Fermi's golden rule applied to the gamma decay in the quasicontinuum of 46Ti

0.53 MB·English

by  M. Guttormsen

Fermi’sgoldenruleappliedtotheγ decayinthequasicontinuumof46Ti
M.Guttormsen,1,∗ A.C.Larsen,1 A.Bu¨rger,1 A.Go¨rgen,1 S.Harissopulos,2 M.Kmiecik,3 T.Konstantinopoulos,2
M.Krtic˘ka,4 A.Lagoyannis,2 T.Lo¨nnroth,5 K.Mazurek,3 M.Norrby,5 H.T.Nyhus,1 G.Perdikakis†,2
A. Schiller,6 S. Siem,1 A. Spyrou†,2 N.U.H. Syed,1 H.K. Toft,1 G.M. Tveten,1 and A. Voinov6
1Department of Physics, University of Oslo, N-0316 Oslo, Norway
2InstituteofNuclearPhysics, NCSR”Demokritos”, 153.10AghiaParaskevi, Athens, Greece
3Institute of Nuclear Physics PAN, Krako´w, Poland
4Institute of Particle and Nuclear Physics, Charles University, Prague, Czech Republic
1 5Department of Physics, A˚bo Akademi University, FIN-20500 A˚bo, Finland
1 6Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA
0 (Dated:January21,2011)
2
Particle-γ coincidencesfromthe46Ti(p,p(cid:48)γ)46Tiinelasticscatteringreactionwith15-MeVprotonsareuti-
n
lizedtoobtainγ-rayspectraasafunctionofexcitationenergy. Therichdatasetallowsanalysisofthecoinci-
a
dencedatawithvariousgatesonexcitationenergy.Formanyindependentdatasets,thisenablesasimultaneous
J
extractionofleveldensityandradiativestrengthfunction(RSF).Theresultsareconsistentwithonecommon
0
leveldensity. ThedataseemtoexhibitauniversalRSFasthededucedRSFsfromdifferentexcitationenergies
2
show only small fluctuations provided that only excitation energies above 3 MeV are taken into account. If

transitionstowell-separatedlow-energylevelsareincluded,thededucedRSFmaychangebyafactorof2−3,
]
x whichmightbeexpectedbecausetheinvolvedPorter-Thomasfluctuations.
e
- PACSnumbers:21.10.Ma,25.20.Lj,27.40.+z,25.40.Hs
l
c
u
n I. INTRODUCTION tainedfromthestudyofphotonuclearcross-sections[4],and
[ thuslimitedonlytoenergiesabovetheparticleseparationen-

ergy.Itwasestablishedthatthegiantelectricdipoleresonance
2 Fermi’s golden rule predicts the transition rate from one
v state to a set of final states in a quantum system. The the- (GEDR)dominatestheRSFinallnuclei. Theinformationon
9 RSF below particle separation energy is significantly less. It
oretical foundation, which has been successfully applied in
0 hasbeenobtainedmainlyfromtheOslomethodand(n,γ)and
manydisciplinesofphysics,wasfirstestablishedbyDiracin
4 (γ,γ(cid:48))reactions.
1927 [1] and emphasized by Fermi in his book in 1950 [2].
3
The Oslo method, which is applicable for excitation ener-
. The rule is based on first-order perturbation theory, where
9 gies below the particle separation, is described in detail in
the transition matrix element is assumed to be small. In nu-
0 Ref. [5]. In this work, we report on results obtained for
clearphysics,thisassumptioniswellfulfilledforβ andγ de-
0 the 46Ti nucleus, which has two protons and four neutrons
1 cay. Thus, wetakethevalidityoftheFermi’sgoldenruleas
outside the doubly-magic 40Ca core. The advantages of the
: granted, rather than testing this rule. In this work, we study
v theγ decaybetweenstatesinthequasi-continuumof46Tiand (p,p(cid:48)) reaction compared to the commonly used (3He,3He(cid:48))
i and (3He,4He) reactions are much higher cross-sections and
X apply Fermi’s golden rule to disentangle the γ strength and
better particle resolution. This allows us to make a detailed
r leveldensity.
a studyofγ-decayasafunctionofexcitationenergy.
TheOslonuclearphysicsgrouphasdevelopedamethodto
It can be discussed if the concept of one unique RSF is
determine simultaneously the level density and the radiative
validforlightnucleiandinparticularatlowexcitationener-
strength function (RSF) from particle-γ coincidences. These
gies. ThisquestiontogetherwiththeapplicabilityoftheOslo
quantitiesprovideinformationontheaveragepropertiesofex-
methodforlightsystemsastitaniumisthemainsubjectofthis
citednucleiandareindispensibleinnuclearreactiontheories
work.
astheyaretheonlyquantitiesneededforcompletedescription
In Sec. II the experimental results are described. The nu-
oftheγ decayathigherexcitationenergies.
clear level density and RSF are extracted in Sec. III, and in
Thenuclearleveldensitycanbedeterminedreliablyuptoa
Sec.IV,theapplicationofFermi’sgoldenruleandtheBrink
fewMeVofexcitationenergyfromthecountingoflow-lying
hypothesisisdiscussed. Summaryandconclusionsaregiven
discreteknownlevels[3]. Thereisalsoreliablelevel-density
inSec.V.
informationfromneutronresonancesathigherexcitationen-
ergies;however,thisinformationisrestrictedinenergyaswell
asspinrange.
II. EXPERIMENTALRESULTS
ThemostrichexperimentalinformationontheRSFwasob-
TheexperimentwasconductedattheOsloCyclotronLabo-
ratory(OCL)witha15-MeVprotonbeambombardingaself-
supporting target of 46Ti with thickness of 1.8 mg/cm2. The
†Currentaddress:NationalSuperconductingCyclotronLaboratory,Michigan
targetwasenrichedto86%46Tiwith48Ti(11%)asthemain
StateUniversity,EastLansing,MI48824-1321,USA.
∗Electronicaddress:[email protected] impurity. Thissmalladmixtureof48Tiisnotexpectedtoplay
2
· 103 )
V
V 2500 (a) Singles 46Ti 2+ 0+ (ke10000
9.3 ke 12500000 4+ E i 110044
3
s / 8000
nt 1000 0+
u
o
C 500
6000 110033
0
· 103
1400 (b) Coincidences 2+ 4000
V 1200
e
k
3 1000 110022
9.
s / 3 680000 2000
ount 400 4+
C 0+
200 0
2000 4000 6000 8000 10000
0
4 6 8 10 12 14 E (keV)
g
Proton energy E (MeV)
FIG. 2: (Color online) The particle-γ coincidence matrix for 46Ti.
FIG.1:(Coloronline)Singles(a)andcoincidence(b)protonspectra
recordedwith15-MeVprotonson46Ti. The γ-ray spectra have been unfolded with the NaI response func-
tions.
a major role. This is supported by the fact that at low exci- informationontheleveldensityandRSFcanbeextractedsi-
tationenergieswheretheleveldensityissmall,wecouldnot multaneously.
identifyanyimportantcontributionsof48Tiintotheγ-spectra. An iterative subtraction technique has been developed to
Inaddition,bothtitaniumsareexpectedtobehavesimilarly. separate out the first-generation (primary) γ-transitions from
Particle-γ coincidencesweremeasuredwitheightSi∆E− the total cascade [8]. The subtraction technique is based on
E particle telescopes and the CACTUS multidetector sys- theassumptionthatthedecaypatternisthesamewhetherthe
tem [6]. The Si detectors were placed in forward direction, levels were initiated directly by the nuclear reaction or by γ
45◦ relativetothebeamaxis. Thefront(∆E)andend(E)de- decay from higher-lying states. This assumption is automat-
tectorshadathicknessof140µmand1500µm,respectively. icallyfulfilledwhenstateshavethesamerelativeprobability
TheCACTUSarrayconsistsof28collimated5”×5”NaI(Tl) to be populated by the two processes, since γ-branching ra-
γ detectorswithatotalefficiencyof15.2%atE =1.33MeV. tios are properties of the levels themselves. If the excitation
γ
Thesingles-protonspectrumandprotonsincoincidencewith binscontainmanylevels,itislikelytofindthesameγ-energy
γ-raysareshowninFig.1. distributionfromthissetoflevelsindependentofthetypeof
In total, 110 million coincidence events were collected in population. However, the assumption is more problematic if
oneweekwithabeamcurrentof1.5nA.Usingreactionkine- the decayinvolves only a few(but not one) levelswithin the
matics,themeasuredprotonenergywastransformedintoex- energybin.
citation energy of the residual nucleus. In this way, a set of Fermi’sgoldenrulepredictsthatthedecayprobabilitymay
γ-rayspectraisassignedtoaspecificinitialexcitationenergy be factorized into the transition matrix element between the
E in 46Ti. Furthermore, the γ-ray spectra are corrected for initialandfinalstates,andthedensityatthefinalstate[2]:
i
theknownresponsefunctionsoftheCACTUSarrayfollowing
2π
theproceduredescribedinRef.[7].Theunfoldedcoincidence λi→f = |(cid:104)f|H(cid:48)|i(cid:105)|2ρf. (1)
h¯
matrix(E,E )of46TiisshowninFig.2.
i γ
Thecoincidencematrixdisplaysverticallinesthatrepresent Realizing that the first generation γ-ray spectra P(Ei,Eγ)
yrast transitions from the last steps in the γ-cascades. How- are proportional to the decay probability from Ei → Ef =
ever, there are also clear diagonal lines where the γ energy Ei−Eγ,i.e.λi→f,wemaywritetheequivalentexpressionof
matchesthedirectdecaytothegroundstate(E =E )andto Eq.(1)as:
i γ
the first and second excited states at 889 keV (2+) and 2010 P(E,E )∝T ρ , (2)
keV (4+), respectively. These γ-rays are primary transitions i γ i→f f
in the γ-cascades. By studying the energy distribution of all where T is the γ-ray transmission coefficient, and ρ =
i→f f
primary γ-rays originating from various excitation energies, ρ(E −E )istheleveldensityattheexcitationenergyE af-
i γ f
3
tertheprimaryγ-rayemission.Thisexpressiondoesnotallow Todemonstratehowwelltheprocedureworks,wecompare
ustoextractsimultaneouslyT andρ fromtheexperimen- inFig.3forsomeinitialexcitationenergiesE theexperimen-
i→f i
tal P(E,E ) matrix. To do that, either one of the factorial talfirst-generationspectraPwiththeonesobtainedbymulti-
i γ
functionsmustbeknown, orsomerestrictionshavetobein- plying the extracted T and ρ functions. The agreement be-
troduced. According to the Brink hypothesis [9], the γ-ray tween calculated and experimental first-generation spectra is
transmission coefficient is independent of excitation energy; excellentfordecayfromhigherexcitations;howeverthereare
onlythetransitionalenergyE playsarole. Thus,wereplace locallystrongdeviationswherethecalculatedspectrafallout-
γ
T withT(E ),giving sideoftheexperimentalerrorbarsforpopulationsoflowerex-
i→f γ
citationsenergies,asseenintheE =5.6and6.4MeVgates.
P(E,E )∝T(E )ρ , (3) i
i γ γ f ThesedeviationscouldbetheconsequenceofPorter-Thomas
which permits a simultaneous extraction of the two multi- fluctuations [11], which will be further discussed in Sec. IV.
plicativefunctions. WethenfitaboutN2/2datapointsofthe The general good agreement holds also for all the other 40
Pmatrixwith2N freeparameters. Aleast χ2 fitisthenpos- spectra (not shown) included in the global fit with the same
sible,sinceitismanymoredatapointsthanfitparameters;in T(E )andρ(E)functions.
γ
thepresentcasewehave150freeparameterstofit2240data The experimental statistical errors are very small as seen
points. in Fig. 3. Thus, the deviations are due to other sources of
At low excitation energy, the γ decay is highly dependent errors. For example for E =6.4 MeV, the T ·ρ prediction
i
ontheindividualinitialandfinalstate;therefore,wehaveex- overestimatesaroundE =3MeVandunderestimatesaround
γ
cluded γ-ray spectra originating from excitation energy bins E =5.5 MeV with several standard deviations. Thus, there
γ
belowEi=5.5MeV. areindicationsthatacommonT andρ functionthatsimulta-
ItiswellknownthattheBrinkhypothesisisviolatedwhen neouslyfitstheP(E,E )matrix,couldnotbefound.Thesys-
i γ
hightemperaturesand/orspinsareinvolved,seeRef.[10]and tematicerrorsbehindthesedeviationscouldbeduetoseveral
referencestherein. However,inthepresentOsloexperiment, factorsasexperimentalshortcomings,assumptionsbehindthe
thetemperaturereachedislow(T ∼1.5MeV)andisassumed Oslo method, the Brink hypothesis [9] and, most probable,
toberatherconstant. Thedependencyonspinisofminorim- Porter-Thomasfluctuations. Keepingallthesepossibilitiesin
portance.Themeasuredratiosoftheγfeedingintotheground mind,theresultsofFig.3areverygratifying.
band indicate a low spin window of I ∼0−6h¯. At excita- There exist infinitely many T and ρ functions making
tionenergyE∼8MeVtheratiosare57:100:9forthe2+,4+ identicalfitstothedata[5]astheexamplesshowninFig.3.
and6+ states,andatE ∼10MeVtheyare55:100:10,which Thesefunctionscanbegeneratedbythetransformations
areequalwithintheerrorbarsforextractingtheseratios. Of
course,atlowexcitationenergythespindistributionfluctuates ρ˜(Ei−Eγ) = Aexp[α(Ei−Eγ)]ρ(Ei−Eγ), (4)
due to a few levels (and spins) present within each 118 keV T˜(E ) = Bexp(αE )T(E ). (5)
γ γ γ
excitationbin.
Inprinciple,theBrinkhypothesisassumedinexpression(3) Inthefollowing,wewilltrytodeterminetheparametersA,α,
could bias the analysis, so that the Oslo method in the next andB. Thisinformationisnotavailablefromourexperiment
turn validates the Brink hypothesis. This issue will be ad- andhastobedeterminedfromotherexperimentalresults.
dressed in Sec. IV, where we find one common level density First, the level density at high excitation energy is nor-
(according to Fermi’s golden rule), which in turn results in malized to the level density at the neutron separation energy
oneuniversalRSFinthequasi-continuumof46Ti. ρ(S ). This data point is calculated from neutron resonance
n
spacingsD (see,e.g.,Ref.[12])withaspindistributiongiven
0
by[13]
III. LEVELDENSITYANDRADIATIVESTRENGTH
FUNCTION g(E,I)(cid:39) 2I+1exp(cid:2)−(I+1/2)2/2σ2(cid:3), (6)
2σ2
In our first investigations, we rely on the Brink hypothe-
whereE isexcitationenergyandIisspin.
sis from the expression (3) and factorize the first-generation Thereexistsnoneutronresonancedatafor46Ti. Wethere-
γ-matrix P(E,E ) into one transmission coefficient T(E )
i γ γ foreestimateρ(S )fromtheparameterizationsofvonEgidy
n
andoneleveldensityρ(E). Sincethedecaybetweenstatesat
andBucurescu[14]usingtheback-shiftedFermigas(BSFG)
lowexcitationenergycannotbetreatedwithinastatisticalap-
model,whichreads
proach,acutofthematrixwithE >1.8MeVand5.5MeV
γ √
<Ei <10.0 MeV was used to exclude clear non-statistical exp(2 aU)
decay routes1. The least χ2 fitting of the two multiplicative ρBSFG(E)=η √ , (7)
12 2a1/4U5/4σ
functionsfollowstheiterativeprocedureofRef.[5].
whereaistheleveldensityparameter,U =E−E1isthein-
trinsic excitation energy, and E1 is the back-shift energy pa-
rameter. Thespincut-offparameterσ isgivenby[14]
1ThelowerexcitationcutconcernsonlytheinitialexcitationenergyEi;one √
stillneedsγ-spectraoriginatingfromexcitationregionsdowntotheground σ2=0.0146A5/31+ 1+4aU, (8)
stateinordertosubtracthigher-ordergenerationsofγ-rays. 2a
4
0.1
V
e
k E = 5.6 MeV V E = 6.4 MeV V E = 7.2 MeV
8 0.08 i ke i ke i
y / 11 0.06 y / 118 y / 118
bilit 0.04 abilit abilit
Proba 0.02 Prob Prob
0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10
V 0.1 Eg (MeV) Eg (MeV) Eg (MeV)
e V V
k E = 8.0 MeVe E = 8.9 MeV e E = 9.7 MeV
y / 118 00..0068 i y / 118 k i y / 118 k i
abilit 0.04 babilit babilit
b o o
Pro 0.02 Pr Pr
0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10
Eg (MeV) Eg (MeV) Eg (MeV)
FIG.3:(Coloronline)Comparisonofexperimentalfirst-generationspectra(squares)andtheonesobtainedfrommultiplyingtheextractedT
andρfunctions(redline).TheinitialexcitationenergybinsEiare118keVbroad.Theerrorbarsrepresenttheexperimentalstatisticalerrors.
Abeingthenuclearmassnumber. [18] is shown in Fig. 5 for comparison (see triangle point at
Figure 4 shows the extracted total level densities for tita- 15.5MeV).
niumisotopeswithknownresonancespacingsD0atSn(filled Oneshouldstressthattheleveldensityfoundfromtheex-
triangles). The resonance spacings for (cid:96)=0 neutrons only perimentisbasedonthespinandparitylevelspopulatedinthe
givetheleveldensitiesforoneortwospinsandonlyonepar- (p,p(cid:48))reaction.Thus,thenormalizationtothetotallevelden-
ity.Inordertoextracttheleveldensityforallspinandparities, sity described above, rests on the assumption that the struc-
weuseEq.(6)andassumeequallymanypositiveandnegative tureoftheleveldensityremainsapproximatelythesameifall
paritystatesatSn. Thisissupportedbycombinatorialquasi- spinsandparitiesareincluded. Thisassumptionisreasonable
particle models [15–17], where the numbers of positive and fulfilled according to Ref. [16], where the level density for
negativeparitystatesatSnarepredictedtobethesame. spinwindowsof2−6h¯ and0−30h¯ havebeencalculatedfor
The points connected with lines are based on the semi- 46Ti within a combinatorial quasiparticle model [17]. From
empirical estimate of von Egidy and Bucurescu [14] with a these estimates, our measurement includes 70−80% of the
common scaling of η =0.5 in order to match qualitatively totalleveldensityandtheleveldensityfinestructuresforthe
theexperimentalρ(S )points. Theestimatedvaluefor46Tiis twospinwindowsareverysimilar.
n
ρ =(4650±1000)MeV−1 atSn=13.189MeV(seeFig.4). The measured level density describes nicely the known
The error bar chosen reflects roughly the general deviation leveldensityuptoaround4MeVofexcitationenergy.Theex-
betweentheglobalestimatesandthepointsderivedfromneu- perimentalresolutionofρ isabout0.3MeVatlowexcitation
tronresonancedata. energy,asseenforthe2+ stateat889keV.Ataround3MeV,
Now the scaling (A) and slope (α) parameters of the level weseeanabruptincreaseinleveldensityduetothebreaking
density can be determined as shown in Fig. 5. The normal- of proton and/or neutron pairs coupled in time-reversed or-
izationoftheleveldensityisfittedtotheknownleveldensity bitals(Cooperpairs[19]).Thisresultsin∼10timesmorelev-
around 3.5 MeV of excitation energy, and to the extrapola- els,makingitdifficulttodeterminetheleveldensityathigher
tionfromρ(Sn)usingtheBSFGmodelwithparameterssum- excitationenergieswithconventionalspectroscopicmethods.
marized in Table I. By choosing other ρ(Sn) values (within ItremainstodeterminethescalingparameterBofthetrans-
the assumed uncertainty of ±1000 MeV−1), the logarithmi- mission coefficient T(E ). Here we use the radiative width
γ
cal slope of the ρ and T will change accordingly, as the α- (cid:104)Γ (cid:105) at S assuming that the γ-decay is dominated by dipole
γ n
parameter of Eqs. (4, 5) has to be adjusted. The ρ curve is transitions. ForinitialspinI andparityπ, thewidthisgiven
wellfixedat∼3.5MeVandwillrotatearoundthispointwith by[20]
different choices of α. Level densities in the 14−19 MeV
excitation region of 46Ti have been measured from Ericson 1 (cid:90) Sn
flthuectvuaalutieosnsre,psoereteRdebf.y[t1h8e]vaanrdioruesfeerxepnecreims ethnetarelignr.ouHposwdeivffeerr, (cid:104)Γγ(cid:105)= 2πρ(Sn,I,π)∑If 0 dEγBT(Eγ)
within a factor of ten. The measurement of the Ohio group ×ρ(S −E ,I ), (9)
n γ f
5
Calculated level density Present experiment
Level density from neutron res. data 104 Known levels
Back-shifted Fermi gas
Estimated level density
From neutron res. data
From Ericson fluctuation data
1) 104 1)
-V -V
Me Me103
(Sn E) (
y at 47Ti 45Ti 48Ti 46Ti r (y
nsit 103 49Ti nsit102
de 50Ti de
el el
ev 51Ti ev
L L 10
52Ti
53Ti
102
1
6 7 8 9 10 11 12 13 14 0 2 4 6 8 10 12 14
Neutron separation energy S (MeV) Excitation energy E (MeV)
n
FIG.4:Deducedtotalρ(Sn)fromneutronresonancespacings(trian- FIG.5:Normalizationofthenuclearleveldensity(filledsquares)of
gles).Thedatapointfor46Ti(square)isestimatedbyextrapolations 46Ti. Atlowexcitationenergies, thedataarenormalized(between
fromtheBSFGmodelwithglobalparameterizationofvonEgidyand thearrows)toknowndiscretelevels(solidline).Athigherexcitation
Bucurescu[14](cirleswithlinestoguidetheeye). energies,thedataarenormalizedtotheBSFGleveldensity(dashed
line)goingthroughthepointρ(Sn)(opensquare),whichisestimated
from the systematics of Fig. 4. For comparison a data point [18]
obtainedfromEricsonfluctuationsareshown(blacktriangle).
wherethesummationandintegrationrunoverallfinallevels
withspinI thatareaccessiblebyγ radiationwithenergyE
f γ
andmultipolarityE1orM1. Furtherdetailsonthenormaliza-
tionprocedurearefoundinRefs.[5,21]. ties in the value of ρ at Sn will change the slope of the RSF,
Sincethereexistsnoneutronresonancedata,weagainhave and thus also the degree of low-energy enhancement. How-
to rely on systematics from other isotopes. From Fig. 6, we ever,thelinesofFig.7showthattheenhancementisnotvery
estimate an average γ-width of (cid:104)Γγ(cid:105)=(1200±500) meV at sensitivetoreasonablechoicesofthevalueρ(Sn).
S . The large uncertainty in (cid:104)Γ (cid:105) gives an absolute normal- Such or similar enhancement has been seen in several
n γ
izationofT(E )andtheRSFwithinabout±50%. However, lightnucleiwithmassA<100, seeRef.[24]andreferences
γ
thisuncertaintydoesnotinfluencetheenergydependence. therein. Thereisstillnotheoreticalexplanationforthisvery
The deduced RSF for dipole radiation can be calculated interestingphenomenon.
fromthenormalizedtransmissioncoefficientT(E )by[22]
γ
1 T(Eγ) TABLE I: Parameters used for the extraction of level density and
f(Eγ)= 2π E3 . (10) radiativestrengthfunctionin46Ti.
γ
ThenormalizedRSFisshowninFig.7. Forcomparison,the Sn a E1 σ ρ(Sn) (cid:104)Γγ(Sn)(cid:105)
(MeV) (MeV−1) (MeV) (MeV−1) (meV)
GEDR data from Ref. [23] are also shown, which have been
translatedfromphotoneutroncrosssectionσ toRSFby[22] 13.189 4.7 -1.0 4.0 4650(1000) 1200(500)
1 σ(Eγ)
f(E )= . (11)
γ 3π2h¯2c2 Eγ
Unfortunately, there is a large energy gap between our data IV. FERMI’SGOLDENRULEANDBRINKHYPOTHESIS
ending at a E =10 MeV and the GEDR data that start at
γ
14MeV. According to Fermi’s golden rule, it is possible to factor-
Our data on the RSF display a minimum near 4−6 MeV ize out the level density from the γ-decay probability. Since
and some structures at lower γ-ray energies. The uncertain- theleveldensityisauniquepropertyofthenucleus,thesame
6
Neutron resonance data Present experiment
3500 High r (Sn)
Estimated Low r (Sn)
(g ,abs) resonance data
3000 3)
-V
e
M
meV) 2500 49Ti 47Ti on (10-7
( 2000 48Ti 46Ti cti
æ g 51Ti 50Ti un
GÆ 1500 h f
t
g
n
1000 e
r
t
s
e 10-8
500 v
ti
a
0 di
6 7 8 9 10 11 12 13 14 Ra
Neutron separation energy S (MeV)
n
FIG.6: Experimentalaverageγ-width(cid:104)Γγ(cid:105)fromneutronresonance 0 2 4 6 8 10 12 14 16 18 20
data(triangles)[22].Onlyaroughestimatefor46Ticanbeachieved g -ray energy E (MeV)
g
(square).
FIG.7: (Coloronline)Experimentalradiativestrengthfunctionfor
46Ti (squares). The RSFs assuming high (5650 MeV−1) and low
extractedleveldensityisexpectedfromthedecayprobability (3650MeV−1)leveldensityatSn arealsoshown. Forcomparison
deduced at different excitation energy regions within the ex- theGEDRdatafromthe(γ,abs)reaction[23]areshown(triangles).
perimental restrictions, in particular the energy resolution of
CACTUS. In order to see if the Oslo method gives a unique
level density, we have divided the data set into three statis-
finedby
tically independent initial excitation-energy regions, namely
Ei=5.5−7.0MeV,7.0−8.5MeVand8.5−10.0MeV,and (cid:82)EidE T(E )ρ(E −E )
applied the same methodology as described in Sec. III. The N (E)= 0 γ γ i γ . (13)
results shown in Fig. 8 are very satisfactory. The different i (cid:82)0EidEγP(Ei,Eγ)
datasetsgiveapproximatelythesameleveldensity. Thus,the
The degree of correctness of the approximation (12) is typi-
disentanglementofleveldensityandtransmissioncoefficient
callyillustratedbythefitsinFig.3.
fromγ-particlecoincidencesseemstoworkverywellaccord-
Inthefollowing,wewillinvestigatethedependencyofthe
ingtoEq.(2).
deduced RSF on initial and final excitation energy. If the
Thefactthatwemeasureapproximatelythesamelevelden-
Brinkhypothesisisvalid,thereexiststhesameRSFforallex-
sity function for primary γ-ray spectra taken at different ex-
citationenergiesinthisnucleus. Wewillcallittheuniversal
citation regions indicates that the RSF depends only on γ-
RSF.InrealitythePorter-Thomasfluctuationsinvolvedinflu-
ray energy and not on excitation energy. This seems to in-
encetheRSFobtainedfromdifferentexcitationregions. We
dicate the validity of Brink hypothesis, which will be tested
willusethetermdeducedRSFinthefollowingforthequan-
morethoroughlyinthefollowing. Inprinciple,thetestcould
tityobtainedfromexperimentaldata. ThededucedRSFsare
be performed by dividing the data set into even more initial
expected to fluctuate around the universal RSF and the fluc-
excitation-energy regions. However, the statistics of the ex-
tuations are expected to be stronger if the number of transi-
periment does not permit such an approach, and a different
tionsusedinthedeterminationofthededucedRSFissmaller.
approachhasbeenused.
Thus, the deduced RSFs extracted from data sets involving
By accepting the level density obtained with the global fit
initialandfinalregionsofhighleveldensityshouldbemuch
ofallrelevantdata(seelowestcurveinFig.8),wemayfurther
closer to the universal RSF than the RSFs deduced from re-
investigatethetransmissioncoefficientindetail,andthusthe
gionsoflowleveldensity.
validityoftheBrinkhypothesis. Wefirstadoptthesolutions
T andρ fromSec.IIIandwrite Since there exists only one unique level density, we con-
structthecounterparttoEq.(12)inthecasethatthetransmis-
N (E)P(E,E )≈T(E )ρ(E −E ). (12) sioncoefficientdependsontheinitialexcitationenergy:
i i γ γ i γ
Thenormalizationfactorforeachinitialexcitationbinisde- N (cid:48)(E)P(E,E )≈T(E ,E)ρ(E −E ), (14)
i i γ γ i i γ
7
-3)eV10-8 FPi/tr r · T Ei = 5.5-7.0 MeV
M
104 SF (
R
5.5-7.0 MeV 10-9
7.0-8.5 MeV
1) x27 8.5-10.0 MeV 0 2 4 6E = 7.0-88.5 MeV10
-MeV103 x9 5.5-10.0 MeV -3)eV10-8 i
( M

ry F (
sit x3 RS
n102
e 10-9
d x1
el 0 2 4 6E = 8.5-810.0 Me1V0
ev 3) i
L 10 -MeV10-8
F (
S
R
10-9
1 0 2 4 6 8 10
E = 5.5-10.0 MeV
3) i
0 2 4 6 8 10 -eV10-8
Excitation energy E (MeV) M
F (
S
R
FIG. 8: Level density extracted from statistically independent data 10-9
sets, taken from various initial excitation energy bins Ei (the three 0 2 4 6 8 10
upper curves). The lower curve is the result for the whole energy Eg (MeV)
regionandisidenticaltotheoneofFig.5.
FIG.9:(Coloronline)Gamma-raystrengthfunctionsextractedfrom
various initial excitation bins Ei. The data points of the three
upper deduced RSFs are from statistically independent data sets.
whereN (cid:48) isdeterminedanalogouslytoEq.(13). Weexpect The data points in the lower panel are identical with the RSF of
thatT(E,E )fluctuatesontheaveragearoundT(E ).Thus, Fig.7. TheRSFsdisplayedasredlines,areevaluatedfromthera-
i γ γ
itisreasonabletoexpectthatN (cid:48)≈N ,whichgivesatrans- tio P(Ei,Eγ)/ρ(Ef) (see text). The resemblance between the two
methods confirms that the level density is common for the various
missioncoefficientof:
excitationregions,inaccordancewithFermi’sgoldenrule.
P(E,E )
T(E,E )≈N (E) i γ . (15)
i γ i ρ(E −E )
i γ
Similarly,thetransmissioncoefficientasafunctionofthefinal
(15). Sincetheapproximation(16)isasimpletransformation
excitationenergyE =E −E isgivenby
f i γ usingE =E −E ,wemayalsoinvestigatethedependencies
f i γ
oftheRSFonfinalexcitationenergyE .
P(E +E ,E ) f
T(Ef,Eγ)≈N (Ef+Eγ) fρ(E γ) γ . (16) In Figs. 10 and 11, the RSFs f(Ei,Eγ) and f(Ef,Eγ) are
f shown for eight excitation regions. For all these deduced
Thevalidityoftheapproximations(14)and(15)isdemon- RSFs, we use one common level density in the evaluation
strated in Fig. 9 by comparing the deduced RSF from this of the approximations (15) and (16). The data points of the
approximation (lines) with the independent fits of T and ρ experimental P matrix cover only a certain region in Ei and
(datapoints). TheRSFsdeterminedforthewholeenergyre- Eγ,makingrestrictionsonthededucedRSFs. Thus, f(Ei,Eγ)
gion 5.5−10 MeV are very similar, especially for low Eγ, is limited to 5.5 MeV <Ei <Sn and 1.8 MeV <Eγ <Ei.
as shown in the lower panels. Also the similarity in the de- The limits for f(Ef,Eγ) are 0 < Ef < Sn−1.8 MeV and
tailed structures of the two methods is recognized, although 1.8MeV<Eγ <Sn−Ef. Foraconsistencycheck, wehave
some differences are present. The deviations are largest for testedthattheaverageRSFforallEf energiesequalstheone
thehighγ-energypartoftheRSFspopulatingthelowest0+, forallEienergies(notshownhere).
2+, and 4+ states, where large Porter-Thomas fluctuations Figure 10 shows that the deduced RSF changes as a func-
are expected (these fluctuations are not included in the error tionofE,whenlowexcitationenergyispopulatedaftertheγ-
i
bars). However,theoverallgoodresemblanceencouragesus emission. Thelower,leftpanel,wheretwoconsecutiveRSFs
to study the detailed evolution of the RSFs as a function of are plotted together, illustrates this. Here, the bumps seen at
initial excitation energies using the approximations (14) and high γ-ray energies are the artifacts of the decay to specific
8
E = 5.4-5.9 MeV E = 6.0-6.5 MeV
i i
10-8 10-8
0 2 4 6 8 10 0 2 4 6 8 10
E = 6.6-7.1 MeEgV (MeV) E = 7.2-7.7 MeEgV (MeV)
i i
10-8 10-8
0 2 4 6 8 10 0 2 4 6 8 10
-3)V Ei = 7.8-8.3 MeEgV (MeV) Ei = 8.4-8.9 MeEgV (MeV)
e
F (M 10-8 10-8
S
R
0 2 4 6 8 10 0 2 4 6 8 10
Ei = 9.0-9.4 MEge (VMeV) Ei = 9.6-10.0 MEge (MVeV)
10-8 10-8
0 2 4 6 8 10 0 2 4 6 8 10
Two last RSFsEg (MeV) All RSFs Eg (MeV)
10-8 10-8
0 2 4 6 8 10 0 2 4 6 8 10
Eg (MeV) Eg (MeV)
FIG. 10: (Color online) Deduced RSFs from various initial excitation bins Ei. The RSFs are evaluated from the ratio P(Ei,Eγ)/ρ(Ef) as
describedinthetext.
isolated levels below 3 MeV of excitation. This is also the andthechangesbetweenthevariousdeducedRSFsarelarge
reason why apparently the fluctuations are typically a factor duetoPorter-Thomasfluctuations.
of 2−3 in the panel where all deduced RSFs are plotted to- InordertoshowthesimilarityofthededucedRSFsinthe
gether (lower, right panel). In the plot of all deduced RSFs, case of strongly suppressed Porter-Thomas fluctuations, we
we see a minimum at about 4−6 MeV and some interest- havecomparedthetwouppermostexcitationenergygatesE
i
ingstructuresatlowγ-rayenergies. Thesestructuresandthe and E in the lower, left panels of Figs. 10 and 11, respec-
f
minimumareindependentoftheuncertaintyinthelog-slope tively. The deduced RSFs are here extracted for γ decay be-
oftheleveldensity,asshowninFig.7. tween states in quasi-continuum, except for the data points
Thevarious f(Ef,Eγ)plotsinFig.11aredifficulttocom- with Eγ >7 MeV in Fig. 10 where the final excitation en-
pare due to different limits appearing at both low and high ergyis<3MeV.ThededucedRSFsextractedfromthequasi-
γ-energies for the various E regions. For example, the first continuumregionbehavesimilartothededucedRSFfromthe
f
three spectra do not reveal the region of low γ-energy en- all-overfitdisplayedinFig.7.
hancement because Eγ > 5.5MeV−Ef > 3.7 MeV. These The good agreement between the deduced RSFs at higher
spectrarepresentthedecaytothe0+,2+and4+groundband energies is consistent with the expectation of suppressed
states,respectively,wheretheexperimentalenergyresolution Porter-Thomas fluctuations due to more initial and final lev-
makessomeoverlapbetweenthesestates. Ingeneralthevar- elsintheevaluationoftheγ strength. Theseresultsstrongly
ious spectra show strong fluctuations in the deduced RSFs indicatethattheconceptofanRSF,whichisindependenton
when the γ emission ends up at low excitation energy, typi- excitationenergy,isvalidalreadyatrelativelylowexcitation
cally Ef <3 MeV. The panel with all deduced RSFs plotted energies in 46Ti. At the same time, the results give us con-
together (lower, right panel) shows approximately the same fidence that Oslo method works reasonably. The differences
scatteringofdatapointsasinFig.10. indeducedRSFsfromlowerenergiesindicatethatthePorter-
It is thus clear that the experimentally deduced RSFs for Thomasfluctuationsaresopoorlysuppressedthatitisdifficult
which states below 3 MeV of excitation energy are involved topredicttheshapeoftheuniversalRSF.Thisisnotverysur-
in the γ decay are very different. Figure 5 shows that this prising and the differences in the deduced RSFs seem to be
excitationregioncoincideswitharegionoffewandwellsep- consistentwiththeexpectedfluctuations.
aratedlow-lyinglevelswithspecificstructures. Thebumpsin In order to display the small differences in the deduced
thededucedRSFsarespecifictothelowenergylevelscheme, RSFs obtained from regions ofhigher level density, we have
9
E = 0.0-0.6 MeV E = 0.6-1.2 MeV
f f
10-8 10-8
0 2 4 6 8 10 0 2 4 6 8 10
E = 1.2-1.8 MEeg V(MeV) E = 1.8-2.4 MeEgV (MeV)
f f
10-8 10-8
0 2 4 6 8 10 0 2 4 6 8 10
-3)V Ef = 2.4-3.0 MEeg V(MeV) Ef = 3.0-3.6 MeEgV (MeV)
e
F (M 10-8 10-8
S
R
0 2 4 6 8 10 0 2 4 6 8 10
Ef = 3.6-4.2 MEge (VMeV) Ef = 4.2-4.8 MeEgV (MeV)
10-8 10-8
0 2 4 6 8 10 0 2 4 6 8 10
Two last RSFsEg (MeV) All RSFs Eg (MeV)
10-8 10-8
0 2 4 6 8 10 0 2 4 6 8 10
Eg (MeV) Eg (MeV)
FIG. 11: (Color online) Deduced RSFs populating various final excitation bins E . The RSFs are evaluated from the ratio P(E +
f f
Eγ,Eγ)/ρ(Ef) as described in the text. It should be noted that there are no final states at Ef =1.2−1.8 MeV, however the experimental
resolution(seeFig.5)isresponsibleforincludingthe2+and4+groundbandstatesinthisgate.
taken the four highest gates shown in Fig. 10 and only con- difficulttodeterminesincetheexperimentalstatisticalerrors
sideredγ energiesupto5.1MeV.Thus,thesestatisticallyin- (seeerrorbars)arealsoofthesameorder.
dependentRSFsareevaluatedinquasi-continuumwithinitial
andfinalexcitationenergiesofroughlyE =8−10MeVand
i
E =3−7 MeV, respectively. The deduced RSFs are pre-
f V. CONCLUSIONS
sentedintheupperpanelofFig.12.
Forthisdatasetweevaluatedtherelativedeviationsby
The level density and radiative strength function for 46Ti
fij−(cid:104)fi(cid:105) have been determined using the Oslo method. Similar level
r = , (17)
ij (cid:104)f(cid:105) densityfunctionshavebeenextractedfromstatisticallyinde-
i
pendentdatasetscoveringdifferentexcitationenergies. This
where the index i represents the γ energy and j is the initial
gives confidence to the Oslo method, since the disentangle-
excitation energy. The average strength at each γ energy is
mentoftheleveldensitybyFermi’sgoldenrulepredictsone
estimatedfromthefourindividualRFSs
andonlyoneuniqueleveldensity,independentofthedataset.
1 ThededucedRSFdisplaysanenhancementatlowγ-rayen-
(cid:104)f(cid:105)= ∑f . (18)
i ij
n ergywhereweseeabumparound3MeVandanotherstruc-
j j
tureatenergiesnear2MeV.Asimilarenhancementhasbeen
InthelowerpanelofFig.12therijvaluesareplottedshowing seen in several other light mass nuclei and is still not ac-
therelativefluctuationsfromthemeanvalueateachγ energy. countedforbypresenttheories.
The average ratio for all data points (ni =28 and nj =4) is AmethodtostudytheevolutionofthededucedRSFsasa
takenas functionofinitialandfinalenergyregionshasbeendescribed.
1 The deduced RSFs are found to display strong variations for
r= ∑|r |, (19)
nn ij different initial and final excitation energies if transitions to
i j ij
thelowestexcitationsareinvolved. Thereasonfortheviolent
givingr∼6%. ThedifferencesinthededucedRSFscaneas- fluctuationsofafactorof2−3isthatonlyafewisolatedlev-
ily be interpreted as remnants of the Porter-Thomas fluctua- els are present at low excitation energies with E <3 MeV.
f
tions. However, a quantitative estimate of the fluctuations is The differences in the RSFs obtained from a few number of
10
· 10-9 transitions,thatcanbeexplainedasaconsequenceofPorter-
16 Thomas fluctuations of individual intensities, show that this
E = 7.8-8.3 MeV
Ei = 8.4-8.9 MeV (a) energyregioncannotbeusedfordeterminationoftheuniver-
14 i
E = 9.0-9.4 MeV salRSF.Eventhough,thededucedRSFsbasedonarestricted
i
E = 9.6-10.0 MeV
3) 12 i numberoftransitionsstillindicatethatthedecayisgoverned
-V byauniversalRSF.
e 10
M
9
-0 8
1
(
F 6 However, the present work shows that it is possible to get
S
R 4 morepreciseexperimentalinformationontheuniversalRSF.
By imposing restrictions on the initial and final excitation
2
energies, the RSFs for the decay between states in quasi-
0 continuumcanbeextracted(i.e.forEf >∼3MeV).Theresults
from this selected data set show that the decay is consistent
) 20 Eg (MeV) (b) withanRSFwhichisidependentofexcitationenergywithin
%
( lessthan∼6%alreadyattheserelativelylowexcitations.
F
S 10
R
n
e i 0 Insummary, theobservationofalmostidenticallevelden-
c
n
e -10 sities and RSFs extracted from different data sets in quasi-
r
e continuum, indicates that the Oslo method works well. Pro-
f
Dif -20 videdthatweusedatafromthequasi-continuum,auniversal
RSF in the light mass region of the 46Ti nucleus can be ex-
tracted.
2 2.5 3 3.5 4 4.5 5
Acknowledgments
2 2.5 3 E (3M.5eV) 4 4.5 5
g
FIG.12: (Coloronline)(a)Radiativestrengthfunctionsforγ transi-
tionsbetweenstatesinquasi-continuum. Datafromthefourhighest
excitationenergygatesofFigs.10havebeenchosen. (b)Ratiosof TheauthorswishtothankE.A.OlsenandJ.Wikneforex-
thedeviationfromtheaverageRSFateachγenergy,seetext. cellentexperimentalconditions. Thisworkwassupportedby
theResearchCouncilofNorway(NFR).
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