Evidence for the pair-breaking process in 116,117Sn

0.13 MB·English

by  U. Agvaanluvsan

Evidence forthe pair-breaking process in116,117Sn
U.Agvaanluvsan1,2,A.C.Larsen3 ,M.Guttormsen3,R.Chankova4,5,G.E.Mitchell4,5,A.Schiller6,S.Siem3,andA.Voinov6
∗
1Stanford University, Palo Alto, California 94305 USA
2MonAme Scientific Research Center, Ulaanbaatar, Mongolia
3Department of Physics, University of Oslo, N-0316 Oslo, Norway
4Department of Physics, North Carolina State University, Raleigh, NC 27695, USA
5Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA and
9 6Department of Physics, Ohio University, Athens, OH 45701, USA
0 (Dated:January6,2009)
0
Thenuclear leveldensitiesof116,117Snbelowtheneutron separationenergyhavebeendeterminedexperi-
2
mentallyfromthe(3He,ag )and(3He,3Heg)reactions,respectively. Theleveldensitiesshowacharacteristic
′
n exponentialincreaseandadifferenceinmagnitudeduetotheodd-eveneffectofthenuclearsystems. Inaddi-
a tion,theleveldensitiesdisplaypronouncedstep-likestructuresthatareinterpretedassignaturesofsubsequent
J
breakingofnucleonpairs.

6
PACSnumbers:21.10.Ma,24.10.Pa,25.55.-e,27.60.+j
]
x
e
- I. INTRODUCTION Section II, followedby a discussion of the analysis and nor-
l malization procedure. The experimentalresults for the level
c
u Nuclear level densities are important for many aspects of densityaregiveninSectionIII,andthedeterminationofvar-
n ious thermodynamic quantities are presented in Section IV.
fundamental and applied nuclear physics, including calcula-
[ ConclusionsaredrawninSectionV.
tionsof nuclearreactioncrosssections. The leveldensityof

2 excited nuclei is an average quantity that is defined as the
v number of levels per unit energy. The majority of data for
6 nuclearleveldensitiesareobtainedfromtwoenergyregions. II. EXPERIMENTALPROCEDUREANDDATAANALYSIS
0
Atlowexcitationenergy,theleveldensityisobtaineddirectly
4
from counting of low-lying levels [1]. As the excitation en- TheexperimentwascarriedoutattheOsloCyclotronLabo-
4
. ergyincreases,theleveldensitybecomeslargeandindividual ratory(OCL)usinga38-MeV3Hebeam.Theself-supporting
0
levels are often not resolved in experiments. Therefore, the 117Sn target had a thickness of 1.9 mg/cm2. The reaction
1
direct counting method becomes impossible. Nuclear reso- channels 117Sn(3He,ag )116Sn and 117Sn(3He,3Heg )117Sn
8 ′
0 nancesatorabovethenucleonbindingenergyprovideanother werestudied.
: sourceofleveldensitydata[2]. Betweenthesetwoexcitation The experiment ran for about 11 days with an average
v
energy regions, the level density is often interpolated using beam current of 1.5 nA. Particle-g coincidence events
Xi phenomenologicalformulas [3, 4, 5]. Itisinthisenergyre- were detected usin≈g the CACTUS multidetector array. The
r gionthepresentmeasurementsfocus. charged particles were measured with eight Si particle tele-
a
Recently,anextensionofthesequentialextractionmethod, scopesplacedat45 withrespecttothebeamdirection.Each
◦
now referred to as the Oslo method, was developed by the telescopeconsistsofafrontSiD Edetectorwiththickness140
Oslo Cyclotron Group. The Oslo method permits a simul- m mandabackSi(Li)E detectorwiththickness3000m m. An
taneous determination of the level density and the radiative arrayof28collimatedNaIg -raydetectorswithasolid-angle
strengthfunction[6, 7]. Forbothofthese quantities, the ex- coverageof 15%of4p wasused.Inaddition,oneGedetec-
≈
perimentalresultscoveranenergyregionwherethereislittle torwasusedinordertoestimatethespindistributionandde-
information available and data are difficult to obtain. How- terminetheselectivityofthereaction. Thetypicalspinrange
ever, the limitation of this methodis thatthe results mustbe isI 2 6h¯.
normalizedtoexistingdata–fromthediscretelevelsandneu- F∼igur−e 1 shows the singles a -particle spectrum (upper
tron resonancespacingsfor the level densityand to the total panel) and the a -g coincidence spectrum (lower panel) for
radiativewidth for the radiativestrength function. Thus, the 116Sn. Thetwopeaksdenotedby0+ and2+ arethe transfer
new and main achievement of the Oslo method is to estab- peaks to the ground state and the first excited state, respec-
lish thefunctionalformoftheleveldensityandtheradiative tively. ThestrongtransferpeakatE =3.2MeViscomposed
strengthfunctionintheabovespecifiedenergyregion. ofmanystatesfoundtobetheresultofpick-upofhigh-jneu-
In this work we present results for the level density in tronsfromtheg7/2 andh11/2 orbitals[9,10]. Anotherstrong
116,117Sn fortheexcitationenergy0<E <S 1MeV.The transferpeakcenteredaroundE =8.0 MeVis newandmay
n
radiativestrength functionsof 116,117Sn have b−eenpublished indicatetheneutronpick-upfromtheg9/2orbital.Thecounts
elsewhere[8]. Theexperimentalset-upisbrieflydescribedin in the coincidencespectrum decrease for excitation energies
higherthanS duetolowerg -raymultiplicitywhentheneigh-
n
boringA 1isotopeispopulatedatlowexcitationenergy.
Thepa−rticle–g -raycoincidencespectrawereunfoldedusing
Electronicaddress:[email protected] the response functions for the CACTUS detector array [11].
∗
2
matrix. The entries of the first-generation matrix P are the
probabilities P(E,Eg) that a g -ray of energy Eg is emitted
a) Singles from an excitation energy E. This matrix is the basis for
60000 E = 3.2 MeV
thesimultaneousextractionoftheradiativestrengthfunction
V andthe leveldensity. Accordingto the Brink-Axelhypothe-
e 50000
k sis [13, 14], a giantelectricdipoleresonancecanbebuilton

20 40000 E = 8.0 MeV everyexcitedstatewiththesamepropertiesastheonebuilton
1 thegroundstate,thatis,theradiativestrengthfunctionisinde-
/
ts 30000 pendentof excitationenergyand thusof temperature. Many
n
theoreticalmodelsdoincludeatemperaturedependenceofthe
u
o 20000 radiativestrengthfunction[15,16].However,thetemperature
C
dependenceisweakandthetemperaturechangeintheenergy
10000 2+ 0+ regionunderconsiderationhereisrathersmall. Therefore,the
temperaturedependenceisneglectedintheOslomethod.
The first-generation matrix is factorized into the radiative
50000 b) Coincidences transmissioncoefficient,whichisdependentonlyonEg, and
ontheleveldensity,whichisa functionoftheexcitationen-
V
e 40000 ergyofthefinalstatesE Eg:
k −
0 Sn
12 30000 P(E,Eg)(cid:181) T(Eg)r (E Eg). (1)
/ −
s
nt 20000 Thefunctionsr andT areobtainediterativelybyaglobalized
u
o fittingprocedure[6]. Thegoaloftheiterationistodetermine
C
thesetwofunctionsat N energyvalueseach;theproductof
10000
the two functionsis kn∼own at N2/2 data pointscontained
∼
in the first-generation matrix. The globalized fitting to the
0
34 36 38 40 42 44 46 48 50 52 datapointsdeterminesthefunctionalformforr andT. The
a energy E (MeV) resultsmustbenormalizedbecausetheentriesofthematrixP
a
areinvariantunderthetransformation[6]
FIG.1:a)Singlesa -particlespectrum,andb)a -g coincidencespec-
trumfromthe117Sn(3He,a )116Snreaction. r˜(E Eg) = Aexp[a (E Eg)]r (E Eg), (2)
− − −
T˜(Eg) = Bexp(a Eg)T(Eg). (3)
The first-generation matrix (the primary-g -ray matrix) con- Inthefinalstep,thetransformationparametersA,B,anda
tainsonlythefirstg -raysemittedfromagivenexcitationen-
whichcorrespondtothemostphysicalsolutionmustbedeter-
ergybin;thismatrixisobtainedbyasubtractionprocedurede-
mined. Details of the normalizationprocedureare described
scribedinRef.[12]. Thisprocedureisjustifiediftheg decay
inseveralpapersreportingtheresultsoftheOslomethod,see
fromanyexcitationenergybinisindependentofthemethod
forexample[17, 18]. In the following, we willfocuson the
offormation–eitherdirectlybythenuclearreactionorindi-
leveldensityandthermodynamicproperties.
rectlybyg decayfromhigherlyingstatesfollowingtheinitial
reaction. This assumption is clearly valid if the same states
arepopulatedviathetwoprocesses,sinceg -decaybranching
ratiosarepropertiesoflevels.Whendifferentstatesarepopu- III. LEVELDENSITIES
lated,theassumptionmaynothold. However,muchevidence
suggests [7] that statistical g decay is governed only by the ThecoefficientsAanda relevantforthenuclearlevelden-
g -rayenergyandthedensityofstatesatthefinalenergy. sityaredeterminedfromnormalizingtheleveldensitytothe
Inthedataanalysis,theparticle-g -raycoincidencematrixis low-lyingdiscrete levelsand the neutronresonancespacings
preparedandtheparticleenergyistransformedintoexcitation just above the neutron separation energy S . For 117Sn, we
n
energyusingthereactionkinematics. Therowsofthecoinci- used s- and p-wave resonance level spacings (D , D ) taken
0 1
dencematrixcorrespondtotheexcitationenergyintheresid- from[2]tocalculatethetotalleveldensityatS (seeRef.[6]
n
ualnucleus,whilethecolumnscorrespondtotheg -rayenergy. formoredetailsonthecalculation). Since thereisnoexper-
Alltheg -rayspectraforvariousinitialexcitationenergiesare imental informationaboutD or D for 116Sn, we estimated
0 1
unfoldedusingtheknownresponsefunctionsoftheCACTUS r (S ) for this nucleusbased on systematics for the other tin
n
array [11]. The first-generation g -ray spectra, which consist isotopes[2,4],andassuminganuncertaintyof50%.Thelevel
ofprimaryg -raytransitions, arethenobtainedforeachexci- spacingsandthefinalvaluesforr (S )aregiveninTableI.
n
tationenergybin[12]. The experimental level density r is determined from the
Thefirst-generationg -rayspectraforallexcitationenergies nuclear ground state up to S 1 MeV. Therefore, an in-
n
∼ −
form a matrix P, hereafter referred to as the first-generation terpolationisrequiredbetweenthepresentexperimentaldata
3
TABLEI:Parametersusedfortheback-shiftedFermigasleveldensityandthecalculationofr (Sn).
Nucleus Epair C1 a D0 D1 s (Sn) Sn r (Sn) h
(MeV) (MeV) (MeV 1) (eV) (eV) (MeV) (105MeV 1)
− −
116Sn 2.25 1.44 13.13 4.96 9.563 4.12(206) 0.46
117Sn 0.99 −1.44 13.23 507−(60) 15−5(6) 4.44 6.944 0.86(25) 0.40
−
andr evaluatedatS . Theback-shiftedFermigaslevelden-
n
sitywiththeglobalparameterizationofvonEgidyetal.[4],
exp(2√aU)
r (E)=h 12√2a1/4U5/4s , (4) -1)V 106 a) 116Sn
isemployedfortheinterpolation1. Theintrinsicexcitationen- Me 105
ergyisgivenbyU =E−Epair−C1,whereC1=−6.6A−0.32 E) ( 104
MeV is the back-shift parameter and A is the mass number.
(

The pairing energy Epair is based on pairing gap parameters ry 103
DcoprdanindgD tnoe[v2a0l]u.aTtehdefrloevmele-vdeenn-soitdydpmaraasmsedtiefrfeareanncdesth[1e9s]painc-- nsit 102
cutoffparameters are givenby a = 0.21A0.87 MeV−1 and de 10
s 2=0.0888A2/3aT,respectively.ThenucleartemperatureT el
isdescribedbyT = U/aMeV.Theconstanth isaparame- v 1
e
terappliedtoensurepthattheFermigasleveldensitycoincides L
withtheneutronresonancedata.Allparametersemployedfor 10-1
0 2 4 6 8 10
theFermigasleveldensityarelistedinTableI.
Figure2showsthenormalizedleveldensitiesof116Snand -1)V 106 b) 117Sn
117Sn. Thefullsquaresrepresenttheresultsfromthepresent Me 105
work. The data points between the arrows are used for nor-
(
malizingtotheleveldensityobtainedfromcountingdiscrete E) 104
levels (solid line) and the level density calculated from the (
neutronresonance spacing (open square). The discrete level ry 103
t
MscheVemieni1s16sSeennbteoyobnedcowmhpiclhetethuepletvoeelxdceintastiitoynoebntaeirngeyd≈fro3m.5 nsi 102
e
discretelevelsstartstodrop. For117Snthediscretelevelden- d 10
sity iscompleteonlyupto 1.5MeV.The newdataofthis el
≈ v 1
workthusfillthegapbetweenthediscreteregionandthecal- e
L
culatedleveldensityatS .
n 10-1
From Fig. 2, we observe that pronounced step-like struc-
0 2 4 6 8 10
turesarepresentintheleveldensitiesofboth116,117Sn. Inthe
Excitation energy E (MeV)
following section, nuclear thermodynamicpropertiesare ex-
tractedusingthepresentleveldensityresults,andthesestruc-
turesareinvestigatedindetail. FIG.2: Normalizedleveldensityofa)116Snandb)117Sn. Results
from the present work are shown as full squares. The data points
betweenthearrowsareusedforthefittingtoknowndata. Thelevel
densityatlowerexcitationenergyobtainedfromcountingofknown
IV. THERMODYNAMICPROPERTIES
discretelevelsisshownasasolidline,whiletheleveldensitycalcu-
latedfromneutronresonancespacingsisshownasanopensquare.
TheentropyS(E)isameasureofthenumberofwaystoar- Theback-shiftedFermigasleveldensityusedfortheinterpolationis
rangeaquantumsystematagivenexcitationenergyE.There- displayedasadashedline.
fore,theentropyofanuclearsystemcangiveinformationon
theunderlyingnuclearstructure. Themicrocanonicalentropy
isgivenby where W (E) is the multiplicity of accessible states and kB is
Boltzmann’s constant, which we will set to unity to give a
S(E)=k lnW (E), (5)
B dimensionlessentropy.
The experimental level density r (E) is directly propor-
tionaltothemultiplicityW (E),whichcanbeexpressedas
1Wechosetoapplytheoldparameterizationof[4]insteadofthemorerecent W (E)=r (E) [2 J(E) +1]. (6)
onein[5]becausenew(g,n)data[21]showedthattheslopeoftheradiative · h i
strengthfunctionin117Sn[8]becametoosteepusingthevaluesof[5]. Here, J(E) is the average spin at excitation energy E and
h i
4
theexcitationenergyincreases,andabove2 3MeVweesti-
matethesingleneutronquasiparticletocarr−yaboutD S 1.6
≃
in units of Boltzmann’s constant. This agrees with previous
16
findingsfromtherare-earthregion[22].
a)
14 Both entropy curves display step-like structures superim-
) posedonthegeneralsmoothincreasingentropyasafunction
B 12
k ofexcitationenergy.Atthesestructures,theentropyincreases
E) ( 10 117Sn abruptlyin a smallenergyintervalbeforeit becomesa more
S( 116Sn slowlyincreasingfunction.
y 8 Thefirstlow-energybumpof116Snisconnectedtothefirst
op 6 excited2+ stateatE =1.29MeVandthesecondexcited0+
tr state at E =1.76 MeV. Similarly, the first bump in the en-
En 4 tropy of 117Sn is connected to the first excited states in this
2 nucleus. The next structuresare probablecandidates for the
pair-breakingprocess. Microscopiccalculationsbasedonthe
0 senioritymodelindicatethatstep structuresin the levelden-
sitycanbeexplainedbytheconsecutivebreakingofnucleon
)B 4.5 b) Cooperpairs[23].
k
( 4 ThebumpspresentintheSnleveldensitiesaremuchmore
S
outstandingthanpreviouslymeasuredforothermassregions
3.5
De by the Oslo group. One explanationof the clear fingerprints
nc 3 could be that since the Z = 50 shell is closed, the break-
e ing of proton pairs are strongly hindered and thus do not
r 2.5
e
f smoothouttheentropysignaturesfortheneutronpairbreak-
dif 2 ing. Therefore,itis verylikely thatthe structuresare due to
y 1.5 pureneutron-pairbreakup.
p
o We have investigatedthe structures furtherby introducing
r 1
nt themicrocanonicaltemperaturegivenby
E 0.5
¶ S −1
0 1 2 3 4 5 6 7 8 9 T =(cid:18)¶ E(cid:19) . (9)
Excitation energy E (MeV)
By taking the derivative, small changes in the slope of the
entropy are enhanced. This is easily seen in Fig. 4, where
FIG.3: a)Experimentalentropiesfor116,117Sn,andb)theentropy
the microcanonical temperatures of 116,117Sn are displayed.
difference. Byfittingastraightlinetotheentropydifferenceinthe
energyregion2.4 6.7MeV,anaveragevalueofD S 1.6kB was The striking oscillations of the temperature is a thermody-
obtained. − ≃ namicsignatureforsuchsmallsystemsasthenucleus.Onlya
fewquasiparticlesparticipateintheexcitationofthenucleus.
Sincethesystemisnotincontactwithaheatbathwithacon-
thefactor[2 J(E) +1]thusgivesthedegeneracyofmagnetic stanttemperature,the system is far fromthe thermodynamic
h i
substates. As the averagespin is notwellknownat all exci- limit.
tation energies, we choose to omit this factor and define the InFig.4, thebumpsbelow 2 MeVareconnectedto the
multiplicityas low-lyingexcitedstates in 116,≈117Sn. However,the peak-like
structurecenteredaroundE =2.8MeV in116Sn andaround
W (E)=r (E)/r 0, (7) E =2.6MeVin117Sncouldbeasignatureofthefirstbreak-
upofaneutronpair.AboveE=4.5and3.6MeVin116,117Sn,
where the denominator is determined from the fact that the
respectively,thetemperatureappearstobeconstantontheav-
groundstateofeven-evennucleiisawell-orderedsystemwith
zero entropy. The value of r =0.135 MeV 1 is obtained erage,indicatingamorecontinuousbreakingoffurtherpairs.
0 −
such that S=lnW 0 for the ground state region of 116Sn.
Thesamer isalso∼appliedfor117Sn. InFig.3theresulting Recently[24],thecriticalityoflow-temperaturetransitions
0
entropiesof116,117Snareshown. was investigated for rare-earth nuclei. We apply the same
method here and investigate the probability P of the system
Theentropycarriedbythevalenceneutroncanbeestimated
atfixedtemperatureT tohaveexcitationenergyE,i.e.,
byassumingthattheentropyisanextensive(additive)quan-
tity[22]. Withthisassumption,theexperimentalsingleneu- P(E,T)=W (E)exp( E/T)/Z(T), (10)
tronentropyisgivenby −
wherethecanonicalpartitionfunctionisgivenby
D S=S(117Sn) S(116Sn). (8)
− ¥
From Fig. 3, we observe that D S becomes more constant as Z(T)= W (E′)exp E′/T dE′. (11)
Z0 −
(cid:0) (cid:1)
5
pair. ThevaluesfoundareT =0.58(2)and0.71(2)MeVfor
c
116,117Sn, respectively. Furthermore, we observe a potential
barrier D F of about 0.25 MeV between the two minima E
2 c 1
V) 1.8 a) 116Sn atondgoEf2r.oTmhoenpeoptehnatsiael(bnaorrpiaeirrsinbdriockaetens)tthoeafnroeteheenre(orgnyenberoedkeedn
e
M 1.6 pair)attheconstant,criticaltemperatureTc.
) ( 1.4 In the process of breaking additional pairs, the structures
E
areexpectedtobelesspronounced.Indeed,inthelowestpan-
T( 1.2
e els of Fig. 5, the free energyis ratherconstantfor excitation
r 1 energies above E 5.0 and 4.2 MeV for 116,117Sn, respec-
u ≈
at 0.8 tively. Instead of a double-minimumstructure, a continuous
per 0.6 minimum of Fc appears for several MeV of excitation ener-
gies. Thisdemonstratesclearly that the depairingprocessin
m
e 0.4 tincannotbeinterpretedasanabruptstructuralchangetypical
T
0.2 ofa firstorderphasetransition. Forthisprocesswe evaluate
thecriticaltemperaturebyaleastc 2fitofF totheexperimen-
0
taldata.Theexcitationenergies(E andE )andtemperatures
1.8 1 2
b) 117Sn forthephasetransitionsaresummarizedinTableII.
) 1.6
V
e
M 1.4
TABLEII:Phasetransitionvaluesdeducedfor116,117Sn.
(
T 1.2
e 116Sn 117Sn
r 1
atu 0.8 Breakingof E(M1−eVE)2 (MTecV) E(M1−eVE)2 (MTecV)
r
e onepair 2.4 4.2 0.58(2) 2.3 3.7 0.71(2)
p 0.6 − −
m twoormorepairs 5.0 8.0 0.69(8) 4.2 6.0 0.63(4)
− −
e 0.4
T
0.2
According to the linearized free-energy calculations, the
0 1 2 3 4 5 6 7 8 first neutron pair is broken for excitation energies between
2.4 4.2MeVin 116Sn. Comparingwith theupperpanelof
Excitation energy E (MeV)
−
Fig. 4, we see that this coincides with the excitation-energy
regionwhereapeakstructureisfound. Thisbumpisthought
FIG.4:Microcanonicaltemperaturesofa)116Sn,andb)117Sn.
to represent the first neutronpair break-upas discussed pre-
viously in the text. The average temperature of the bump is
0.59(2) MeV, in excellent agreement with the deduced crit-
Lee and Kosterlitz showed [25, 26] that the function
ical temperature T = 0.58(2). Similarly, for 117Sn in the
c
A(E,T)= lnP(E,T),forafixedtemperatureT inthevicin-
− lowerpanelof Fig.4, we findthata peak-likestructurewith
ity of a critical temperature T of a structuraltransition, will
c an average temperature of 0.74(2) MeV is present between
exhibit a characteristic double-minimum structure at ener-
E = 2.3 3.7 MeV. Compared to the critical temperature
gies E1 and E2. For the critical temperature Tc, one finds T =0.71−(2)MeV,thevaluesagreesatisfactory.
c
A(E ,T)=A(E ,T ). ItcanbeeasilyshownthatAisclosely
1 c 2 c Forthecontinuousbreak-upprocesscharacterizedbyazero
connected to the Helmholtz free energy, and that this condi- potential barrier (D F 0 MeV), we estimate for 116Sn an
c
tionisequivalentto ≈
average microcanonical temperature of 0.75(6) MeV in the
excitation-energy region E = 5.0 8.0 MeV, in reasonable
Fc(E1)=Fc(E2), (12) agreement with T =0.69(8) calc−ulated from the linearized
c
Helmholtz free energy. In the case of 117Sn, the average
which can be evaluated directly from our experimentaldata.
microcanonical temperature for excitation energies between
ItshouldbeemphasizedthatF isalinearizedapproximation
c
4.2 6.0MeVisfoundtobe0.64(3)MeV,whichagreesvery
totheHelmholtzfreeenergyatthecriticaltemperatureT ac-
c −
wellwithT =0.63(4)MeV.Thisgivesfurtherconfidencein
cordingto c
ourinterpretationofthedataastwoindependentmethodsgive
F(E)=E TS(E). (13) verysimilarresults.
c c
−
Linearizedfree energiesF forcertaintemperaturesT are
c c
displayed in Fig. 5. In the upper panels, 116,117Sn data are V. CONCLUSIONS
shownwheretheconditionF(E )=F(E )=F isfulfilled.
c 1 c 2 0
Each nucleus shows a double-minimumstructure, which we Thenuclearleveldensitiesfor116,117Snareextractedfrom
interpretasthecriticaltemperaturesforbreakingoneneutron particle-g coincidencemeasurements. New experimentalre-
6
1.5 a) 116Sn -1 b) 117Sn
T = 0.58(2) MeV T = 0.71(2) MeV
C C
)
V 1 F = -0.44(3) MeV F = -2.45(2) MeV
e 0 -1.5 0
M
(
C0.5
F
y -2
g 0
r
e
n
E
-0.5 -2.5
-1
-3
2 -0.8
c) 116Sn d) 117Sn
1.5
-1
T = 0.69(8) MeV T = 0.63(4) MeV
1 C C
V) F0 = -1.3(6) MeV -1.2 F0 = -1.7(5) MeV
e 0.5
M
( -1.4
C 0
F
y -0.5 -1.6
g
r
e
n -1 -1.8
E
-1.5
-2
-2
-2.2
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
Excitation energy E (MeV) Excitation energy E (MeV)
FIG.5:LinearizedHelmholtzfreeenergyFcatcriticaltemperatureTcfora)116Sn,andb)117Sndisplayingacharacteristicdouble-minimum
structure. TheconstantlevelF0,whichisconnectingtheminima,isindicatedbyhorizontallines. AcontinuousminimumofFcisshownfor
c)116Sn,andd)117Sn.
sultsarereportedfortheleveldensityin116,117Snforexcita- Alliances program through DOE Research Grant No. DE-
tionenergiesabove1.5MeVand3.5MeVuptoS 1MeV. FG52-06NA26194.U.A.andG.E.M.alsoacknowledgesup-
n
−
The level densities for both nuclei display prominent step- port from U.S. Departmentof EnergyGrant No. DE-FG02-
like structures. Thestructureshavebeenfurtherinvestigated 97-ER41042.FinancialsupportfromtheNorwegianResearch
by means of thermodynamicconsiderations: microcanonical Council(NFR)isgratefullyacknowledged.
entropyandtemperature,andthroughcalculationsofthelin-
earizedHelmholtzfreeenergy. Bothmethodsgiveconsistent
results, instrongfavorofthepair-breakingprocessasanex-
planationofthestructures.
Acknowledgments
ThisresearchwassponsoredbytheNationalNuclearSecu-
rityAdministrationundertheStewardshipScienceAcademic
7
[1] R. Firestone and V. S. Shirley, Vol. II, Table of Isotopes, 8th [13] D.M.Brink,Ph.D.thesis,OxfordUniversity,1955.
edition(Wiley,NewYork,1996). [14] P.Axel,Phys.Rev.126,671(1962)
[2] S.F.Mughabghab,AtlasofNeutronResonances,FifthEdition, [15] S.G.Kadmenski˘ı,V.P.Markushev,andV.I.Furman,Yad.Fiz.
ElsevierScience(2006). 37,277(1983),[Sov.J.Nucl.Phys.37,165(1983)].
[3] A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43, 1446 [16] G.Gervais,M.Thoennessen,andW.E.Ormand,Phys.Rev.C
(1965). 58,R1377(1998).
[4] T.vonEgidy,H.H.Schmidt,andA.N.Behkami,Nucl.Phys. [17] A.Schiller,M.Guttormsen,E.Melby,J.Rekstad,andS.Siem,
A481,189(1988). Phys.Rev.C61,044324(2000).
[5] T. von Egidy and D. Bucurescu, Phys. Rev.C 72, 044311 [18] U. Agvaanluvsan, A. Schiller, J. A. Becker, L. A. Bernstein,
(2005);73,049901(E)(2006). P. E. Garrett, M. Guttormsen, G. E. Mitchell, J. Rekstad,
[6] A.Schiller,L.Bergholt,M.Guttormsen,E.Melby,J.Rekstad, S.Siem,A.Voinov, andW.Younes, Phys.Rev.C70, 054611
and S. Siem, Nucl. Instrum. Methods Phys. Res. A 447, 498 (2004).
(2000). [19] G.AudiandA.H.Wapstra,Nucl.Phys.A595,409(1995).
[7] L. Henden, L. Bergholt, M. Guttormsen, J. Rekstad, and [20] J.Dobaczewski, P.Magierski, W.Nazarewicz, W.Satuła,and
T.S.Tveter,Nucl.Phys.A589,249(1995). Z.Szyman´ski,Phys.Rev.C63,024308(2001).
[8] U.Agvaanluvsan,A.C.Larsen,R.Chankova,M.Guttormsen, [21] H.Utsunomiya,privatecommunication.
G.E.Mitchell,A.Schiller,S.Siem,andA.Voinov,LosAlamos [22] M. Guttormsen, M. Hjorth-Jensen, E. Melby, J. Rekstad,
preprintserver: A.Schiller,andS.Siem,Phys.Rev.C63,044301(2001).
http://xxx.lanl.gov/abs/0808.3648,submittedtoPhys. [23] A. Schiller, E. Algin, L. A. Bernstein, P. E. Garrett, M. Gut-
Rev.Lett. tormsen, M. Hjort-Jensen, C. W. Johnson, G. E. Mitchell,
[9] E. J. Schneid, A. Prakash, and B. L. Cohen, Phys. Rev. 156, J.Rekstad, S.Siem, A.Voinov, andW.Younes, Phys.Rev.C
1316(1967). 68,054326(2003).
[10] K.Yagi,Y.Sagi,T.Ishimatsu,Y.Ishizaki,M.Matoba,Y.Naka- [24] M. Guttormsen, M. Hjorth-Jensen, J. Rekstad, S. Siem,
jima,andC.Y.Huang,Nucl.Phys.A111,129(1968). A.Schiller,andD.Dean,Phys.Rev.C68,034311(2003).
[11] M.Guttormsen, T.S.Tveter, L.Bergholt, F.Ingebretsen, and [25] Jooyoung Lee and J. M. Kosterlitz, Phys. Rev. Lett. 65, 137
J. Rekstad, Nucl. Instrum. Methods Phys. Res. A 374, 371 (1990).
(1996). [26] Jooyoung Lee and J. M. Kosterlitz, Phys. Rev. B 43, 3265
[12] M. Guttormsen, T. Ramsøy, and J. Rekstad, Nucl. Instrum. (1991).
MethodsPhys.Res.A255,518(1987).