Experimental investigation on the temperature dependence of the nuclear level density parameter

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by  Balaram Dey

Experimental investigation on the temperature dependence of the nuclear level
density parameter
Balaram Dey,1 Deepak Pandit,1 Srijit Bhattacharya,2 K. Banerjee,1 N. Quang Hung,3 N. Dinh
Dang,4 Debasish Mondal,1 S. Mukhopadhyay,1 Surajit Pal,1 A. De,5 and S. R. Banerjee1,∗
1Variable Energy Cyclotron Centre, 1/AF-Bidhannagar, Kolkata-700064, India
2Department of Physics, Barasat Govt. College, Barasat, N 24 Pgs, Kolkata - 700124, India
5
3School of Engineering, Tan Tao University, Tan Tao University Avenue,
1
Tan Duc Ecity, Duc Hoa, Long An Province, Vietnam
0
4Theoretical Nuclear Physics Laboratory, RIKEN Nishina Center for Accelerator-Based Science,
2
RIKEN, 2-1, Hirosawa, Wakocity, Saitama 351-0198,

n Japan and Institute for Nuclear Science and Technique, Hanoi, Vietnam
a 5Department of Physics, Raniganj Girls’ College, Raniganj-713358, India
J
(Dated: January 8, 2015)

7
Theeffect of temperature(T) andangular momentum(J) on theinverseleveldensityparameter
] (k)hasbeenstudiedbypopulatingthecompoundnucleus97Tcin thereaction 4He+93Nbat four
x incidentbeam energies of 28, 35, 42 and 50 MeV.Forall thefour energies, thevalueof kdecreases
e with increasing J. The T dependenceof k has been compared for two angular momentum windows
-
l with different theoretical predictions as well as with FTBCS1 calculation which takes into account
c the quasiparticle-number fluctuations in the pairing field. Interestingly, the experimental data are
u
ingoodagreementwiththetheoreticalcalculationsathigherJbutdeviatefromallthecalculations
n
at lower J.
[

1
v I. INTRODUCTION of non-interacting fermions. The backshifted Fermi gas
9 model of NLD for a spherical nucleus of mass number
∗
6 A at excitation energy E and angular momentum J is
Thenucleusisamanybodyquantumsystemwhichex-
3 given by
periences many different configurations even when fairly
1
0 small excitation energy is provided. It is now a very
1. wstealtle-ksnionwcrneafasecstrtahpaitdtlyhewditehnseixtyciotaftqiounanetnuemrgymeacnhdansoicoanl ρ(E∗,J)= 2J +1√aexp(p(aU)). (1)
0 12I3/2 U2
5 becomes very large. As a result, the nucleus leaves the
1 discrete region and enters the region of quasi-continuum HereU=E*-Erot-δpistheavailablethermalenergy.
v: and continuum. Due to this complexity, statistical con- Erot, δpandI arethe energyboundin the rotation,par-
cept and models are not only appropriate but also cru- ing energy and effective moment of inertia, respectively.
i
X cial for comprehension and prediction of various nu- The nuclear level density parameter (a) is related to the
r clear phenomena. Intriguingly, the nuclear level densi- single particle level density in the region of the Fermi
a ties are indispensable in the study of nuclear reaction energy and is correlated to the mass of the nucleus as a
crosssections,nuclearreactionrates[1]whichareneeded = A/k, where k is the inverse level density parameter.
for astrophysical calculations (inputs in modeling stel- The temperature dependence of the NLD parameter has
lar evolution and nucleosynthesis), fissionor fusion reac- been investigated by various theoretical approaches [6–
tor design, transmutation of nuclear waste and produc- 10] and experimental methods [11–15]. It is found from
tionofradioactiveisotopesintherapeuticusesinnuclear theexperimentalanalysisofnuclearresonancesandevap-
medicine. Alongwiththat,theyalsoprovideinformation oration spectra that for cold nuclei k 8 MeV but for
∼
about the thermodynamic quantities such as tempera- hightemperatures like T = 5 MeV, k is around13 MeV.
ture and entropy as well as pairing correlations in the The understanding of this behavior has been attempted
nuclear structure study [2]. The knowledge of nuclear by taking into account the effects of correlation, i.e. the
level density is also very crucial while extracting the pa- T dependence of the frequency dependent effective mass
rameters of the giant dipole resonance built on highly as well as the finite size effect, the momentum depen-
excitedstatesofthenuclei[3,4]. However,thecharacter- dence of the effective mass, the effects of continuum and
ization of the nuclear level density (NLD) in the regions the shell effects [6, 9, 10].
of high excitation energy, angular momentum and dif- Itmaybenotedthatthe leveldensityformalismgiven
ferentnuclearshapesareinlargepartphenomenological. inEq.(1)isforasystemofnon-interactingfermionswith
Themostcommonlyusedanalyticalexpressionforcalcu- equidistantsingleparticlestatesanddoesnotincludethe
latingNLDisbasedontheworkofBethe[5]forasystem collectiveenhancementofNLDduetothecouplingofro-
tationalaswellasvibrationaldegreesoffreedomwiththe
single-particle degrees of freedom. The enhanced level
density is expressed as ρ(E*, J) = ρ (E*, J) K (E*),
int coll
∗e-mail:[email protected] whereKcoll(E*)isthecollectiveenhancementfactorcon-
2
sisting of both vibrational and rotational contributions 6 (a) E = 28 MeV
[16]. Very recently, neutron evaporation spectra from lab
201Tl*, 185Re*, and 169Tm* compound nuclei have been 5 F = 2
F = 3
measured at two excitation energies (E* 37 and 26 4 F > 3
∼
MeV)to studythe effect ofcollectivitybyextractingthe 3
inverse level density parameter (k) [17]. It was observed
2
that for large ground state deformed nuclei (185Re* and 3
0
169Tm*) the value of k decreased substantially at the 1 1
x
lower excitation energy, while for near spherical nucleus s 0
(201Tl) it remains the same at the two excitation ener- ount 6 (b) Elab = 50 MeV
gies. Theresultsindicatedtowardsthestrongcorrelation C 5 F = 2
between collectivity and ground-state deformation. F = 3
4 F > 3
Recently,therehavebeenampleofexperimentalefforts
in order to comprehend the spin dependence of the level 3
density parameter. In few measurements of angular mo- 2
mentum gated neutron evaporation spectra in A 119,
∼ 1
97 and 62, it was seen that the k value decreased with
0
increase in J which indicated that level density increases
0 10 20 30 40
with J [18, 19]. On the other hand, the inverse level
Angular Momentum (h)
density parameter extracted from the alpha evaporation
spectrainA 180andA 120massregionshowedthat
∼ ∼ FIG. 1: Angular momentum distribution for different folds
thevalueofkiseitherconstantorincreaseswithincrease
for the4He+93Nb system at 28 and 50 MeV incident energy.
in angularmomentum [20, 21]. However,theoreticalcal-
culationforsimilarmassesshowsthatthe kvalueshould
increase with increase in angular momentum for all the
systems [22]. Interestingly, the γ-multiplicity gated pro-
tion energies of 29.3, 36.0, 43.0 and 50.4 MeV, respec-
ton spectra in A 105 showed a drastic dependence on
∼ tively. The maximum populated angular momenta for
fold. The spectra acquired a broad structure at higher
fusionwere16,18,19and20¯h, respectively. The evapo-
multiplicity folds for proton energies beyond 15 MeV
∼ ratedneutronsfromthecompoundnucleusweredetected
which was explained by a prescription of a localised en-
by a liquid organic scintillator (BC501A) based neutron
hancement of NLD [23]. Thus, extremely exciting but
detector [24] that was placed at a distance 1.5 m from
conflicting experimental results on the spin dependence
the target position and at an angle of 900 to the beam
of the level density parameter motivate one to carry out
axis. AlongwiththeBC501Aneutrondetector,a50ele-
further investigations.
mentlowenergyγ-multiplicityfilter[25]wasalsousedto
Inthisworkwereportontheangularmomentumgated
estimate the angular momentum populated in the com-
neutron evaporation spectra at different excitation ener-
pound nucleus as well as to get a fast start trigger for
gies (30-50 MeV) for the reaction 4He + 93Nb. The spe-
neutron time-of-flight (TOF) measurement. The multi-
cific advantageofusinglightionreactionis that the ma-
plicityfilterwassplitintotwoblocksof25detectorseach,
jorresiduesareofsimilarnatureandinourcasetheyare
in a staggeredcastle type geometry to equalize the solid
95Tc, 94Tc and 93Tc depending on the excitation energy
angle for each multiplicity detector element, and placed
of the compound nucleus. However, for excitation ener-
at a distance of 5 cm above and below the centre of the
gies above 42 MeV, another channel contributes ( 20%)
due to (α,2n) populating 91Nb. Nevertheless, the∼defor- target. The efficiency of the multiplicity set-up was 56%
as calculated using GEANT4 simulation. A level-1 trig-
mations of all the nuclei populated in the decay chain
ger (A) was generated from the multiplicity filter array
aresimilarandareofthe orderofbeta 0.05. Theshell
∼ when at least one detector each from the top and bot-
effects are also very small and similar. Therefore, the
tom blocks fired in coincidence above a threshold of 250
neutron evaporation spectra will have the contribution
keV. Another trigger (B) was generated when the sig-
from similar kind of nuclei only.
nal in BC501A detector crossed a threshold of 250 keV.
An on-line coincidence of these two triggers (A and B)
ensured the selection of neutron events and rejected the
II. EXPERIMENTAL DETAILS
backgrounds. TheTOFtechniquewasemployedforneu-
tron energy measurement. The neutron-γ ray discrimi-
TheexperimentwasperformedattheVariableEnergy nation was achieved by both pulse shape discrimination
Cyclotron Centre, Kolkata using alpha beam from the (PSD) and TOF techniques. To keep the background
K-130 cyclotron. A self supporting 1 mg/cm2 thick tar- of the detectors at a minimum level, the beam dump
get of 99.9% pure 93Nb target was used. Four different was heavily shielded with lead bricks and borated paraf-
beam(4He) energiesof 28,35,42and50 MeVwereused fin in both the experiments. A CAMAC electronics and
to populate the compound nucleus 97Tc at the excita- VME based data acquisition system were used to simul-
3
(a) (b) 11
)
104 k (a) Elab = 28 MeV (b) Elab = 35 MeV
(
F=2 28 MeV F=2 er 10
103 t
e
F=3 F=3 m 9
102 F>4 F>4 ra
a
p 8
101 y
t
104 si 7
F=2 35 MeV F=2 en (c) Elab = 42 MeV (d) Elab = 50 MeV
d
103 F=3 F=3 el 10
V v
Me102 F>4 F>4 le 9
5 se
0. er 8
d / 104 42 MeV nv
el F=2 F=2 I 7
Yi103
F=3 F=3 12 14 16 18 20 12 14 16 18 20
102 F>4 F>4 Angular momentum (h)
FIG.3: Angularmomentumdependenceofnuclearlevelden-
104 sity parameter at four incident energies.
F=2 50 MeV F=2
103 F=3 En (MeV) F=3
F>4 F>4
102 at high excitation energy. The transmission coefficients
for statistical calculation were obtained from the opti-
calmodel. The potentialparametersfor neutron,proton
2 4 6 8 2 4 6 8 and alpha were taken from [29], [30] and [31], respec-
E (MeV) tively. The experimental fold distribution measured us-
n
ing the 50-element γ-multiplicity filter was converted to
thespindistributionusingGEANT4simulationapplying
FIG. 2: (a) Neutron evaporation energy spectra (filled trian-
gles) alongwith theCASCADEpredictions(continuousline) the approach discussed in Ref [25]. The simulated spin
fordifferentfolds(F)atincidentenergiesof28,35,42and50 distributions deduced from the experimental fold distri-
MeV for 4He+93Nb system. (b) Neutron evaporation energy butions were used as inputs for different folds. The an-
spectra (filled triangles) along with the Maxwellian fitting gular momentum distributions for different folds at 28
(continuous line) for different folds (F) at incident energies and 50 MeV incident energies are shown in Fig 1. The
of 28, 35, 42 and 50 MeV for 4He+93Nb system. F=2 and momentofinertiaoftheCNwastakenasI =I *(1+
eff 0
F=3 data havebeen multiplied by 100 and 10, respectively. δ J2 + δ J4) where I is the moment of inertia of spher-
1 2 0
ical nucleus. The role of the deformation parameters δ
1
and δ were found to be inconsequential and the shape
2
of neutron energy spectra was mostly dependent on the
taneously record the energy and time information of the
inverse level density parameter. The value of k has been
detectors.
extracted from the best fit statistical model calculations
usingaχ2 minimizationintheenergyrangeof3 7MeV
−
(Fig 2). The extracted inverse level density parameters
III. DATA ANALYSIS AND RESULTS aregiveninTable Ifordifferentangularmomentaand
−
excitation energies.
The TOFspectrumwasconvertedtoenergyspectrum It is very interesting to note that the values of inverse
using the prompt γ peak as a time reference. The effi- leveldensityparameterdecreasewithincreaseinangular
ciencycorrrectionforBC501Aneutrondetectorwasper- momentum for all the incident energies (Fig 3). The re-
formed using GEANT4 simulation [26]. In order to ex- sultisincontrasttothetheoreticalcalculationsobtained
tractthe inverseleveldensity parameterfromthe exper- under the frameworkof statistical theory of hot rotating
imental neutron spectra, the theoretical neutron energy nucleiwhichpredictsthatkshouldincreasewithincreas-
spectra were calculated employing the statistical model ing angular momentum [22]. However, the trend of the
code CASCADE [27]. The level density parameter pre- angular momentum dependence measured in this work
scriptionofIgnatyuketal. [28]wasadoptedwhichtakes is consistent with the previous measurements for 62Zn
into account the nuclear shell effects at low excitation [19] and 119Sb [18]. The inverse level density parameter
energy and connects smoothly to the liquid drop value as a function of temperature, for two angular momen-
4
14 11
(a) J = 12-15 h Lestone k) Experimental Data
)12 Shlomo r (10 FTBCS1
k Mughabghab e
er ( met 9 FTBCS
et10
m a
a ar 8
ar 8 p
y p ty 7 (a) J = 12-15 h
nsit 6 nsi
de (b) J = 16-20 h de10
vel 12 el 9
e v Col 1 vs Col 2
se l10 e le 8 CCooll 1126 vvss CCooll 1137
er s
v r
In 8 ve 7 (b) J = 16-20 h
n
I 6
6
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Temperature (MeV) Temperature (MeV)
FIG. 4: Temperature dependence of nuclear level density FIG.5: Temperature dependenceofthenuclearleveldensity
parameter compared with different theoretical calculation. parametercomparedwiththecalculationsofFTBCSandFT-
Green dashed line represents the calculation by Shlomo, red BCS1.
dashed line represents the calculation by Lestone and black
dot-dashed line represents thecalculation by Mughabghab.
populated compound nucleus for different excitation en-
ergies were calculated by fitting the experimental data
TABLEI:Averagetemperatureandaverageangularmomen-
tumalongwithleveldensityparameterfor4He+93Nbsystem with Maxwellian function (√Eexp(−E/T)). It needs to
be mentionedthatnoneofthe theoreticalpredictionsin-
at different beam energies.
clude J effect. As can be seen from Fig 4, the inverse
level density parameter increases with increase in tem-
perature for all the predictions. However, the data at
Systems E∗ hJi hTi A/k higher angular momenta match very well with the pre-
MeV ¯h MeV MeV−1 diction but are at disparity at lower angular momenta.
12± 5 1.18 A/(8.5± 0.5) This outcome was unexpected since the theoretical pre-
4He+93Nb 29.3 14± 6 1.06 A/(8.3± 0.3) dictions do not include J effect and, thus, experimental
data and theoretical calculation should have matched at
@Elab = 28 MeV 16± 5 1.03 A/(8.0± 0.5)
lowerJ.Theresultsindicatethattheleveldensityissup-
13± 4 1.47 A/(9.7± 0.5)
pressed at lower angular momenta. This deviation can-
4He+93Nb 36.0 15± 5 1.41 A/(9.5± 0.3)
notbe attributed to the effect of collectivity because the
@Elab = 35 MeV 18± 5 1.35 A/(8.2± 0.4) level density is not enhanced but suppressed compared
14± 5 1.52 A/(9.0± 0.4) to the Fermi gas model.
4He+93Nb 43.0 16± 5 1.43 A/(8.1± 0.4)
Recently,thetemperaturedependenceofthelevelden-
@Elab = 42 MeV 19± 6 1.38 A/(7.8± 0.5) sity was studied in hot medium-mass nuclei, which un-
14± 5 1.60 A/(9.2± 0.5) dergo a non-collective rotation about the symmetry axis
4He+93Nb 50.4 16± 5 1.55 A/(8.5± 0.4) [32]. Thenumericalcalculationswithinthefinitetemper-
@Elab = 50 MeV 20± 5 1.50 A/(8.2± 0.4) ature BCS (FTBCS) and FTBCS1 theories have shown
the pairing re-entrance in the pairing gapat finite angu-
lar momentum M (M is the z-projectionof total angular
momentum) and temperature T. The FTBCS1 theory
tum windows,has also been comparedwith the different includes the effects due to quasiparticle-number fluctua-
theoreticalcalculationspredictedby Shlomo [6], Lestone tions in the pairing field and the z projection of angu-
[7] and Mughaghab [8] (Fig 4). It should be mentioned lar momentum at T = 0 MeV. The pairing re-entrance
6
that the prediction of Shlomo has been taken for A = changestheTdependenceoftheleveldensityfromacon-
110 mass region [6] whereas the other two predictions vex function to a concave one. Similar calculation was
have been calculated for A=97 mass using the formula performed to observe wheather the pairing re-entrance
given in Ref [7, 8]. The average temperatures of the plays any important role in the case of 97Tc. It was ob-
5
servedthattheFTBCSgapscollapseatacertaincritical behavior.
temperature,whereastheFTBCS1gapsdonot. Starting
from M=12 ¯h the FTBCS1 proton gap shows the pair-
ing re-entrance effect, that is the pairing gap reappears
IV. SUMMARY AND CONCLUSIONS
and remains finite at T>0.3 MeV and M 12 h¯. The
≥
value of k was extracted from the excitation energies us-
ing the relations k = 4AE∗/S2. Since, the nucleus 97Tc Angular momentum gated neutron evaporation spec-
is not spherical but slightly prolate and the calculations tra have been measured in the reactions 4He + 93Nb at
werecarriedout under the assumptionofa sphericalnu- E = 28, 35, 42 and 50 MeV to study the T and J
lab
cleus for which the z-projection M of the total angular dependence of the inverse level density parameter. It
momentum J coincides with J, the results of theoretical wasobservedthattheinverseleveldensityparameterde-
calculationshavebeenre-normalizedto matchthecorre- creased with increasing J for all the excitation energies.
spondingdata pointatthe lowestJforboththe selected The T dependence ofthe inverseleveldensity parameter
angular momentum window. As can be seen from Fig wasstudiedbyselectingtwoangularmomentumwindows
5, the increase of k with T is observed in the data are andcomparedwithdifferenttheoreticalcalculations. In-
also reproduced by the results of theoretical calculations triguingly, the results of all the theoretical calculations
although the latter agree better with the data for the agree well with the data at higher J window but differ
higherJwindowwhereasforthelowerJwindowthethe- from the experimental data at lower J window.
oryunderestimatesthetwodatapointsatT=1.41MeV
(J=15¯h) and 1.47 MeV (J=13¯h). The FTBCS and FT-
BCS1 predict similar results indicating that pairing re-
Acknowledgements
entrance does not have much effect on the inverse level
density parameter in this case. As could be discerned,
collectivity and pairing re-entrance are not responsible The authors greatfully acknowledge helpful discussion
for decrease in k with increase in angular momentum. with Professor J. N. De. The theoretical calculations
Moreover, it may be noted that the effect of angular within FTBCS and FTBCS1 were carried out using
momentum on the k is not observed for higher masses RIKENIntegratedClusterofClusters(RICC).NQuang
[20,21,33]butisonlyapparentinlowandmediummass HungacknowledgesthesupportbytheNationalFounda-
A 120nuclei[18,19]. Moreexperimentaldata atboth tion for Science and Technology Development (NAFOS-
≤
highandlowmassregionarerequiredtounderstandthis TED) of Vietnam through Grant No. 103.04-2013.08.
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