasrotation-
ergyandB(M1)strengthwiththesystematicsforthe1+ like out-of-phaseoscillations of the deformed protonand
sc
as a function of nuclear deformation and mass number. neutron subsystems.
Notethatthecalculated2.71-MeVstateisnotpurelyor- The nature of the 2.71-MeV state is further inspected
bital but has some spin admixture, which is typical [12] in Fig. 8, where distributions (proton, neutron, isoscalar
for light and medium deformed nuclei. Despite a notice- and isovector) of the orbital transition current of this
ableenergydifference,the2.71-MeVstatecorrespondsto state are exhibited. The distributions represent the pro-
the experimentally observedstate at 3.628MeV because ton (p) and neutron (n) transition densities of the con-
both of them deliver experimentally and theoretically a vection current, as well as their sums (∆T=0) and dif-
maximal B(M1) strength in the energy range 2-4 MeV. ferences (∆T=1). For the convenience of the view, the
Thescissors-likecharacterofthecalculated2.71-MeV, currents in all the panels are equally scaled so as to ex-
1+ state can be additionally confirmed by comparison hibit mainly the strong flows. The figure shows that the
of the proton, isoscalar (∆T=0) and isovector (∆T=1), orbital motion is basically represented by rotation-like
7
out-of-phaseoscillationsofprotonsandneutrons. Indeed
TABLE II: The energies E, B(M1) values and two maximal
in the regions of most intense flow (at the left and right
2qp components of particular Iπ QRPA states with K=1
nuclear surface), the protons and neutrons move in the K,ν
and 0. The components are given in Nilsson quantum num-
opposite directions (compare plots a) and b)). In these bers [NnzΛ]. N denotes the contribution of the component
regions, the ∆T=1 motion dominates over the ∆T=0
to thestate norm.
one. Itisremarkablethatthe∆T=1motionismostpro-
E B(M1) p/n 2qp N
nouncedintheregionofthenuclearequatorandweakat
MeV µ2
thenuclearpoles. Thisissurprisingsincethescissors-like N
K=1
picture suggests the opposite. One may suggestthat the
2.71 1.52 n [303]7/2 [312]5/2 69%
orbital motion in 2.71-MeV state actually corresponds
p [312]5/2 [321]3/2 27%
not to the scissors-like flow proposed by N. Lo Iudice
5.01 1.20 n [321]3/2 [321]1/2 54%
and F. Palumbo [13] but to an alternative picture from p [321]3/2 [321]1/2 33%
E. Lipparini and S. Stringari [29], where the isovector 7.70 1.30 n [312]5/2 [301]3/2 40%
motion takes place inside the rigid surface and is maxi- n [321]3/2 [310]1/2 29%
mal in the equator region. These two pictures are analo- K=0
gous to the Goldhaber-Teller [45] and Steinwedel-Jensen 11.0 1.56 p [312]5/2 [303]5/2 32%
[46] treatments of the electric isovector giant dipole res- n [321]3/2 [301]3/2 25%
onance, respectively. Our self-consistent Skyrme-QRPA
resultsmorecloselycorrespondtotheLipparini-Stringari
picture for the low-energy orbital M1 mode. where the M1 strength function
Now let’s briefly discuss the states at E > 4 MeV,
i.e. above the scissors-like mode. As seen from Fig. 6, S(M1;E)=X|h1+K,ν|Mˆ(M1,K)|0i|2ξ∆(E−E1+ )
these states are mainly of a spin nature with only a few ν K,ν
exceptions at ∼4 and 6.5-7.5 MeV. The spin states form (6)
twogroupsat∼5and∼8MeV,thatconstitutealtogether is presented. Here |0i is the ground state wave func-
the K=1 branch (with both K=±1 contributions) of the tion, |1+K,νi and E1+ are QRPA states and energies,
K,ν
M1 spin giant resonance. Mˆ(M1,K) is the M1 transition operator with the pro-
jection K, ξ (E−E )=∆/(2π[(E−E )2−∆2/4]
6 (a) p (b) n is the Loren∆tz smoo1t+Kh,iνng with the averag1in+Kg,νparameter
4 ∆=0.5 MeV.
Figure 9 shows that the spin resonance basically con-
2
) sists of two bumps at ∼5 and ∼8 MeV from the K=1
m
(f 0 branchandthebumpat∼11MeVfromthebranchK=0.
z
-2 Altogether spin excitations fill a broad region from 5-13
MeV,whichwellmatchesaresonanceregioninLSSMcal-
-4
culations, see Fig. 10 below. In our QRPA calculations,
-6 thespinresonanceisnotconcentratedinonebroadbump
6 (c) ∆T=0 (d) ∆T=1 butlookslikeasequenceoffewwellseparatedstructures.
This may be expected from the fact that QRPA omits
4
parts of the spin correlations. As seen from Fig. 10, the
2 inclusionofmorecorrelationsinthe LSSMallowstomix
)
m the separate structures and to form a broad regular res-
f 0
z( onance.
-2 Figure 9a) shows that the interference of the orbital
-4 andspinexcitationsisconstructiveatE<7MeVandde-
structive at higher energies. As seen from Fig. 9b), the
-6
excitations in the energy interval of our interest, E<10
-6 -4 -2 0 2 4 6-6 -4 -2 0 2 4 6 MeV, almost completely belong to the K=1 branch. In
x(fm) x(fm)
the energy interval 4.5-14 MeV where the spin M1 res-
onance is localized, the calculated total M1 strength is
FIG. 8: Proton (a), neutron (b), isoscalar (c), and isovec- ∼5.9 µ2 . This is in accordance with the experimental
tor (d) orbital currents in x-z plane for the 2.71 MeV-state strengthNs of 5-7 µ2 found in 24Mg and 28Si [2]. For the
calculated in the Skyrme QRPA with the force SGII. In all N
interval 2-9.7 MeV covered by the present experiment,
thepanels,thecurrentsareequallyscaledtodemonstratethe
the calculations give a total strength of ∼6.2 µ2 which
most strong flows. The magnitude of the current is deter- N
somewhat overestimates the total observedM1 strength
mined by thearrow length.
∼ 4.01 µ2 .
N
ItisinstructivetoseethestructureoftheQRPAstates
AmoregeneralviewofthisresonanceisgiveninFig.9 providing the dominant contribution to the orbital and
8
4 1.2 50 total
Cr
50
Skyrme QRPA K=0+1 )
3 (a) Cr N (a)
2
µ 0.8
orbital
1 ) 2 spin ) (
- eV 1 total M1
M ( 0.4
B
2 N 0
4 6 8 10 12 14
1) 3 (b) 0
M K=0 1.2
( K=1 orbital
S 2
total )
N
1 2µ 0.8 (b)
(
)
0 1
2 4 6 8 10 12 14 16 M
0.4
E (M eV) B(
FIG. 9: (Color online) Strength functions in 50Cr smoothed
by the Lorentz weight with ∆=0.5 MeV. Panel (a): orbital
0
(red dashed curve), spin (blue solid curve) and total or- 1.2
spin
bital+spin (black bold curve) strengths. Contributions from
alltheprojections(K=0and1)areincluded. Panel(b): K=0 )N (c)
(red dashed curve), K=1 (blue solid curve) and total (black 2
µ 0.8
bold curve) strengths. (
)
1
M
0.4
(
B
spin bumps in Fig. 9. This information is given in Ta-
ble II. The table shows that the calculated 1+ at 2.71
sc 0
MeV is formed by ∆K=1 transitions between the levels
0 2 4 6 8 10 12 14 16 18 20
[303]7/2, [312]5/2 and [321]3/2. They are neighboring
Energy (MeV)
levelsarisingduetothedeformationsplittingofthe1f
7/2
subshell. Thus one encounters here a typical scheme for
orbitalexcitations. Notethatthecollectivityofthe2.71- FIG. 10: Total (a), orbital (b), and spin (c) M1 excitation
MeV state is low. Its two largest components exhaust strengthscalculatedwithinalarge-scale shell-modelwiththe
96% of the norm. KB3G interaction.
Table II also shows the structure of the calculated
statesat5.01,7.70and11.0MeVthatstronglycontribute
to the main parts of the spin M1 resonance in Fig. 9. It tational procedures used in [37], which we follow here.
is seen that these states are rather collective. Their two In the spherical shell model, 50Cr is described in a 0~ω
largest components give altogether only 57-87% of the space, i.e., ten particles are allowed to occupy all the
norm. Some of the components, e.g. [312]5/2-[301]3/2 states available in the fp shell. The KB3G [47] interac-
in the 7.70 MeV-state and [321]3/2-[301]3/2, [312]5/2- tion is used. The single-particle energies are taken from
[303]5/2inthe11.0MeV-statedirectlycorrespondtothe the experimental spectrum of 41Ca.
1f7/2−1f5/2spin-fliptransition. Thestrongdeformation Figure 10 shows B(M1) values obtained for the total
splitting spreadsthestrengthof1f7/2−1f5/2 transitions (orbital+spin),orbital,andspinM1transitions. Inthe
overdifferentQRPAstates. Since the QRPAdescription spin part, a quenching factor of 0.75 [48] is used. The
does not include all correlations, the spin M1 resonance calculations give the state at 3.5 MeV with a sizeable
is yet represented by a small number of states and looks strengthB(M1)=0.826µ2 . Theorbitalstrengthdomi-
N
like a sequence of a few separated structures. The more natesthe 3.5MeVstate,whichindicatesthe scissors-like
comprehensivepicture ofthe spinM1 resonanceis given character of the state. It should be pointed out that in
below in terms of the shell model which includes more theLSSMthe3.5-MeVstateisnotpurelyorbitalbutalso
correlations. has some spin admixture, which appears in the QRPA
calculations. The calculated 3.5-MeV state matches re-
markably well with the experimentally observed state at
2. Large-scale shell model 3628.2 keV.
The LSSM calculations show that M1 transitions
The results of the LSSM calculations are presented above 3.6 MeV are mainly of spin character (see lower
in Figure 10. We start with a reminder of the compu- panelofFigure10). ThecalculatedsummedM1strength
9
Random-Phase-Approximation (QRPA) [26] and the
TABLE III: Calculated B(E2,0+1 → 2+1)↑ value and charac- Large-ScaleShellModel(LSSM)[8]. TheQRPAallowed
teristics of the lowest 1+1,ν=1-state of 50Cr (energy E, total to highlight the basic features and origin of 1+ states
excitation strength B(M1)↑, orbital strength B(M1) , spin
O whiletheLSSMdemonstratedtheimportantroleofcom-
strength B(M1) , and ratio R=B(M1) /B(M1) ) as com-
S O S plex configurations. Both models reproduce the similar
pared tothe present experimental data.
B(E2) values.
Exp. SkyrmeQRPA LSSM Followingourtheoreticalanalysisandcomparisonwith
B(E2)↑e2b2 0.1052(32)a 0.11 0.108 other available results for fp-shell nuclei the lowest 1+-
E [MeV] 3.628 2.71 3.54
state at 3.6 MeV was identified as an isovector orbital
B(M1) [µ2 ] 1.113(49) 1.52 0.826
N mode withsomespinadmixture. Inagreementwithpre-
B(M1) [µ2 ] - 0.60 0.316
O N viousstudiesforfp-shellnuclei[9],aconstructiveinterfer-
B(M1) [µ2 ] - 0.21 0.120
S N ence of the orbital and spin contributions for this state
R - 2.9 2.6
was confirmed. This augments the previous rare data
aFromRef.[28]. (46,48Ti [1] and 56Fe [9]) on this orbital mode in the fp-
shell region. The obtained data match the estimations
and trends for the scissors-like mode in a satisfactory
in the experimentally investigated energy range from 2 - way.
9.7 MeV amounts to ∼5.6 µ2N as compared to the mea- It is also interesting that distributions of the orbital
sured value of 4.1(1) µ2N. The total calculated strength current,computedwithintheSkyrmeQRPAforthelow-
in the resonance, energy interval 4.5 - 14 MeV, is ∼6.6 est1+-state,favortheschematicpictureofLippariniand
µ2N in good correspondence to the QRPA result of 5.9 Stringari [29] (isovector rotation-like oscillations inside
µ2N. the rigidsurface)rather than the scissors-likeview of Lo
InTableIII,theLSSMresultsarecomparedtoQRPA Iudice and Palumbo [13].
ones and present experimental data. The table shows
The spin excitationsabovethe scissors-likemode were
thattheLSSMbetterdescribesthelevelenergythanthe
also inspected. The QRPA calculationconfirms that the
SkyrmeQRPA.Mostprobablythis is because the LSSM spin M1 resonance in 50Cr is provided by spin-flip tran-
takes into account the coupling to complex configura-
sitions (mainly 1f −1f ) inside the fp-shell.
7/2 5/2
tions, omitted in the QRPA.
IV. SUMMARY AND OUTLOOK Acknowledgments
The low-lying M1-strength in the open-shell nucleus The effort of the S-DALINAC crew at TU Darmstadt
50Cr has been determined with the method of nuclear to provide an excellent electron beam is gratefully ac-
resonance fluorescence using bremsstrahlung at the su- knowledged. We also thank the operators team of Duke
perconducting Darmstadt linear electron accelerator S- Free Electron Laser Laboratory for providing excellent
DALINAC and Compton backscattered photons at the beams. This work was supported by the DFG under
High Intensity γ-ray Source (HIγS) facility. Fifteen 1+ grant No. SFB 634 and by the Helmholtz Interna-
stateshavebeenobservedbetween3.6and9.7MeV.Fur- tional Center for FAIR. V.O.N. thanks the DFG RE
ther,14γ-raytransitionsand13B(M1)valueshavebeen 322/14-1, Heisenberg-Landau (Germany-BLTP JINR),
measured for the first time. and Votruba-Blokhintsev (Czech Republic-BLTP JINR)
The experimental results were compared to cal- grants. J.K.and A.R. are gratefulfor the support of the
culations in the self-consistent Skyrme Quasiparticle Czech Science Foundation (P203-13-07117S).
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