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E. Clément, A. Görgen, W. Korten, E. Bouchez, A. Chatillon, J.-P. Delaroche,
M. Girod, H. Goutte, A. Hürstel, Y. Le Coz, et al.
To cite this version:
E. Clément, A. Görgen, W. Korten, E. Bouchez, A. Chatillon, et al.. Shape coexistence in
neutron-deficient krypton isotopes. Physical Review C, American Physical Society, 2007, 75, pp.054313.
�10.1103/PhysRevC.75.054313�. �in2p3-00141102�
E. Clement, A. Gorgen, W. Korten, E. Bou hez, A.Chatillon, J.-P.Delaro he, M.Girod, H.Goutte, A.Hurstel, 1 Y. LeCoz, 1,z A.Obertelli, 1 S.Peru, 2 Ch.Theisen, 1 J.N.Wilson, 1,x M.Zielinska, 1,3 C.Andreoiu, 4,{ F.Be ker, 5 P.A. Butler, 4 J.M.Casandjian, 6, W.N.Catford, 7 T.Czosnyka, 3,yy G.de Fran e, 6 J.Gerl, 5 R.-D.Herzberg, 4 J.Iwani ki, 3,4 D.G. Jenkins, 4,zz G.D. Jones, 4 P.J.Napiorkowski, 3 G.Sletten, 8 andC.Timis 7 1
CEA Sa lay, DAPNIA/SPhN, F-91191 Gif-sur-Yvette, Fran e 2
CEA/DIF, DPTA/SPN, B.P. 12, F-91680 Bruyeres-le-Ch^atel, Fran e 3
Heavy Ion Laboratory, Warsaw University, Warsaw, PL-02097, Poland 4
Oliver Lodge Laboratory, University of Liverpool, Liverpool, L69 7ZE, United Kingdom 5
Gesells haft f ur S hwerionenfors hung, D-64291 Darmstadt, Germany 6
GANIL, BP-5027, F-14076 Caen Cedex, Fran e 7
Department of Physi s, University of Surrey, Guildford, GU2 7XH, United Kingdom 8
Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen , Denmark
(Dated: February2,2007)
Shape oexisten einthelightkryptonisotopeswasstudiedintwolow-energyCoulombex itation
experiments using radioa tive 74
Kr and 76
Kr beams from the SPIRAL fa ility at GANIL. The
ground-state bands in both isotopes were populated up to the 8 +
state via multi-step Coulomb
ex itation,andseveralnon-yraststateswereobserved.Largesetsofmatrixelementswereextra ted
for both nu lei from the observed -ray yields. Diagonal matrix elements were determined by
utilizing the reorientation e e t. Inboth isotopesthe spe tros opi quadrupole moments for the
ground-statebandsandthebandsbasedonex ited0 + 2
statesarefoundtohaveoppositesigns. The
experimental dataare interpretedwithin aphenomenologi al two-bandmixingmodeland
model-independentquadrupoleinvariantsarededu edfortherelevant0 +
statesusingthe ompletesetsof
matrixelementsandtheformalismofquadrupolesumrules. Con gurationmixing al ulationsbased
ontriaxialHartree-Fo k-Bogolyubov al ulationswiththeGognyD1Se e tiveintera tionhavebeen
performedandare omparedbothwiththeexperimentalresultsandwithre ent al ulationsusing
theSkyrmeSLy6e e tiveintera tionandthefullgenerator- oordinatemethodrestri tedtoaxial
shapes.
PACSnumbers: 21.10.Ky,21.60.-n,23.20.Js,25.70.De,27.50+e
I. INTRODUCTION
Theshapeofanatomi nu leusisafundamental
prop-erty re e ting the spatial distribution of the nu leons.
Closed-shell nu lei are always spheri alin their ground
statesin eallorientationsinspa eofthenu leonorbitals
areequallyprobable.Innu leiwithopenshellsthe
o u-pationof ertainshape-drivingorbitalstendstopolarize
thenu leus. To rstorder,whendes ribingthe nu lear
potentialbyasimpleharmoni os illator,thebinding
en-ergyisindependentof thesignoftheelongation
param-eter, and ompressed ellipsoidal (oblate) and elongated
Presentaddress:ISOLDE,CERN,CH-1211Geneva,Switzerland
y
Presentaddress: Gesells haftfurS hwerionenfors hung,D-64291
Darmstadt,Germany z
Present address: DEN/DTN/SMTM,CEA Cadara he, F-13108
Saint-Paul-lez-Duran e,Fran e x
Present address: Institut dePhysiqueNu leaire,IN2P3-CNRS,
F-91406OrsayCedex,Fran e {
Present address: Departmentof Physi s, Universityof Guelph,
Guelph,Ontario,N1G2W1,Canada
Presentaddress:DAPNIA/SAp,F-91191Gif-sur-Yvette,Fran e yy
De eased. zz
Presentaddress:DepartmentofPhysi s,UniversityofYork,
Hes-(prolate)shapesshouldbeequallyprobable[1℄.
Cal ula-tionsperformedwithmorerealisti potentialsshowthat
prolategroundstatesaremu hmoreabundant. This
ten-den yisalso on rmedbyexperimentsshowingastrong
dominan eofprolateground-stateshapes. This
observa-tion anberelatedtotheshellstru tureofnu leiandin
parti ulartothestrengthofthespin-orbittermrelative
totheradialtermof thenu learpotential[2℄. Both
ex-perimentsandtheoryshowthattheprolatedominan eis
parti ularlyevidentin heavynu lei(Z ;N >50), where
theshell stru turehas hangedfrom aharmoni
os illa-tortype to aMayer-Jensen typewith ahigh-j intruder
orbitalinea hmajorshell. Inlighternu lei(Z ;N <40)
prolateand oblate solutionsappear more evenly in the
groundstates.
Manyneutron-de ientnu leiinthemassA=70 80
region, espe ially loseto the N =Z line, havea large
quadrupole deformation in their groundstate. In
addi-tion,oblate and prolateshapesare predi tedto oexist
withinaverysmall energyrangeof afew hundredkeV.
Thisshape oexisten eisduetothe ompetitionoflarge
shell gaps in the parti le level s heme for both oblate
andprolatedeformationatproton/neutronnumbers34,
36,and38. Theneutron-de ientSeandKrisotopesare
hen eideal andidatesforthestudyofshapepolarization
andshape-mixingproperties. Deformedshell-model
have oblate ground states with a prolate on guration
oexistingat lowex itation energy[3℄. The situation is
predi tedtobeinversedforheavierisotopes,wherea
pro-late groundstateis expe tedto oexist withanex ited
oblate on guration.
A rst experimental indi ation of shape oexisten e
in even-evennu lei is the observation of alow-lying0 + 2
state,whi h anbeinterpretedasthegroundstateofa
di erent shape. If both 0 +
statesare (intrinsi ally)
de-formed,onewouldexpe ttworotationalbandsrelatedto
thedi erentshapes. If the on gurations ome losein
energy,thewavefun tionsofstatesofthesamespinand
parity an mix and ause adistortion of the rotational
bands. Shape oexisten ein light kryptonisotopeswas
rst suggested by Pier ey et al. [4℄ in order to explain
theirregularitiesin the ground-statebandsat lowspin.
Ametastablelow-lying0 +
2
state,i.e.ashapeisomer,was
rstreported for 74
Kr[5,6℄. Morere ently,anisomeri
0 + 2
statewasobservedin 72
Kr[7℄,extendingthe
system-ati s to the N = Z line. The ex itation energy of the
0 +
2
states is de reasing from 78
Krto 74
Kr and then
in- reasingagain for 72
Kr. Themeasured strengthsof the
ele tri monopoletransitions 2
(E0), ontheotherhand,
ismaximalfor 74
Kr. Themixingamplitudesofthewave
fun tions were derived from the distortion at low spin
oftheotherwiseregularrotational bands,and were also
foundmaximalfor 74
Kr[7℄. Theseobservationswere
in-terpretedaseviden eforaninversionoftheground-state
deformationwithde reasingneutronnumber: 78
Krand
76
Kr are assumed to be prolate in their ground state,
whileanoblateshapebe omesthegroundstateof 72
Kr.
For 74
Krtheintrinsi statesofoppositedeformationare
assumedtobealmostdegenerate,andthemeasured
dis-pla ementofthetwophysi al0 +
statesismostlydueto
therepulsionofthestronglymixedstates[7℄. Thisshape
oexisten es enariois on lusive,butitisonlybasedon
indire tmeasuresofthenu learshapes. Neitherthesign
ofthedeformationnorthemixingofthelow-spinsstates
havebeendeterminedexperimentally.
Low-energy Coulomb ex itation is a well-established
methodtostudy olle tiveex itationsinnu lei[8℄. When
two nu lei are passing ea h other on lose traje tories,
but without oming into therangeof the nu lear
inter-a tion, olle tive states anbeex ited in a purely
ele -tromagneti pro ess. Sin e the intera tion time is
rel-atively long, several su essiveex itations are possible,
populatingstatesup to relativelyhigh spins. From the
measured(di erential)Coulomb ex itation rossse tion
the orresponding ele tromagneti matrix elements an
be extra ted. Diagonal matrix elements an be
deter-minedandthe(intrinsi )shapeparametersextra tedvia
se ond-orderinterferen etermsintheex itationpro ess.
Sensitivitytothediagonalmatrixelements omesfor
ex-ample from the reorientation e e t, whi h is aused by
se ond-ordertransitionsbetweenthemagneti sub-states
of an ex ited state[9℄. Coulomb ex itation is thus the
onlymethodthat andire tlydistinguishbetween
di er-the shape oexisten e s enario in the light Se and Kr
isotopes.
Untilveryre entlyCoulombex itationexperimentsat
lowenergy were limitedto stable orverylong-lived
nu- lei, both for proje tile or target ex itation. With the
availability of radioa tive ionbeams (RIBs) from ISOL
fa ilities, su h experiments are now starting to be ome
possible for proje tileex itation of short-lived unstable
nu lei. In- ightfa ilities annotdeliverpurelow-energy
radioa tive beams with a eptable emittan e and are
thereforeonly suitable for intermediate-energy, but not
for low-energy Coulomb ex itation experiments. For a
pre ise determination of diagonal matrix elements it is
ne essarytomeasurethedi erentialCoulombex itation
ross se tion (or the ex itation probability) very
a u-rately at safe energies well below the Coulomb barrier
overalargerangeofs atteringangles. Avariationofthe
atomi numberZofthetargetnu leusin reasesthe
sen-sitivityofthemeasurement. TheSPIRALfa ility[10℄at
GANIL (Caen, Fran e) delivers high-quality RIBs, and
espe ially rare gases are produ ed with relatively high
intensities. Two proje tile Coulomb ex itation
exper-iments were performed with beams of
74
Kr and
76 Kr,
respe tively. The highpre isionof thedata allowed
ex-ploiting the reorientation e e t for the rst time with
radioa tivebeams.
II. EXPERIMENTALDETAILS
Theradioa tive 74
Krand
76
Krbeams were produ ed
attheSPIRAL fa ility[10℄at GANILbyfragmentation
of an intense primary beam of stable 78
Kr of 10 12
par-ti lesperse ond at 68:5A MeVonathi k arbon
pro-du tiontarget. The radioa tivespe ies were extra ted,
ionized in an ECR sour e, and post-a elerated in the
K = 265 CIME y lotron to 4:4A MeV for
76
Kr and
4:7A MeV for 74
Kr. The average se ondary beam
in-tensitywas510 5 and10 4 ppsfor 76 Krand 74 Kr,
respe -tively. Duetotheex ellentmassseparationoftheCIME
y lotronisobari ontaminantsarestronglysuppressed;
only in the ase of 74
Kr a small 74
Se ontamination of
1%wasobserved. The
76 Krand 74 Krproje tiles were Coulombex itedon 208 Pbtargetsof0.9and1.0mg/ m 2
thi kness,respe tively. Theprodu tions hemeofthe
ra-dioa tivebeamsand theexperimental set-upare shown
s hemati allyinFig.1.
Bothexperimentsusedthesameexperimentalset-up.
Thes attered proje tiles and there oiling targetnu lei
were dete ted in an annular, highly segmented
double-sidedsili onstripdete tor(DSSD)of300m thi kness.
The dete tor was pla ed 25 mm downstream from the
208
Pbtarget. Thea tiveareawithinnerandouterradii
of11and35mm,respe tively,wassegmentedinto16
on- entri ringsand16azimuthalse tors. Theenergy
reso-lutionofthesili ondete torwassuÆ ienttodistinguish
Ion Source
Injector
78
Kr
CSS1
CSS2
SPIRAL
Production target
ECR Ion Source
68.5 MeV/u
10
12
ions/s
CIME
Cyclotron
74
Kr
10
4
ions/s
4.7 MeV/u
Pb
EXOGAM
DSSD
FIG.1: Produ tions hemeoftheradioa tive 74
Krand 76
Kr
beamsands hemati viewofthedete tionset-up.
theCoulombex itationeventswheneithertheKrorthe
Pbnu leuswasdete ted. The DSSD overeds attering
angles between23.8 Æ
and 54.5 Æ
in the laboratory frame
orrespondingtoa ontinuousrangeofs atteringangles
between24 Æ
and 145
Æ
in the enter-of-massframe. For
thes attering angles overedby theDSSD thedistan e
of losestapproa hdbetweenproje tileandtargetnu lei
always orrespondedto\safe"valuesto ensureapurely
ele tromagneti ex itation,ful llingthe ondition[11℄
d>1:25(A P +A T ) 1=3 +5fm: (1)
Thesegmentationofthesili ondete torallowed
measur-ingthedi erentialCoulombex itation rossse tionasa
fun tion ofs atteringangle. Uns atteredproje tiles left
thetargetareathroughthe entralholeinthedete tor,
redu ingtheradioa tiveba kgroundfrom thebeam.
The rays depopulating the Coulomb-ex ited states
were dete tedin the EXOGAM array[12℄ of large
seg-mentedgermanium loverdete torswithes ape
suppres-sionshields. Ea h loverdete tor omprisesfour
individ-ual germanium rystals, and ea h rystal is ele tri ally
segmented into four longitudinal segments. The array
omprisedsixfull-sizeandonesmaller loverdete torfor
the 76
Kr experiment, and seven large and four smaller
dete tors for the 74
Kr experiment. The dete tors were
pla ed at 90 Æ
and 135
Æ
with respe t to the beam axis,
andthedistan e betweenthefrontfa eof thedete tors
and thetarget was11.2 mfor thelarge and14 m for
the smallerdete tors. TheeÆ ien y for full-energy
ab-sorption of a 1.3 MeV ray was measured to be 12%
duringthe 74
Krexperiment. Eventswerere ordedwhen
at least one raywasdete ted in oin iden e withone
of the ollision partners. The oin iden e requirement
suppressedthe verylargeba kgroundfrom the
radioa -tivebeamalmost ompletely. Thesegmentationofboth
0
200
400
600
800
1000
1200
1400
E
γ
(keV)
1
10
100
1000
counts / keV
2
1
+
→ 0
1
+
4
1
+
→
2
1
+
0
2
+
→
2
1
+
Pb x-rays
2
2
+
→
0
1
+
2
2
+
→
2
1
+
6
1
+
→
4
1
+
8
1
+
→
6
1
+
4
2
+
→
4
1
+
2
3
+
→
0
2
+
(4
2
+
→
2
2
+
)
FIG. 2: Total -ray spe trum in logarithmi s ale after
Coulombex itationofthe4:4 AMeV 76 Krbeamona 208 Pb targetof0.9mg/ m 2
thi knessin oin iden ewitheitherthe
s atteredbeamparti leorre oilingtargetnu leus.
thegermaniumandthesili ondete torsallowedapre ise
determinationoftherelativeanglebetweens atteredKr
proje tilesandtheemitted rays. AfterDoppler
orre -tionaresolutionof8keVwasobtainedfora rayof500
keV.
III. DATAANALYSISANDRESULTS
A. 76
Kr Experiment
The total Doppler orre ted -ray spe trum in
oin- iden ewith eitherthes attered 76
Krproje tiles orthe
re oiling 208
Pb target nu lei is shown in Fig. 2. The
spe trumisvery leanandneitherba kgroundfromthe
radioa tivede ayofthebeamnorfromisobari
ontam-inants ofthebeamare present. Data were olle tedfor
50hourswithase ondarybeamintensityof510 5
pps.
Theground-statebandwasobserveduptothe8 +
state,
populatedin multi-stepCoulombex itation,andseveral
non-yraststateswereex ited. A partial levels hemeof
76
Krispresentedin Fig.3, showingallstatesthat were
in ludedintheCoulombex itationanalysisandall
tran-sitionsthatwereobserved. All statesand transitionsof
thelevels heme in Fig.3had beenobservedpreviously
[13℄.
The0
+ 2
stateat770keV,whi hisa andidatefor
hav-ing a shape di erent from that of the ground state, is
populated and itsde ay to the 2 +
1
stateobserved. The
2 + 3
stateat 1687 keV is feedingthe 0 + 2
state. However,
thistransition of 918keV is notfullyresolvedfrom the
4 + 2 !4 + 1
transitionwith923keV.TheJ=2sequen e
abovethe2 +
2
stateat1222keVhasbeeninterpreted[14℄
asaK =2quasi-gammaband togetherwith aJ =2
sequen e on top of a 3 +
state at 1733 keV, whi h was
notpopulatedin thepresentCoulomb ex itation
exper-iment. The 2
+
FIG.3: Partiallevels hemesof 74
Kr(left) and 76
Kr(right)
showingalltransitionsobservedinthemeasurementandtheir
energies in keV and all states that were in luded in the
Coulombex itationanalysis.
0
30
60
90
120
150
180
θ
c.m.
(deg)
0.01
0.1
1
10
d
σ
/d
Ω
(b/sr)
A
B
C
D
2
1
+
4
1
+
6
1
+
FIG.4: Di erentialCoulombex itation rossse tionto
pop-ulatethe2 + 1 ,4 + 1 ,and6 + 1 statesin 76
Kr. Theangularranges
(inthe enter-of-masssystem) overed by theworkingrings
ofthesegmentedsili ondete torarelabeledA-D.
waspopulatedandpossiblyalsothe4 + 2
state,sothatall
even-spinmembersofthebandhavebeenin ludedinthe
Coulomb ex itationanalysis.
Inordertoextra t matrixelementsfromthe
di eren-tialCoulombex itation ross se tionsand theobserved
-rayyields, the data wasdivided into several sub-sets
orresponding to di erent ranges of s attering angles.
Be ause theinnermost ringsandsomerings in the
en-terofthesegmentedsili ondete torwerenotfun tioning
properly,onlyapartialrangeofs atteringangles overed
by the dete tor wasused in the analysis. This is
illus-trated in Fig.4, whi h shows the rossse tion to
pop-ulate thestates of the ground-stateband asa fun tion
of s attering angle (in the enter-of-mass frame). The
ranges that were used in the analysis are indi ated by
the verti al lines and are given in Table I. The
ex i-TABLEI:Observed -raytransitionsin 76
Krwiththeir
inten-sities(withouteÆ ien y orre tion) forfour di erent ranges
of enter-of-masss atteringangles.
Dataset I i I f E
(keV) Counts Error
A 2 + 1 0 + 1 424 18426 190 [39:5 Æ ;49:0 Æ ℄ 4 + 1 2 + 1 610 1122 50 6 + 1 4 + 1 825 41 9 2 + 2 2 + 1 797 132 16 2 + 2 0 + 1 1221 77 14 0 + 2 2 + 1 346 154 40 B 2 + 1 0 + 1 424 11595 140 [61:4 Æ ;71:8 Æ ℄ 4 + 1 2 + 1 610 2141 61 6 + 1 4 + 1 825 171 18 2 + 2 2 + 1 797 211 18 2 + 2 0 + 1 1221 113 15 0 + 2 2 + 1 346 314 35 C 2 + 1 0 + 1 424 14123 168 [71:0 Æ ;87:7 Æ ℄ 4 + 1 2 + 1 610 3343 82 6 + 1 4 + 1 825 503 31 8 + 1 6 + 1 1019 91 14 2 + 2 2 + 1 797 413 29 2 + 2 0 + 1 1221 203 20 0 + 2 2 + 1 346 706 51 2 + 3 0 + 2 918 a 4 + 2 4 + 1 923 a 81 18 D 2 + 1 0 + 1 424 5924 83 [107:0 Æ ;121:5 Æ ℄ 4 + 1 2 + 1 610 2308 68 6 + 1 4 + 1 825 493 50 8 + 1 6 + 1 1019 115 21 2 + 2 2 + 1 797 251 40 2 + 2 0 + 1 1221 177 34 0 + 2 2 + 1 346 789 75 2 + 3 0 + 2 918 a 4 + 2 4 + 1 923 a 118 23 a
Unresolveddoublet;yieldofthesumofbothtransitions.
the rossse tiontopopulatethe2 + 1
stateislargest, ould
notbemeasured. However,therangeswherethe
higher-lyingstatesarepopulatedwiththehighest rossse tions
are overedbythedete tor. Moreover,thesensitivityto
higher-ordere e ts, whi h allowdetermining the
diago-nalmatrix elements, omes mostly from the large
s at-teringangles. Thedivisionofthetotaldataintothefour
rangesshownin Fig.4wasfoundto beagood
ompro-mise between the largest possible number of data
sub-setsfor di erent angularrangesand the minimumlevel
ofstatisti srequiredtoextra tthe -rayyieldsfromthe
individualspe tra.
The -rayspe traforthefour data setsare shown in
Fig. 5 and the -ray yields extra ted from these
spe -tra are summarized in Table I. As the enter-of-mass
s atteringangleisin reasingforthedatasetsfromAto
D,theimpa tparameterandthedistan eof losest
ap-proa harede reasing. Asa onsequen e,theprobability
topopulatestatesathigherex itationenergyandhigher
0
400
800
1200
E
γ
(keV)
1
10
100
1000
counts / kev
A
4
1
+
→
2
1
+
2
1
+
→ 0
1
+
6
1
+
→
4
1
+
2
2
+
→
2
1
+
0
2
+
→
2
1
+
2
2
+
→
0
1
+
8
1
+
→
6
1
+
4
2
+
→
2
2
+
2
3
+
→
0
2
+
4
2
+
→
4
1
+
400
800
1200
E
γ
(keV)
B
400
800
1200
E
γ
(keV)
C
400
800
1200
E
γ
(keV)
D
FIG.5: Spe traafterCoulombex itation of 76
Kron 208
Pbfor the foursub-setsofdata orrespondingtodi erentranges of
s atteringanglesasde nedinTableI.
40
60
80
100
120
θ
c.m.
(deg)
10
-3
10
-2
10
-1
10
0
I
γ
(I
i
→
I
f
) / I
γ
(2
1
+
→
0
1
+
)
4
1
→ 2
1
0
2
→ 2
1
6
1
→ 4
1
2
2
→ 2
1
8
1
→ 6
1
FIG.6: Intensityofsomeofthe -raytransitionsin 76
Kras
afun tionofs atteringangleinthe enterofmassframe. All
intensitieshavebeennormalizedtothe2 + 1 !0 + 1 transition.
inFig.6,wherethe -rayyieldsareplottedasfun tions
ofthes atteringangleforseveraltransitionsnormalized
to the 2 + 1 ! 0 + 1
transition. The -ray yields show not
only a strong dependen e on the s attering angle, but
thisdependen e alsodi erssigni antlyforthedi erent
transitions, whi h illustrates the sensitivity of the data
tothematrixelements.
B. 74
Kr Experiment
The produ tion rate of 74
Kr with SPIRAL is at the
limit of feasibility for a measurement of spe tros opi
quadrupole moments with the low-energy Coulomb
ex- itation te hniqueutilizing the reorientation e e t. An
averagese ondarybeamintensityof10 4
ppswasa hieved
during the experiment. The lowerbeamintensity
om-paredtothe 76
Krexperimentwaspartly ompensatedby
alongerrunning time of150 hours. Furthermore,the
experimental diÆ ulties on erning the sili on dete tor
ouldberesolvedandalargerrangeofs atteringangles
was overed. Thenumberofgermanium loverdete tors
intheEXOGAMarraywasalsoin reased,resultingina
0
200
400
600
800
1000
1200
1400
E
γ
(keV)
1
10
100
1000
counts / 2 keV
2
1
+
→ 0
1
+
4
1
+
→
2
1
+
2
3
+
→
0
2
+
Pb x-rays
2
2
+
→
0
1
+
2
2
+
→
2
1
+
6
1
+
→
4
1
+
8
1
+
→
6
1
+
0
3
+
→
2
1
+
2
2
+
→
0
2
+
4
2
+
→
2
2
+
74
Se
2
3
+
→
2
1
+
FIG.7: Total -rayspe trumafterCoulombex itationofthe
4:7 AMeV 74 Krbeamona 208 Pbtargetof1.0mg/ m 2
thi k-nessin oin iden ewitheitherthes atteredbeamparti leor
re oilingtargetnu leus.
higherfull-energydete tioneÆ ien y of 12%at a -ray
energyof1.3MeV.Eventhoughthese ondarybeam
in-tensity for 74
Kr was 50 times smaller ompared to the
76
Krexperiment,thelevelofstatisti swasonlyredu ed
by afa tor of ve. The total -rayspe trumin
oin i-den ewith either s attered 74
Krproje tiles orre oiling
target nu lei is shown in Fig. 7. Besides transitions in
74 Kr the 2 + 1 ! 0 + 1 transition of 74 Se is also visible in
thespe trum. Itsstrengtha ountsfor1.2%ofthetotal
beamintensity. Noother ontaminantsofthebeamwere
observed.
Apartiallevels hemeof 74
Krwiththeobserved
tran-sitionsispresentedin Fig.3. As wasthe asefor 76
Kr,
theground-stateband of 74
Krwaspopulatedup tothe
8 +
state. The metastable 0 + 2
state at 508 keV was
in-terpreted as a shape isomer orresponding to a shape
di erentfromthat oftheground state[7℄. Thisstateis
populatedviathe694keVtransitionfrom the2 +
2 state.
Itsde aypro eedsviaanenhan edE0transitiontothe
groundstateandastrongly onvertedE2transitionof52
keVtothe2 + 1
state[5{7℄andisthereforenotobservedin
thisexperiment. The2 +
statealsode aystothe2 +
dire tlytothegroundstate. Thebran hingratioforthe
de ayofthe2 + 2
stateisknownfroman earlier
measure-ment[15℄. A4 + 2
stateis expe tedabovethe2 + 2
stateas
partofarotationalstru ture,buthasnotbeenreported
previously. A new transition is observed at 910 keV in
thespe trumofFig.7,whi hdoesnot orrespondtoany
knowntransitionin 74
Krorneighboringnu leithat ould
potentially ontaminate the beam. Be ause the energy
and,aswill beshown,thematrixelementofthis
transi-tionagreewiththeexpe tedrotationalstate,a4 +
2 state
istentativelypla edat2112keV.Wehavealsoobtained
weak oin iden e data on rming the above
assign-ment, but an alternative interpretation of the 910 keV
peakasa4 + 2 !4 + 1
transition annot be ompletely
ex- luded. In this ase the4 +
2
statewould belo ated 188
keVlower.
Athird 0 +
stateat 1654keVand athird 2 +
stateat
1741 keVex itation energy have been observed after
de ay[15℄. The0 + 3 ! 2 + 1
transition of 1198keVis not
resolvedfrom the2 +
2
!0
+
1
transition of1202 keV. The
largerwidthofthepeakat1200keV,however,showsthat
the0 + 3
statewasalsopopulated. The2 + 3 !0 + 2 transition
of 1233 keV is visible asa shoulderof this peak, and a
very weak transition at 1285 keV is interpreted as the
2 + 3 !2 + 1 transition.
Thefulldatasetfor 74
Krwasdividedintofourranges
of s atteringangles asshown in Table II. The rsttwo
ranges, A and B, orrespond to the dete tion of the
74
Krproje tileinthesili ondete tor,whilethelasttwo
ranges,CandD, orrespondtothedete tionofthe
re oil-ing 208
Pbnu leifrom thetarget. Theindividualspe tra
fromthefour sub-setsofdataareshownin Fig.8. This
division is again a ompromise between the maximum
numberofdatasets andtheminimumlevelofstatisti s
requiredtoextra tthe -rayyields. Adivisionintoonly
tworangesofs atteringanglesimprovestheun ertainty
of the -ray yields, but was found to result in an
in-suÆ ient number of data points to determine the large
numberofmatrixelementsneededtodes ribethedata.
Theyieldsextra tedfromthespe traofFig.8are
sum-marizedinTableII.
Duringthedataanalysisitwasfoundthat thesili on
dete torwasnotfullyalignedwiththebeamaxisin the
experiment. The ount rates are not isotropi ally
dis-tributed overtheazimuthalse tors of the dete tor. By
measuring the Rutherford s attering ross se tion
indi-vidually for the azimuthal se tors and omparing to a
Monte Carlo simulation, the displa ement of the
dete -torwithrespe ttothebeamaxiswasfoundtobe3.0(5)
mm. Thismisalignmentbreaksthe ylindri alsymmetry
of the set-up and introdu es an azimuthal dependen e
ofthes attering angle( ), whi h hadto betakeninto
a ountfortheDoppler orre tionofthe raysandthe
Coulombex itationanalysis. Thesmalloverlapbetween
the ranges of s attering angles B and C (see Table II)
is also due to this misalignment. The Doppler
orre -tionimprovessigni antlywhentakingthedispla ement
76
TABLEII:Observed -raytransitionsin 74
Krwiththeir
in-tensities(withouteÆ ien y orre tion) forthe fourdi erent
rangesofs atteringangles.
Dataset I i I f E
(keV) Counts Error
A 2 + 1 0 + 1 456 4550 200 [24:0 Æ ;54:5 Æ ℄ 4 + 1 2 + 1 558 400 80 6 + 1 4 + 1 768 27 10 8 + 1 6 + 1 967 11 6 2 + 2 2 + 1 746 36 6 0 + 3 2 + 1 1198 a 2 + 2 0 + 1 1202 a 82 10 2 + 2 0 + 2 694 26 5 B 2 + 1 0 + 1 456 2044 100 [54:5 Æ ;73:9 Æ ℄ 4 + 1 2 + 1 558 445 30 6 + 1 4 + 1 768 55 10 8 + 1 6 + 1 967 15 5 2 + 2 2 + 1 746 55 10 0 + 3 2 + 1 1198 a 2 + 2 0 + 1 1202 a 55 15 2 + 2 0 + 2 694 22 5 2 + 3 0 + 2 1233 17 10 C 2 + 1 0 + 1 456 1775 100 [67:1 Æ ;97:3 Æ ℄ 4 + 1 2 + 1 558 630 50 6 + 1 4 + 1 768 140 25 8 + 1 6 + 1 967 35 20 2 + 2 2 + 1 746 103 15 0 + 3 2 + 1 1198 a 2 + 2 0 + 1 1202 a 112 10 2 + 2 0 + 2 694 35 15 2 + 3 0 + 2 1233 25 10 (4 + 2 ) 2 + 2 b 910 8 5 2 + 3 2 + 1 1285 16 5 D 2 + 1 0 + 1 456 1090 100 [97:3 Æ ;144:5 Æ ℄ 4 + 1 2 + 1 558 440 30 6 + 1 4 + 1 768 130 30 8 + 1 6 + 1 967 53 20 2 + 2 2 + 1 746 90 30 0 + 3 2 + 1 1198 a 2 + 2 0 + 1 1202 a 59 15 2 + 2 0 + 2 694 25 15 2 + 3 0 + 2 1233 25 15 (4 + 2 ) 2 + 2 b 910 8 4 2 + 3 2 + 1 1285 12 4 a
Unresolveddoublet;yieldofthesumofbothtransitions.
b Alternativeassignment(4 + 2 )!4 + 1 .
experimentis a hieved. This is shown in Fig. 9, where
thedi erentstepsof theDoppler orre tion pro essare
illustrated.
IV. COULOMBEXCITATIONANALYSIS
A. GOSIA analysis
0
400
800
1200
E
γ
(keV)
1
10
100
1000
counts / 2 keV
A
400
800
1200
E
γ
(keV)
B
400
800
1200
E
γ
(keV)
C
400
800
1200
E
γ
(keV)
D
4
1
+
→
2
1
+
2
1
+
→ 0
1
+
74
Se
4
1
+
→
2
1
+
2
2
+
→
0
2
+
2
2
+
→
2
1
+
6
1
+
→
4
1
+
4
2
+
→
2
2
+
8
1
+
→
6
1
+
0
3
+
→
2
1
+
2
2
+
→
0
1
+
2
3
+
→
0
2
+
2
3
+
→
2
1
+
Pb x-rays
FIG.8: Spe trafromthe 74
Krexperiment orrespondingtothedi erentrangesofs atteringanglesasde nedinTableII.
0
1000
2000
0
10
20
30
20
40
60
counts / 2 keV
600
800
1000
1200
1400
E
γ
(keV)
0
20
40
60
2
2
+
→
0
2
+
0
3
+
→
2
1
+
2
2
+
→
0
1
+
2
2
+
→
2
1
+
6
1
+
→ 4
1
+
2
3
+
→
0
2
+
2
3
+
→
2
1
+
4
2
+
→
2
2
+
8
1
+
→
6
1
+
74
Se
a)
b)
c)
d)
FIG.9: Gamma-rayspe trafromthe 74
Krexperiment
illus-tratingthedatapro essing: Spe truma)hasno onditionon
the parti le dete tor; onlythe radioa tive ba kgroundfrom
the beam and the room is visible. Spe trum b) has a
o-in iden e ondition withs atteredparti les, butnoDoppler
orre tion is applied. Spe trum )is Doppler orre ted
as-suming thatthe sili on dete tor was alignedwiththe beam
axis. Themisalignmentofthesili ondete torwasin ludedin
theDoppler orre tionfor spe trumd),resulting inthe nal
spe trumof Fig. 7(forthefull energyrange andona
loga-rithmi s ale). Allspe traareshownwithoutanyba kground
subtra tion.
2
fun tionis onstru tedfromthemeasured -rayyields
andthose al ulatedfroma ompleteset of
ele tromag-neti matrix elements, both transitional and diagonal,
betweenallknownstatesinvolvedin theex itation
pro- ess (see Fig. 3). As longas the \safe" onditionfrom
eq.1isful lled,theele tromagneti ex itation
probabil-ity anbe al ulatedwithveryhighpre isionin a
semi- lassi alway[9℄. The 208
Pbtargetisassumedtobeinert
to the high ex itation energy of the rst ex ited state
(I
=3 at2.6MeV).
In order to exploit the dependen e of the ex itation
probability on the s attering angle, the data is divided
intoseveralsub-sets orrespondingto di erentrangesof
s attering angles, as des ribed in the previous se tion.
An alternative(or additional) method would be to use
di erenttarget materialsand exploit theZ dependen e
oftheCoulombex itation rossse tion. Duetotheweak
intensitiesoftheradioa tivebeams,thelevelofstatisti s
of the -ray spe tra is limited, and the bins of
s atter-inganglesarerelativelywide. Asthenumberof matrix
elements, i.e. the number of degrees of freedom in the
tting pro ess, is similar to the number of data points
( -ray yields), the tting problem is under-determined.
However, the onvergen e of the t an be improved
by using experimentallyknown spe tros opi data su h
as lifetimes, bran hing and mixing ratios as additional
datapointsin the t. Theexperimental un ertaintiesof
the respe tive spe tros opi information enter into the
2
t of the matrixelements. In experimentswith
sta-blebeams,whi hdonotsu er fromlowstatisti s,these
quantities anbetreatedasfreeparameters,andallthe
spe tros opi information an in prin iple be extra ted
dire tlyfromtheCoulombex itation data.
Inorder to al ulate the -rayyieldsfromthe matrix
elements,thepositionandgeometryofalldete torsand
theirrelativeeÆ ien yhavetobetakenintoa ount. In
viewofthe losegeometryitisalsoimportantto orre t
forthesmalldispla ementofthesili ondete torwith
re-spe ttothebeamaxisinthe 74
Krexperiment. Theexa t
reprodu tion of theexperimental yields requires the
in-tegration over the range of s attering angles as de ned
inTablesIandII,andovertherangeofbombarding
en-ergiesresultingfromtheenergylossoftheproje tiles in
theleadtarget. The -rayintensitiesare orre tedfor
in-ternal onversion. Theangulardistributionofthe rays
istakenintoa ount;itis orre tedforrelativisti e e ts
andtheattenuation aused bythenu leardeorientation
e e tduringre oilinto va uum.
Itisofgreat advantagethatthelevels hemesofboth
74
Kr and
76
transi-elementsthatareimportantintheex itationpro essare
wellde ned. Thenon-observationofatransition,i.e.an
upperlimitofitsintensity,isalsousedinthe tting
pro- ess and has to be reprodu ed by the matrixelements.
Insome asesitispossibletoextra tE2matrixelements
betweenknownstatesevenifnotransitionwasobserved.
Thenu learlevelsare rstgroupedintoband stru tures
andthematrixelementsareinitializedassuminga
rota-tional relationbetweenthem. This servesonlythe
pur-poseof ndingrealisti startingvalues. Noassumptions
on erningtherotationalstru tureofthestatesaremade
duringtheminimizationpro ess.
Theresult of the 2
minimizationis aset of redu ed
matrixelementshI 2
kM(E2)kI 1
ithatreprodu esthe
ex-perimental data. The transitional matrix elements are
relatedtotheredu edE2transitionprobabilitiesas:
B(E2;I 1 !I 2 )= jhI 2 kM(E2)kI 1 ij 2 2I 1 +1 :
Thetransitionalquadrupolemomentintheintrinsi
ref-eren e frame an be expressed in the rotational model
as: eQ t 0 = r 16 5 1 p 2I 1 +1 hI 2 kM(E2)kI 1 i hI 1 K 1 20jI 2 K 2 i :
Note that one has to make assumptions about the K
quantumnumbersoftheinitialand nalstatesinorderto
expressthematrixelementasatransitionalquadrupole
moment,whereastheB(E2)valueismodelindependent.
Thesignsofthetransitionalmatrixelementsarerelative
andgivenoadditionalinformationabouttheshape.
Thediagonalmatrixelementisadire tmeasureofthe
spe tros opi quadrupole moment Q
s
(I) of the statein
the laboratory frame and, using the sign onvention of
AlderandWinther[9℄, anbewrittenas:
Q s = r 16 5 hII20jIIi p 2I+1 hIkM(E2)kIi:
The diagonal matrix elements are related to the stati
quadrupole moments of the nu lear state in the
intrin-si frameandthereforeto the hargedistributionofthe
nu leusinthat state:
eQ s 0 = r 16 5 1 p 2I+1 hIkM(E2)kIi hIK20jIKi :
In order to extra t the stati quadrupole momentfrom
the diagonal matrix element an assumption on the K
valueofthestateisagainneeded. Thesignofthe
diag-onalmatrixelementisrelatedto thetypeofquadrupole
deformation: forK =0anegativevalue orrespondsto
aprolate and apositive valueto an oblatedeformation
intheintrinsi frameofthenu leus.
B. Matrix elements for
76 Kr Inthe aseof 76 Krthe 2
minimizationwasperformed
TABLEIII:RelativeintensitiesI
[13 ,17 ℄andmixing
param-etersÆ[13 ℄formixedE2=M1transitionsin 76 Kr. I i I f E (keV) I Æ(E2=M1) 2 + 2 2 + 1 797 1 :0 0:2(1) 2 + 2 0 + 1 1221 0 :69(4) 2 + 3 0 + 2 918 1 :0 2 + 3 0 + 1 1688 0 :288(10) 2 + 3 2 + 1 1264 0 :212(7) 2 + 3 4 + 1 653 0 :092(3) 2 + 3 2 + 2 467 0 :046(16) 4 + 2 4 + 1 923 1 :0 0:84(5) 4 + 2 2 + 2 736 0 :56(15) 4 + 2 2 + 1 1533 0 :28(8)
andE2=M1mixingratiosfrompreviousworkwereused
as additional data for the t; they are summarized in
TableIII. Thelifetimes of allstatesthat enterintothe
Coulombex itation analysis ex eptthose ofthe2 + 3 and the6 + 2
stateswereexperimentallyknown. Theyare
sum-marizedinTableIV. Totestthe onsisten ybetweenthe
lifetime and Coulomb ex itation measurements, the 2
minimizationwas rstperformedwithoutusing the
life-timesasadditionalinputdata. Theresultinglifetimesof
thestatesin theground-statebandagree withthe
mea-suredvalues from Ref. [18, 19℄ within the experimental
un ertainties. The analysis for 74
Kr, however, revealed
signi antdis repan ies betweenthelifetimes extra ted
fromtheCoulombex itationdataandthosefoundinthe
literature,aswillbedis ussedinthenext hapter. These
in onsisten iespromptedanewlifetimemeasurementfor
severalstatesinboth 74
Krand 76
Krwithimproved
a u-ra y[20℄,theresultsofwhi harealsopresentedinTable
IV.Thelifetimefoundforthe2 + 1
stateislongerthanthat
ofRef. [18℄, while that for the4 + 1
state isshorter. The
results from the Coulomb ex itation experiment are in
betweenthevaluesofthetwomeasurements. The
mea-suredlifetimes were then used as additionalinput data
intheGOSIAanalysis,whi henhan edthesensitivityto
thediagonalmatrixelementsandthetransitionalmatrix
elementsbetweenhigher-lyingstates. Thisenhan ed
sen-sitivityallowedextra tingalifetimealsoforthe2 +
3 state,
whi hwaspreviouslyunknown.
ThetransitionalE2matrixelementsfoundinthe
min-imizationwithGOSIAarepresentedinTableVtogether
withthetransitionalquadrupolemomentsandtheB(E2)
values. ThediagonalmatrixelementsaregiveninTable
VI together with the dedu ed stati and spe tros opi
quadrupole moments. The M1matrix elements forthe
mixed transitions between states of the same spin and
parityareshowntogetherwiththe orrespondingB(M1)
values in Table VII. All matrix elements are treated
equallyin t;theyareshownseparatelyfor larityofthe
presentation. Theresultsare omparedtotheoreti al
al- ulations,whi hwillbedis ussedinChapterV. Someof
TABLE IV: Lifetimes of the relevant states in 76
Kr. The
valuesfromvariousmeasurementsare omparedto are ent
re oil-distan e lifetime measurement [20 ℄ and to the results
from the present Coulomb ex itation experiment with and
without the independentlymeasured lifetimes as additional
inputdatafortheGOSIA tofthematrixelements.
(ps)(GOSIA) I (ps) Ref. (ps)[20 ℄ free onstr. 2 + 1 36.0(10) [18 ℄ 41.5(8) 38.0(22) 41.2(6) 4 + 1 4.9(4) [18 ℄ 3.67(9) 4.4(2) 3.9(1) 6 + 1 0.86(10) [19 ℄ 0.97(29) 0.82(5) 0.76(6) 8 + 1 0.29(3) [19 ℄ 0.25(3) 10 + 1 0.14(2) [19 ℄ 0.15(3) 0 + 2 61.0(80) [17 ℄ 68.3(25) 2 + 2 1.4(2) [13 ℄ 1.6(1) 4 + 2 1.3(4) [14 ℄ 1.3(2) 2 + 3 0.47(5)
someofthe higher-lyingstates ouldnotbeestablished
withinmeaningfulerrorsandareomittedinTablesVand
VI. Neverthelesstheyentered intothe 2
minimization.
ThesameistrueforsomeoftheM1matrixelementsof
thetransitionsbetweenstatesofthesamespin.
Startingvaluesand signsofthematrixelementshave
beensystemati ally hangedintheinitializationofthe t
and alsoduring theminimization pro edure in order to
avoidtrappinginlo al 2
minima. Thesignsofthe
tran-sitionalmatrixelementsarerelative. Multiple
ombina-tionsof signs anresult in exa tlythesame population
of the states. Positive signs were hosen for the
tran-sitional matrix elements in the ground-state band and
other in-bandtransitions. The signsof allother
transi-tional matrixelements aredetermined relativeto these.
Thesignsofthediagonalmatrixelementsareobservables
and annot be hosen. Changing the signof anyof the
diagonalmatrixelementsshowninTableVIresultsina
higher 2
valueofthe t. Inaddition,therelativephases
ofthetransitionalmatrixelementsbetweenthedi erent
0 +
and 2
+
stateswere investigated. Changing the sign
ofa losedloopofthree matrixelements
P 3 (I 1 ;I 2 ;I 3 )= hI 1 kM(E2)kI 2 ihI 2 kM(E2)kI 3 ihI 3 kM(E2)kI 1 i
always resulted in a higher 2
value, showing the
sen-sitivity to thesigns ofthe transitionalmatrixelements.
On theother hand,the signsof thediagonal matrix
el-ementsremained the samewhen the signof P 3
was
in-verted,showingtherobustnessofthe tforthediagonal
matrixelements. As an example,the 2
in reasesfrom
1.7to 13 (afterminimization) when hangingthephase
P 3 (0 + 2 ;2 + 1 ;2 + 2
)frompositivetonegative.Inthat asethe
diagonalmatrixelementsforthe2 + 1 and2 + 2 states hange
from-0.9to-0.6ebandfrom-1.0to-0.5eb,respe tively.
Tofurtherillustratethesensitivityofthe ttothe
di-agonalmatrix elements, the 2
variation is shown asa
fun tion ofthe diagonalmatrixelements forthe2 + ,4 +
-3
-2
-1
0
1
2
< I || M(E2) || I > (eb)
1
1.1
1.2
1.3
1.4
1.5
χ
2
2
3
+
2
1
+
4
1
+
FIG.10: Normalized 2urvesasafun tionofthe diagonal
matrixelementsofthe2 + 1 ,4 + 1 ,and2 + 3 statesin 76 Kr.Onlythe
onematrixelementinquestionwasvariedinthe al ulation
ofthe -rayyieldsinordertoillustratethesensitivityofthe
2
ttothismatrixelement.
and2 +
3
states in Fig.10. In order to nd these 2
val-uesonlytheonediagonalmatrixelementinquestionwas
varied,andthe -rayyieldswere al ulatedfromthe
en-semble ofall matrixelements. The 2
fun tions forthe
di erentstates are normalizedto thesame valuein
or-dertoallowaquantitative omparisonoftheirbehavior.
The
2
urvesresulting from the variation of the
diag-onalmatrixelements for the2 +
1
and 4
+
1
statesare very
narrow,showingthatthe tisverysensitivetothese
ma-trixelements,whi hhave onsequentlyarelativelysmall
un ertainty. The urve for the2 + 3
stateis mu h wider,
so that this matrix element is less well de ned and the
error is larger. It should be noted, however, that this
one-dimensionalvariationof the matrixelements serves
onlythepurposeofillustratingthesensitivityofthe t.
To ndthevaluesanderrorsofthematrixelements,all
matrixelementsarevariedinamulti-dimensional t
in- ludingafullerroranalysis ofthe orrelatedparameters
[22℄.
Both the angular distribution of the rays and the
deorientationofthe nu learalignmenthaveto betaken
intoa ountin the 2
minimization ofthe -rayyields.
Thedeorientationis due to theintera tion betweenthe
re oilingnu leusand u tuatinghyper ne elds reated
by atomi ele trons. The e e t attenuates the angular
distribution of the rays. The omplex deorientation
e e t is treatedin asimplifying phenomenologi al
two-statedeorientationmodel [23, 24℄. Themost important
parametersinthismodelarethespinandlifetimeofthe
state,aswellasitsgyromagneti fa tor. Whilethe
life-timesof the states areeither takenfrom a
omplemen-tarymeasurementor omedire tlyoutoftheGOSIA t,
theg fa torsaremostlyunknown. Inthese asesthe
g-fa tor valuesfrom the general approximation g = Z =A
anbeused, eventhoughthere anbesigni ant
TABLE V: Redu ed E2 matrix elements for in-band and inter-band transitions in 76
Kr. The dedu ed B(E2) values are
omparedtotheoreti alresults.
I 1 I 2
hI2kM(E2)kI1i(eb) Q
t 0
(eb) B(E2;I1!I2)(e
2 b 2 ) Skyrme[21 ℄ Gogny 2 + 1 0 + 1 0 :849 +0:006 0:006 2:69 +0:02 0:02 0.144 +0:002 0:002 0.202 0.117 4 + 1 2 + 1 1 :49 +0:01 0:01 2:94 +0:03 0:03 0.247 +0:006 0:006 0.281 0.234 6 + 1 4 + 1 1 :90 +0:11 0:03 2:98 +0:17 0:06 0.28 +0:03 0:01 0.39 0.30 8 + 1 6 + 1 2 :25 +0:16 0:10 3:02 +0:22 0:14 0.30 +0:05 0:03 0.44 10 + 1 8 + 1 2 :19 +0:22 0:14 2:60 +0:26 0:15 0.23 +0:05 0:03 2 + 3 0 + 2 0 :87 +0:04 0:02 2:77 +0:12 0:07 0.15 +0:01 0:01 0.04 0.08 4 + 2 2 + 2 0 :89 +0:10 0:13 1:77 +0:20 0:26 0.09 +0:02 0:03 0.11 2 + 3 0 + 1 0 :121 +0:004 0:005 0.0029 +0:0002 0:0002 0.0033 0.0002 2 + 2 0 + 1 0 :183 +0:008 0:006 0.0067 +0:0005 0:0004 0.0010 0 + 2 2 + 1 0:490 +0:011 0:008 0.241 +0:011 0:009 0.0001 0.234 2 + 3 2 + 1 0:200 +0:009 0:008 0.0080 +0:0007 0:0007 0.0829 0.00006 2 + 2 2 + 1 0:09 +0:04 0:04 0.002 +0:002 0:002 0.148 4 + 2 2 + 1 0 :09 +0:01 0:19 0.0010 +0:0003 0:0010 0.0015 2 + 3 4 + 1 0 :52 +0:05 0:05 0.055 +0:012 0:010 0.095 0.049 2 + 2 4 + 1 0:62 +0:04 0:05 0.079 +0:014 0:014 0.023 4 + 2 4 + 1 0 :43 +0:03 0:03 0.021 +0:003 0:003 0.076 2 + 2 0 + 2 1 :22 +0:08 0:04 0.30 +0:04 0:02 0.025 2 + 3 2 + 2 0 :81 +0:10 0:24 0.13 +0:03 0:07 0.04
TABLEVI:Diagonal matrixelementsfor 76
Kr. Thededu edintrinsi andspe tros opi quadrupolemomentsare ompared
totheoreti alvalues. I hIkM(E2)kIi(eb) Q s 0 (eb) Q s
(eb) Skyrme[21 ℄ Gogny
2 + 1 -0.9 +0:3 0:3 2.5 +0:8 0:8 -0.7 +0:2 0:2 -0.78 -0.50 4 + 1 -2.3 +0:4 0:4 4.7 +0:8 0:8 -1.7 +0:3 0:3 -1.25 -0.85 6 + 1 -2.9 +0:4 0:4 5.1 +0:7 0:7 -2.0 +0:3 0:3 -1.44 -1.01 2 + 3 1.3 +0:5 0:5 -3.4 +1:3 1:3 1.0 +0:4 0:4 0.25 0.04 2 + 2 -1.0 +0:5 0:5 -2.6 +1:3 1:3 a -0.7 +0:3 0:3 0.26 a
assumingK=2,oppositesignin aseofK=0
TABLE VII: Redu ed M1 matrix elements between states
of the same spinand parity and the orresponding B(M1)
values. I 1 I 2 hI2kM(M1)kI1i(N) B(M1;I1!I2)( 2 N ) 2 + 2 2 + 1 -0.42(1) 0.035(2) 4 + 2 4 + 1 -0.39(3) 0.017(3)
state. The gfa tor ofthe2 +
1
statein 76
Krwasre ently
measuredtobeg=+0:37(11)[25℄. Thein uen eoftheg
fa torandthedeorientatione e tonthematrixelements
wasinvestigatedby omparingtheresultsobtainedusing
themeasuredgfa torwiththoseusingg =Z =A=0:47,
orignoringthedeorientatione e tentirely. Nodi eren e
wasfoundfor thetransitional matrixelementswhen
in- luding the deorientation e e t ornot, and when using
Z =Aorthemeasuredvalue. Thediagonalmatrixelement
ofthe2 +
1
statedi eredby8%whenthedeorientationwas
notin luded in the al ulation. Thedi eren e between
using g = 0:37 and g = 0:47, however, was negligible.
Consequently, theapproximationg =Z =A wasused to
modelthedeorientatione e t forthehigher-lyingstates in
76
Krandforallstatesin 74
Kr.
The fa t that the 2 + 3
! 0
+ 2
transition with 918 keV
annot be fully resolved from the 4 + 2 ! 4 + 1 transition
with923 keV requires parti ular attentionfor the
eval-uationof the respe tive matrixelements. It is possible
toin lude su h doubletsin theGOSIA analysisbyonly
xingthesumofboth -rayintensitiesratherthanusing
the individual ones. This results ne essarily in less
a - uratevaluesfor the matrixelements. Thetransitional
matrixelements that are foundwithout any further
as-sumptions about the doublet show that the transition
strengthforthe2 + 3
!0
+ 2
transitionismu hhigherthan
forthe4 + 2 !4 + 1
transition. Inadditiontothat,the
pop-ulation of the 4 +
2
state is expe ted to be mu h weaker
thanforthe2 +
3
state,be auseitrequiresatleasta
two-stepex itation,while therather olle tive2 + 3
stateat a
similarex itation energy anbe rea hed in one step. If
the 4 + 2
state were populated, not only its de ay to the
4 +
1
state,butalsothe736keVtransitiontothe2 +
2 state
should beobserved. This bran hing ratio wasreported
tobe0.56[13℄ and 0.82[19℄. Sin ethere is no
onvin -ingeviden einthespe trafora736keVtransition(see
Figs. 2 and 5) it an be assumed that the 4 +
only weakly populated,if at all, andthat the ontribu-tionof the 4 + 2 !4 + 1
transition to the peak observedat
920 keV an be negle ted. This assumption improves
the a ura y for both the transitional matrix elements
oftherespe tivetransitionsandfor thediagonalmatrix
element of the 2 + 3
state. Followingthis argumentation,
node ayfromthe4 + 2
stateisobserved,and onsequently
thediagonalmatrixelementforthis state annotbe
de-termined.
Thematrix elements of Table V aregrouped into
in-band transitions in the upper part of the table and
inter-band transitions in the lower part. The in-band
transitions omprise all transitions of the ground-state
band and the 2
+ 3 ! 0 + 2 and 4 + 2 ! 2 + 2 transitions (see
Fig. 3). All in-band transitions have large matrix
el-ements, i.e. they are olle tive, as expe ted for
well-deformed rotational bands. The dedu ed B(E2)
val-ues are fully ompatible with those extra ted from the
measuredlifetimes[19,20℄. Thedropofthetransitional
quadrupolemomentsQ
t 0
inthelowerpartofthe
ground-stateband hasbeen interpretedas asign formixing of
prolate and oblate on gurationsin thelow-spin states
[20℄. This interpretation is further supported by the
large matrix elements for some of the inter-band
tran-sitions. The fa t that the0 +
2
state isstrongly linkedto
allthreeobserved2 +
states,ontheotherhand,indi ates
thatagroupingofthestatesintorotationalbandsisnot
straightforward and that the situation might be more
omplexthanthedes riptionwithtworotationalbands
builtonstronglydeformed prolateandoblate states
to-getherwith aquasi-gamma band. Thetransitional
ma-trixelementsallowtodeterminepreviouslyunknown
life-times,forexampleforthe2 + 3
state,ortoimprovethe
pre- isionofpreviouslyknownlifetimes, asissummarizedin
TableIV.
The diagonal matrix elements for the three lowest
statesof theground-stateband and forthe twoex ited
2 +
statesaregiveninTableVItogetherwiththededu ed
stati andspe tros opi quadrupolemoments. The
nega-tivesignofthediagonalmatrixelementsforthestatesin
the ground-state band provestheir prolateshape. The
absolute size of the quadrupole moments de reases
to-wardsthebottom oftheband. Thisshowsthatthe
de-formationforthe2 +
1
stateisindeedsmallerthanforthe
states above. In addition, the values of the stati and
transitionalquadrupolemomentsforthisstatearerather
similar, asexpe tedforarotationalnu leus. Verylarge
stati quadrupolemomentsarefoundforthehigher-spin
states in the ground-state band, whi h di er from the
transitionalquadrupolemoments. Thismightbedueto
apossible ouplingtootherunknownstates,whi h ould
notbetakenintoa ountin theanalysis.
The diagonal matrix element of the 2 + 3
state has a
relatively large un ertainty, mainly be ause no
transi-tionabovethisstatewasobserved. Neverthelessthereis
nodoubt aboutthepositivesignofthematrixelement,
whi h is onsistent with the assumption of an
oblate-TABLEVIII:Bran hingratiosforseveraltransitionsin 74
Kr
measuredafter de ayof 74 Rb[15 ℄. I i I f E (keV) I 2 + 2 0 + 1 1202 1 :0 2 + 2 2 + 1 746 0 :73(58) 2 + 2 0 + 2 694 0 :30(35) 2 + 3 0 + 2 1233 1 :0 2 + 3 2 + 1 1285 0 :31(21) 0 + 2 0 + 1 508 1 :0 0 + 2 2 + 1 52 1 :50(36) 0 + 2 2 + 1 52 1 :2(5) a a
Bran hingratiofromRef.[7℄
the0 + 2
statefurther supports theassumption ofa
rota-tional hara ter of the state, so that the asso iation of
the2 + 3
statewithanoblateshapeseemswelljusti ed.
The negative sign of the matrix element for the 2 + 2
stateis morediÆ ultto understand. In aseofa
quasi-gammaband (withK =2), thequadrupole moment in
the body- xed frame Q
s 0
be omes negative, indi ating
an oblate shape, in ontradi tion to the assumption of
a gamma vibration based on the prolate ground state.
Indeed, the oupling of the 2 + 2 state to the 0 + 2 state is
mu h strongerthan thatto thegroundstate. Ifone
as-sumesK =0for the2 + 2
state, the quadrupole moment
Q s 0
be omespositive,indi atingaprolateshape,inwhi h
asetheinterpretationofaquasi-gammabandisalso
ex- luded, besidesthefa t that thereshould beathird 0 +
stateforwhi h there isnoeviden e. Aswasargued
be-fore,thestrong ouplingbetweenallthreebands andin
parti ularbetweenthe 0 + 2
state andall three 2 +
states
blurs a lear separation and grouping into band
stru -tures,aswill bedis ussedin moredetailin se tionV.
C. Matrix elementsfor
74 Kr
The
2
minimization of the -rayyieldsfor 74
Krwas
performed with 31 E2 and 5 M1 matrixelements
on-ne tingthe knownstatesasshown in Fig.3. As in the
aseof 76
Kr,experimentallyknownbran hingratiosand
lifetimeswereusedasadditionalinputdatainthe tting
pro edure.Thebran hingratiosthatwerere ently
mea-suredafter de ayof 74
Rb[15℄aresummarizedinTable
VIII. Thede ayofthe0 + 2
staterepresentsaspe ial ase
be ausetheE0bran htothegroundstatepro eeds
ex- lusively via onversionele tronsand the E2 bran h to
the2 +
1
statehasverylowenergyandis onsequentlyalso
highly onverted. Eventhough theCoulomb ex itation
experimentwasnotsensitiveto onversionele trons,the
de ay of this state with its bran hing ratio and partial
lifetimes[7℄ anneverthelessbeusedto onstrainthe t.
Nomixing ratiosareknownin 74
Kr.
Lifetimes in 74
Kr have been measured for the 2
+ 1
and4 + 1
TABLEIX:Lifetimesofstatesin 74
Kr. Thevaluesfrom
vari-ousmeasurementsare omparedtoanewre oil-distan e
life-timemeasurement[20 ℄ andto theresultsfromthe Coulomb
ex itation experiment with and without the lifetimes from
Ref.[20 ℄as omplementaryinputdata.
(ps)(GOSIA) I (ps) Ref. (ps)[20℄ free onstr. 2 + 1 23.5(20) [26 ℄ 33.8(6) 29.6(2.1) 33.8(6) 4 + 1 13.2(7) [27 ℄ 5.2(2) 5.9(5) 5.3(2) 6 + 1 1.08(14) [29 ℄ 1.09(23) 1.4(5) 1.01(9) 8 + 1 0.35(5) [29 ℄ 0.32(6) 10 + 1 0.16(3) [29 ℄ 0.16(3) 0 + 2 33.8(50)10 3a [7℄ 36.2(43)10 3a 2 + 2 2.0(2) 0 + 3 0.09(2) a a
PartiallifetimeofE2bran htothe2 + 1
state.
higher-lyingstatesoftheground-stateband[26,28,29℄.
Thelifetimeoftheisomeri 0 +
2
statewasestablishedina
onversion-ele tronmeasurement[7℄. Thelifetimevalues
aresummarizedintable IX.
Before using the literature values for the lifetimes as
omplementaryinputinthe ttingpro edure,their
on-sisten ywiththeCoulombex itationdatawasexamined.
Theresults,alsogiveninTableIX,showastrong
devia-tionfrom thevaluesreportedin Ref.[26,27℄. Espe ially
the lifetime of the 4 + 1
state is found to be signi antly
shorter than previously reported. On the other hand,
usingthelifetimesofthe2 +
1 and4
+
1
statesintheGOSIA
tinordertoenhan ethesensitivitytothediagonal
ma-trixelementsisevenmoreimportantin 74
Krthanitwas
in 76
Kr,asthelevelofstatisti sismorelimited.
The in onsisten y between the earlier lifetime
mea-surementsandthe Coulomb ex itation data anbe
fur-ther investigated by omparing the experimental -ray
intensities as a fun tion of s attering angle with those
al ulatedfrom the matrixelementsobtainedusing the
lifetimes. This is shown in Fig.11 for the 2 + 1 !0 + 1 and 4 + 1 !2 + 1
transitions. The data points represent the
ob-served -rayintensitiesnormalizedbythenumberof
s at-tered proje tiles for several angularranges. The
un er-taintiesarelargeforsmallanglesbe auseofdire tbeam
hitting the innermost dete tor rings. The dashed lines
showthe -rayyields al ulatedfromthefullset of
ma-trixelementsobtainedina tthatwas onstrainedbythe
lifetimes reportedbyTaboretal.forthe2 + 1
[26℄andby
Rothetal.forthe4 +
1
state[27℄. Therangeofs attering
angles for this omparison was limited to small values
m
< 70 Æ
, where higher-order e e ts and in
parti u-larthein uen e ofthediagonalmatrixelementsare
ex-pe tedtobesmall,sothatthis omparisonof -rayyields
representsarelatively leantestoftheB(E2) strengths
and of the lifetimes of the respe tive states. The fa t
that the -ray yields annot be reprodu ed
simultane-ouslyforbothtransitionsshowsthat theCoulomb
ex i-20
30
40
50
60
70
θ
c.m.
(deg)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
N
γ
/ N
Kr
2
1
+
4
1
+
old τ
new τ
FIG.11: Gamma-rayintensitiesnormalizedbythenumberof
s atteredproje tilesasafun tionofthe enter-of-mass
s at-tering angle for the 2 + 1 and 4 + 1 statesin 74 Kr. The dashed
lines show the -ray yields al ulated from the matrix
ele-mentsthat wereobtained using the lifetimes of the 2 + 1 [26 ℄ and4 + 1
states[27 ℄intheGOSIA t. Usingthelifetimesfrom
Ref.[20 ℄insteadresultsinthefulllines.
inRef.[26, 27℄.
Thisin onsisten y, whi h makesthedetermination of
thediagonal matrix elements diÆ ult, prompted anew
re oil-distan elifetime experimentwith mu h improved
pre ision [20℄. The new measurement found slightly
longer and shorter lifetimes for the 2 + 1 and 4 + 1 states,
respe tively,thanexpe tedfromtheCoulombex itation
data,butespe iallytheresultforthe4 +
1
stateisdeviating
signi antlyfromthevaluereportedbyRothet al.[27℄.
Possible reasonsforthis are dis ussedin Ref. [20℄. The
new results are ompared to the earlier measurements
andto the resultsfrom the Coulomb ex itation data in
Table IX. When the lifetimes of the new measurement
areusedintheGOSIA t,thein onsisten iesdisappear.
The -rayyields al ulatedfromthere-evaluatedmatrix
elementsbasedonthenewlifetime resultsareshownas
fulllinesinFig.11. Eventhoughthenewlifetimes have
small un ertaintiesand leave verylittle roomfor
varia-tionsofthe al ulated -rayyieldsintheshownangular
range, the agreement is ex ellent, whi h shows the full
onsisten ybetweenthere oil-distan eandtheCoulomb
ex itationmeasurements.
Using the new pre ise lifetimes in the GOSIA
anal-ysis enhan es the sensitivity to the higher-order e e ts
signi antly, so that many more transitional matrix
el-ementsbetweenhigher-lyingstatesandseveraldiagonal
matrix elements ould be determined. The results for
thetransitionalmatrix elements aresummarized in
Ta-ble X; those for the diagonal matrix elements and the
stati quadrupole moments an be found in Table XI.
Thein onsisten ywiththepreviouslyreportedlifetimes
illustratesageneraldiÆ ulty that anarisein Coulomb
ex itation experiments with weak radioa tive beams of
deter--3
-2
-1
0
1
2
< I || M(E2) || I > (eb)
1
1.1
1.2
1.3
1.4
1.5
χ
2
2
2
+
2
1
+
4
1
+
FIG.12: Normalized 2urvesasafun tionofthediagonal
matrixelementsofthe2 + 1 ,4 + 1 ,and2 + 2 statesin 74 Kr. Onlythe
onematrixelementinquestion wasvariedinthe al ulation
ofthe -rayyieldsinordertoillustratethesensitivityofthe
2
ttothismatrixelement.
alone,andwhereatthesametimethea essto
spe tro-s opi informationwith omplementarymethodsis
diÆ- ult.
ToillustratethesensitivityoftheCoulombex itation
analysis to the diagonalmatrix elements, 2 urves for the2 + 1 ,4 + 1 ,and 2 + 2
statesareshown in Fig.12as
fun -tions of the diagonal matrix elements of the respe tive
states. The urves are normalized in exa tly the same
wayasin Fig.10 toallow adire t omparison. The 2
minimaarewiderthaninthe 76
Kr ase,whi hismostly
due to the lower level of statisti s in the 74
Kr
experi-ment. Againitshouldbenotedthatthese urves annot
beusedforanerroranalysis,asonlyonematrixelement
wasvariedatatime,whiletheun ertaintiesgivenin
ta-bles X and XI are based onthe simultaneous t of the
orrelatedmatrixelements.
The high level of statisti s for the transitions in the
ground-statebandallowsextra tingthe -rayintensities
for smallerangular binsthan the four rangesthat were
usedin theGOSIA analysis(see TableII). The
intensi-tiesofthe4 + 1 !2 + 1 andthe6 + 1 !4 + 1
transitionsareshown
intheupperpartofFig.13asfun tionsofthes attering
angle. Theintensitiesarenormalizedtothe2 + 1 !0 + 1
tran-sitioninordertominimizethesystemati errorsfromthe
eÆ ien iesof thegermaniumand sili ondete tors. The
full lines orrespond to the -ray intensities al ulated
fromthe ompletesetofmatrixelementsfoundinthe 2
minimization (see Tables X and XI). Even though the
matrixelements wereextra ted fromthe -rayyieldsof
only four angular ranges, the agreementwith the more
detailedanalysisisremarkableandshowsthe onsisten y
of the analysis. The dashed line results from inverting
thesignofthediagonalmatrixelementsforthestatesof
theground-stateband. Theinversionof thesignsleads
toadisagreementespe iallyforthelarge enter-of-mass
Center-of-Mass Scattering Angle (deg)
20
40
60
80
100
120
140
Intensity
0
0.1
0.2
0.3
0.4
0.5
0.6
)
+
1
0
→
+
1
(2
γ
I
)
+
1
2
→
+
1
(4
γ
I
)
+
1
0
→
+
1
(2
γ
I
)
+
1
4
→
+
1
(6
γ
I
prolate
oblate
Center-of-Mass Scattering Angle (deg)
20
40
60
80
100
120
140
Intensity
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
)
+
1
0
→
+
1
(2
γ
I
)
+
1
2
→
+
2
(2
γ
I
FIG.13: Gamma-rayyields asafun tionofs atteringangle
for the4 + 1 !2 + 1 ,6 + 1 !4 + 1 (top)and2 + 2 !2 + 1 (bottom) transi-tionsin 74 Krnormalizedtothe2 + 1 !0 + 1
transition. Thelines
representthe orrespondingyieldsbasedonthefullsetof
ma-trixelements. The dashedline in theuppergraphresulted
fromaninversionofthesignsofthediagonalmatrixelements
forthestatesintheground-stateband.
loseapproa h of proje tile and target, while the small
s attering angles are mostly sensitive to the rst-order
ex itationpro ess,i.e. tothe B(E2) values, aswas
dis- ussedin the ontextofFig.11. Thedi eren ebetween
the yields al ulated for di erent signs of the diagonal
matrixelementsillustratesthesensitivityofthe
measure-mentto the reorientation e e t. Again, this annot be
used for a quantitative analysis, be ause only some
se-le ted matrix elements were arbitrarily hanged, while
both the transitional and thediagonal matrix elements
ofthenon-yraststateswerekeptun hanged. These ond
graphshowsthe equivalent plot for the 2 + 2 !2 + 1
transi-tion,alsonormalizedto the2 + 1 !0 + 1 transition. Be ause
theyield issmaller forthis transition, the intensity has
to be divided into fewer and wider angular bins. The
agreementwiththe yield al ulatedfrom thematrix
el-ementsisyetanotherexampleforthe onsisten yofthe
analysis.
ThetransitionalmatrixelementsinTableXareagain
