APS/123-QED

Probing Nuclear Shapes Close to the Fission Limit with the GDR in 216Rn

M. Kmiecik1, A. Maj1, B. Million2, M. Brekiesz1, W. Krolas1, W. Meczynski1, J. Styczen1,

M. Zieblinski1, A. Bracco2, F. Camera2, G. Benzoni2, S. Leoni2, O. Wieland2, S. Brambilla2,

B. Herskind3, M. Kicinska-Habior4, and N. Dubray5, J. Dudek5 and N. Schunck6

1The Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences,ul. Radzikowskiego 152, 31-342 Krakow, Poland

2Dipartimento di Fisica, Universita di Milano and INFN Sez. Milano, Via Celoria 16, 20133 Milano, Italy3The Niels Bohr Institute, Blegdamsvej 15-17, 2100, Copenhagen, Danmark

4Institute of Experimental Physics, Warsaw University, Warsaw, Poland5Institut de Recherches Subatomiques and Louis Pasteur University,23 rue du Loess, BP28 F-67037, Strasbourg Cedex 2, France and

6Department of Physics, 30BC04, University of Surrey, Guildford GU2 1XH, UK(Dated: October 22, 2004)

The gamma-ray decay of the Giant Dipole Resonance in the compound nucleus 216Rn formed withthe reaction 18O+198Pt at the bombarding energy of 96 MeV was investigated. High energy gamma-ray spectra in coincidence with both prompt and delayed low energy transitions were measured. Theobtained GDR width at the average temperature 〈T 〉 ≈ 1 MeV was found to be larger than thatat T = 0 MeV and to be approximately constant as a function of spin. The measured width valueof 7 MeV is found to be consistent with the predictions based on calculations of the nuclear shapedistribution using the newest approach for the treatment of the fission barrier within the liquid dropmodel. The present study is the first investigation of the Giant Dipole Resonance width from thefusion-evaporation decay channel in this nuclear mass range.

PACS numbers: 21.60.Ev, 24.10.Lx, 24.30.Cz, 24.60.Dr

I. INTRODUCTION

The mechanism of nuclear fission induced by fast ro-tation and the nuclear shape evolution on the way toscission is an interesting and still open question. TheGiant Dipole Resonance (GDR) has proven to be sensi-tive to the average nuclear shapes at finite temperatureand therefore it represents an adequate tool to study thisproblem [1–3]. Nuclear deformations corresponding tothe saddle point configurations are often expected to bevery large. The related experimental information is par-ticularly interesting as it provides a sensitive test of boththe macroscopic methods of calculating the total nuclearenergy at high temperatures as well as the microscopicones taking into account thermal excitations.

The term ’fission limit’ used throughout this paper ap-plies by definition to the spin value, say If , for which thefission-barrier penetrability from the static equilibriumdeformation equals 50%. Within a model, such a spincan be relatively easily determined in the calculationsbut much less so through measurement. In no place inthis paper the precise value of such an experimental pa-rameter will be needed. Calculation results and theircomparison with experiment suggest that in the presentcontext If ∼ 40 ~, within a few ~ inaccuracy.

In order to obtain information about the nuclear shapeevolution along the fission path, it is possible to use atleast two different approaches. One of them consists instudying the GDR in fissioning nuclei, by measuring theGDR decay during the fission process. From the high-energy γ-ray spectra in coincidence with the fission prod-ucts, in principle one may expect to obtain the informa-

tion about the shape of fissioning nuclei and thus abouttheir shape evolution along the fission path. However,the existing data show that this is not at all an easy taskbecause of the presence in the spectra of the γ-rays fromthe fission products [4, 5].

Another approach consists in measuring the GDR de-cay in the fusion-evaporation process by selecting veryhigh spins around the critical spin-value at which fissionprocess is beginning to dominate. Such measurementsare also in general very difficult, because one has to se-lect a very narrow spin window and at the same time toreject the fission contamination. This can be achievedby tagging either on the fusion-evaporation products oron the γ-rays from specific residual nuclei. So far, in-vestigations of this type have provided high-energy γ-rayspectra through gating on different residual nuclei (seee.g. the case of 194Hg [6]). However, they have not al-lowed for the spin selection in a narrow region close tothe fission limit.

In this paper we present the results of an experimenton the GDR γ-decay associated with high spins aroundthe fission limit. For this purpose, we choose the fusion-evaporation reaction 18O+198Pt at the bombarding en-ergy of 96 MeV, populating with a rather high cross-section residual nuclei characterized by the presence oflong-lived high-spin isomeric states. The decaying 216Rncompound nucleus, selected in the present study, stronglyfeeds the isomeric states of I = 30 in 212Rn and I = 63/2in 211Rn. In addition, since only two feeding transi-tions to these isomeric states were found [7], one expectsthat the fission limit in 216Rn is below or around 40 ~.Therefore, the selection of the GDR γ-decay in coinci-

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dence with delayed γ-ray transitions is expected to probemainly the compound nuclei which survive fission, yet,with angular momenta close to the fission limit.

It should be noted that the isomer tagging techniqueused in the present work has been successfully employedin lighter-mass nuclei to select the particular areas ofthe phase space of the decay, leading to specific nucleardeformations [8].

In the following we present the GDR data concerningthe compound 216Rn nucleus, and discuss the resultinginclusive and exclusive (isomer tagged) spectra. We findit attractive to perform the investigations in this heavymass region, in particular for 216Rn for which the theorypredicts a rather unexpected shape evolution as a func-tion of angular momentum up to the fission limit, namely:almost no evolution at all. More precisely, this nucleus ispredicted to proceed from small oblate deformations tofission directly i.e. without passing through intermediateequilibrium shapes, in contrast to many other nuclei inwhich the presence of the high angular momentum in-duces either the super-deformed, intermediate equilibriaat high temperatures, and/or the Jacobi-type shape tran-sition. We find it instructive to test the actual predictionexperimentally.

II. THE EXPERIMENT

The experiment was performed at LNL in Legnaro(Italy) using 96 MeV 18O beam bombarding a self-supporting target 198Pt (1 mg/cm2 thick). The bom-barding energy was chosen to lead to the angular mo-mentum distribution with lmax ≈ 42, which is slightlylarger than the critical spin for fission and therefore al-lows for a good population of the isomeric states in 211Rn(T1/2 = 201 ns, I = 63/2−) and 212Rn (T1/2 = 154 ns,

I = 30+) through the intense channels 5n and 4n. Thecorresponding excitation energy of the compound 216Rnnucleus is E∗ = 56 MeV. The main residues of the formedcompound nucleus are 212,211,210Rn covering 99 % of totalevaporation residues. The predicted cross sections for thepopulation of the 212,211,210Rn residual nuclei are shownas functions of spin in Fig. 1. In the left panels the totalfusion-evaporation cross section for 212,211,210Rn is dis-played while in the right panels the fraction (in percent)of the population of each residual nucleus is given. Allthe calculated results were obtained using the statisticalmodel and the Monte Carlo technique. The curves in thetop panels correspond to no condition on the high energygamma-rays required. In contrast, the curves displayedin the bottom panels are obtained with the additionalcondition of having in the decay leading to the resid-ual nuclei a high energy gamma-ray with Eγ > 8 MeV.One can see that requiring the presence of a high energygamma-ray in the GDR region (as it is done in the ex-periment) implies the selection of events correspondingmostly to the 212Rn residual nucleus. It is character-ized by high spins (experimentally selected by the isomer

decay). Therefore on the basis of these results and forthe chosen experimental conditions we do not expect anysizable background due to fission events.

FIG. 1: Calculated cross sections for the population of evap-oration residues in 212,211,210Rn as functions of spin: with nocondition on γ-ray energy (top) and with the gate on γ-rayenergy in the GDR region (bottom).

In the experiment the HECTOR array [9] was em-ployed. High-energy γ-rays were measured in 8 largeBaF2 detectors, while the prompt, low-energy transitionswere detected in 38 small BaF2 detectors arranged inhoney-comb mode above and below the target locationand provided the sum-energy and the γ-ray fold. In ad-dition, a Mylar foil (8 µm) with a central hole (6 mm)placed downstream at 40 cm from the target was em-ployed as a catcher to stop the recoiling residual nuclei.The direct beam and most of the scattered beam passedtrough the central hole in the catcher. Because of thishole, the stopping efficiency was only about 28 %. Thechosen distance ca. 40 cm was appropriate for the life-time of the isomeric states of interest, namely 201 ns and154 ns. The delayed gamma radiation of the stoppedresidues (deexciting the isomeric states) was measured ina BGO detector. This BGO detector was the standardCompton shield usually used for Ge-detectors. Becauseof its geometry the BGO shield surrounded the beam-linearound the catcher, covering a large fraction of the solidangle. In addition, a Ge-detector with its BGO shieldwas placed near the reaction chamber at 146o to mea-sure discrete transitions at the target position. In orderto detect the delayed gamma transitions the beam waspulsed with 10 ns pulses separated by 400 ns intervals.

The trigger condition was the logical sum of the fol-lowing events: i) the coincidence of the high-energy γ-rays with the multiplicity filter; ii) the coincidence of thehigh-energy γ-rays with the multiplicity filter and theBGO detector; iii) single Ge-detector firing; iv) the coin-cidence of the Ge-detector with the BGO detector.

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III. EXPERIMENTAL RESULTS

In this section we are going to present various aspectsof the experimental analysis and the obtained results,including the analysis of the time spectra, of the folddistributions as well as the statistical model analysis andthe corresponding parametrisations of the GDR curves.

A. Time Spectra

In Fig. 2 (top panel), the time spectrum of the gammaradiation emitted by nuclei stopped in the catcher - withrespect to a prompt radiation detected with the multi-

FIG. 2: Top panel: Time of flight spectrum measured by thecatcher detector around the catcher foil. Regions correspond-ing to various processes taking place during the measurementare indicated. The region corresponding to the emission fromrecoils at the catcher is shown in detail in the inset; this spec-trum is background subtracted. It corresponds to detectionof the γ-rays from the isomeric states in 211Rn and 212Rn.Bottom panel: The γ-ray spectrum measured with the Gedetector as obtained without any gate on the time spectrum(gray histogram) compared to the spectrum gated with isomerpart of time of flight (black histogram). The gated spectrumshows an enhancement of the transitions above the isomericstates. The inset to this panel shows the parts of the levelschemes of 211Rn and 212Rn with transitions above the iso-meric states.

plicity filter around the target - is shown. This spec-

trum has a rather complicated structure, which resultsfrom many different components of various decays in-volved. In particular, we identify in the spectrum theprompt γ-rays from the target, the γ-rays produced bythe scattered beam in the catcher, and the γ-rays emit-ted by the nuclei with the mass similar to the targetmass stopped in the catcher. The events of interest arein the region indicated by the horizontal full arrow whichcorresponds to the time of flight of fusion-evaporationrecoils (ca. 150 ns) convoluted with the isomeric-decaytime-distribution. One can see an excess of counts inthis region as compared to the overall background. Thiscan be seen more clearly in the inset in Fig. 2, where theregion of interest, after background subtraction, is shownin linear scale. In order to investigate the quality of theisomeric-decay selection by using the time gating, in thebottom part of Fig. 2 we compare two Ge-spectra: theone without any conditions and the other one, gated bythe delayed γ-rays (background subtracted) measured bythe BGO around the catcher, with the time condition asindicated in Fig. 2 (top). The spectrum gated by thedelayed radiation shows mainly the lines correspondingto the prompt transitions feeding the isomeric states (seeinset to the Fig. 2, bottom panel). This demonstrates thesensitivity of our gating condition for selecting most ofthe contributions from the decay chains populating thehigh-spin isomers. It should also be noted that the ob-served intensity ratio of 924 and 1117 keV transitionssuggests that the transition order should be reversed,compared to that given in Ref. [7].

Figure 3 shows the high energy spectra obtained bygating on ”recoils at catcher” and on ”background” re-gions (these regions are shown in the top panel of Fig. 2).The total spectrum denoted as ”recoils” contains back-ground in addition to the data corresponding to the iso-meric states. The background was eliminated by sub-tracting the background spectrum from the total one af-ter normalization. The difference is the isomer gatedhigh-energy gamma spectrum used for analysis with thestatistical model (see Sect. III C).

B. Fold Distributions

The experimental fold distributions measured in thepresent experiment are shown in Fig. 4. Open squaresshow the total fold distribution, the full triangles the folddistribution gated by the isomeric decay, while the fullcircles - the difference between these two spectra. Thelatter gives the fold distribution of the total prompt decaywhich is compared to the calculated ones (lines) assum-ing a particular lmax value, ranging from 33 to 45 ~. Thecalculations took into account the multiplicity filter effi-ciency and the scattering probability (using the methoddescribed in [9]). We may remark that: a) the value oflmax that gives the best reproduction of the experimentaldata corresponds to 39 ~, the spin-value very close to thepredicted fission limit; and b) the fold distribution popu-

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FIG. 3: The spectra of high energy γ-rays for 3 different con-ditions: recoils and background corresponding to the gatesshown in Fig.2 and the difference of these two spectra shownwith filled squares.

lating the isomeric state agrees well with the assumptionof 5 transitions (on the average) above the isomeric state,as suggested by the good reproduction of the data withthe calculated curve. These two remarks provide an ad-ditional proof of the good time gating sensitivity to selectthe cascades leading to the isomers.

FIG. 4: The experimental coincidence fold-distributions ofthe low energy transitions. The data without any conditionson the time of flight spectrum of Fig. 2 are marked with thesquares, while the triangles correspond to the isomer gateddata. The difference of these two distributions (after normal-ization in the total counts in the fold interval 1-4) is shownwith the circles. The lines show the calculated fold distribu-tions, each corresponding to a value of lmax in the interval33 to 45 ~. The best fit to the data is for lmax = 39 ~ (solidline), while the distribution with the average γ-multiplicity,Mγ = 5, corresponds to the isomer gated data.

High-energy γ-ray spectra were obtained with the con-ditions set either on the gamma coincidence-fold or on

the time-spectrum, in order to select the delayed radia-tion emitted and detected around the catcher foil. Thenon-isomer gated spectra are taken for folds > 4 only, asthe lower fold spectra can be contaminated by the reac-tions other than fusion. The isomer gated high-energyspectrum corresponds to the whole fold region.

C. Statistical Model Analysis

In this section we present the measured high-energyγ-ray spectra and discuss the comparison with the sta-tistical model predictions which are needed to extract theGDR parameters. All calculations were performed by us-ing the Monte-Carlo version of the CASCADE code. Infact, as we will discuss below, only by using the Monte-Carlo technique, one can obtain predictions correspond-ing to the evaporation chains leading to the specific resid-ual nuclei.

The first step of the analysis was to obtain the in-formation about the GDR from the high-energy γ-rayspectrum corresponding to the 5-30 range of the γ-foldmeasured by the multiplicity filter. As discussed in theprevious section, in connection with the multiplicity fil-ter data of Fig. 4, this fold interval is expected to se-lect the high energy γ-rays mainly coming from fusion-evaporation events. This spectrum, denoted ’total’, isshown in Fig. 5 and compared to the best-fit statistical-model calculation in which the experimental yrast lineand Reisdorf level density model [10, 11] were used. Withthis Monte-Carlo code we produced 1 million events, eachconsisting of a chain including evaporated-particle type,energies and spins as well as the γ-ray properties. Thisenabled us to sort those events simulating in the bestpossible way the experimental conditions of selecting thespin of the residual nucleus. The spin distribution ofthe compound nucleus, given as an input to the code,was deduced from the measured low-energy γ-ray fold-distribution adopting the same procedure described inRef. [9] which is based on the measured response functionof multiplicity filter. To convert the multiplicity to spinwe applied the empirical relation I = 2M − 3 which as-sumes quadrupole stretched transitions and a correctionfor the E1 transitions and angular momentum removedby a decay through particle emission. This resulted inlmax = 39 ~ which reproduces very well the measuredfold distribution (see Fig. 4).

The GDR parameters (centroid and width) are de-duced from the chi-square minimization in the region ofγ-ray energy of 8-14 MeV for the isomer gated-, and of 8-16 MeV in the total-data case. We use two approaches forthe parameterization of the GDR line-shape. The first,similar to that in many previous works, assumes a singleLorentzian function. In this case the best fit of the GDRwidth is 7 MeV. The second approach takes into accountthe fact that the dipole resonance mechanism involvesthree degrees of freedom usually interpreted in terms ofoscillations in three orthogonal directions, and therefore

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FIG. 5: Top panel: The experimental high energy γ-ray spec-trum corresponding to the folds 5-30 represented with fulldots. The line is the result of the best-fit statistical-modelcalculations.Bottom panel: The data in the linearized form described inthe text and the best fit GDR line-shape (solid line) consistingof the sum of three Lorentzian functions. The one-Lorentzianfunction drawn with the dashed line is compared to the sum(full line) of three Lorentzian components.

we parameterize its line-shape by a superposition of 3-Lorentzians (below abbreviated to 3-L). With this ap-proach we obtain a better description of the experimentaldata with an overall GDR width nearly identical to thatof the single Lorentzian fit. This is illustrated in the bot-tom part of Fig. 5, where we compare the data in the lin-ear form and the best fitted GDR strength function rep-resented by the quantity: F3L(Eγ)∗Y expγ (Eγ)/Y calγ (Eγ).

In that expression Y expγ (Eγ) and Y calγ (Eγ) are the exper-imental and calculated spectra, respectively. The bestfitting GDR strength function, F3L(Eγ), consists of 3-component Lorentzian function with parameters (energy,width and strength) taken from the best fit to the exper-imental spectrum. F3L(Eγ) and its individual compo-nents are also presented in the figure.

It should be noted that the best-fit F3L(Eγ)-line, isa superposition of three similar Lorentzians positionedat EGDR = 11.2, 13.2 and 15 MeV with the widths ofΓGDR = 3.5, 4.5 and 5 MeV, respectively. It has a widthof 7 MeV - the result coinciding with the one obtained

when using the single-Lorentzian approach. The 3-L ap-proach suggests that the susceptibilities in generating thethree orthogonal modes are different thus implying theobserved differences in the Lorentzian factors. Our par-ticular result does not give an evidence of the triaxiality;we rather conclude that the shapes involved are not verydifferent from the spherical one, while in terms of theamplitudes of the oscillations related to the three modessome differences do appear.

FIG. 6: The experimental spectra (points) corresponding tothe selected low- and high-fold regions, and the best fit MonteCarlo cascade calculations (lines) performed with a value ofthe GDR width of 7 MeV.

In the next step of the analysis we interpret the high-energy γ-ray spectra associated with two different spinregions of the compound nucleus. Fig. 6 shows two spec-tra, the one obtained by gating on the fold 5-8 and theother one by gating on the fold 9-30. They correspondto 10 ~ wide regions of angular momentum, with the av-erage spin of 23 ~ and 29 ~, respectively. The statisticalmodel fits, corresponding to these spin regions, are alsodisplayed in the figure. The 3-L approach, applied here,results in the GDR parameters (centroids and widths)equal to those obtained from total spectrum. The fittedsingle Lorentzian GDR width is also 7 MeV, in both casesstudied.

The most interesting high energy γ-ray spectrum isthe one corresponding to the selection of the delayed lowenergy gamma transitions measured around the catcher.In particular, such a spectrum is obtained by gating onthe isomer related part of the time spectrum, namelythat indicated by the double-end long arrow in Fig. 2,and using as the background the spectrum gated by thepart indicated by the dotted arrow. The selection on thepopulation of the isomeric states leads to a high-energyγ-ray spectrum with rather poor statistics, dying out atabout the GDR centroid energy.

The isomer gated spectrum is shown in Fig. 7. To seewhether or not its shape is very different from that ofthe total spectrum, the isomer gated data are compared

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to the best-fit calculation-result for the total spectrum(shown with thin solid line in Fig. 7). There are alsoshown results of the calculations performed for two otherextreme GDR width values viz. 4 and 11 MeV (dashedand dash-dotted lines, respectively). One can see thatnone of them can describe the experimental data.

FIG. 7: The experimental high-energy spectrum gated by theisomer decay is shown with the full circles. Four statisticalmodel Monte Carlo calculations are shown with lines. Thecalculation using the fold interval 5-30 and GDR width of7 MeV, which corresponds to the best-fit curve of Fig. 5, isdisplayed as thin solid line. The calculations corresponding toGDR widths of 4 and 11 MeV are shown with the dashed anddash-dotted lines, respectively. The fit to the isomer gatedspectrum obtained for Γ = 7.3 ± 1 MeV and assuming onlythe high-spin region (〈I〉 = 35 ~) is presented with thick solidline. The shaded area indicates the possible deviations of thecalculated GDR spectrum from the best fit curve within the1 MeV error bars of the GDR width.

Indeed, the isomer gated spectrum has the shape re-markably different from that of the total spectrum inthe whole region of Eγ > 6 MeV. In this context it isimportant to stress that the isomer-gated spectrum cor-responds to a different spin distribution as compared tothe total spectrum and, therefore, it is related to a differ-ent region of the phase space of the nuclear decay. Con-sequently, before trying to fit the spectra by changingthe GDR parameters, we have introduced in our sim-ulation the condition of gating on the isomeric decay.This was achieved by performing the Monte-Carlo calcu-lations. We used the same parameters as those used inthe analysis of the total spectrum. We selected only thecascades leading to the nuclei having the average entryspin of 35 ~, that is the average of the angular momen-tum distribution ranging from 30 ~ to the lmax. Thisresulted in quite good overall reproduction of the data.

Subsequently, in order to fit the isomer-gated spectrumthe calculations for different GDR parameters were made.The best fit spectrum shown with the thick full line inFig. 7 was obtained for the GDR width Γ = 7.3 ± 1 MeVusing χ2 minimization. It indicates that the GDR widthat these very high spins does not differ appreciably fromthe widths obtained at lower spins. In Fig. 7 a shadedarea is shown which is limited at the lower and upperextremes by the values corresponding to the GDR widthof 8.3 and 6.3 MeV, respectively.

IV. CALCULATION METHOD

It is known from several other studies that the role ofthe shell effects diminishes with temperature. In doublymagic nuclei the temperature at which the strong shelleffects are smeared out is expected to be of the order of2 MeV. Outside the doubly closed-shell nuclei, these lim-iting temperatures are lower and the calculations basedon the cranking Strutinsky method (cf. e.g. [12]) indi-cate that at T ∼ 1 MeV the corresponding shell effectsare usually negligible. Consequently, the results of themacroscopic model calculations for the 216Rn nucleus inthe form of the potential energy surfaces shown in Fig. 8,are a realistic approximation of the nuclear energies atthe 1 MeV temperature that corresponds to the condi-tions of the present experiment.

To understand better the shape fluctuation mechanismunderlying the experimental conditions of the presentstudy, we would like to obtain the shape distributionwith the help of the newest version of the liquid dropmodel that accounts explicitly for the nuclear-surfacecurvature effects. These effects are particularly impor-tant at large elongations, especially for the neck-formingnuclear configurations and the corresponding approachis expected to provide a better description, in particular,of the fission barriers. Such a model was recently devel-oped in Strasbourg within the Lublin-Strasbourg collab-oration and is further referred to as Lublin-StrasbourgDrop (LSD) approach [13, 14]. It has been shown thatthe LSD approach provides indeed a more realistic pa-rameterization of the fission barriers as compared to theearlier calculations of this type and one may expect it togive a more realistic distribution of shapes close to thefission limit. In fact, in the case of the light nucleus 46Ti,the model does predict a Jacobi transition in accordancewith the experimental results of Ref. [15].

The technique of calculating the distributions of proba-bility to find a given nucleus at the deformation specifiedby an ensemble of deformation parameters, say {α}, isfairly standard nowadays (cf. Ref. [16]) and below wegive only a few mathematical expressions rendering thepresented information reproducible by an independentstudy.

The probability distribution to find the nucleus at agiven deformation is given by the so-called Boltzmannfactor, P = P ({α}, I, T ); in nuclear physics applications

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FIG. 8: (Color online) The 216Rn potential energy maps calculated using the LSD model. As one can see, this approach predictsfor the 216Rn nucleus an equilibrium shape (cf. the minima of the potential energy) which is approximately spherical up to thespins around I = 40 ~. At I = 40 ~ the fission barrier is about 3 MeV high, so that for I ≥ 40 ~ the fission decay mode isexpected to dominate. Note that the only evolution with spin is the decreasing in the barrier height and no Jacobi transitionis expected in this nucleus according to the calculations.

it is a function of the ensemble of deformation parame-ters, {α}, spin, I , and the temperature, T . To shortenthe notation, we omit those three arguments and write

P ∼ e−F/kT ; F ≡ E − TS, (1)

where F denotes the free energy, E the energy of the sys-tem (in the present application set equal to the macro-scopic energy obtained from the LSD approach), and Sthe entropy given by the single-particle level-occupationfactors nν

S = −∑

ν

nν lnnν −∑

ν

(1− nν) ln(1− nν). (2)

The occupation factors depend in fact on the deforma-tion, spin and temperature as well, but as before, thesearguments are omitted and we have

nν =1

1 + exp[(eν − λ)/T ]. (3)

The Fermi energy λ is found from the usual particle-number condition

(N or Z) =∑

ν

nν . (4)

The single-particle energies eν are calculated using thedeformed Woods-Saxon potential with the Universal pa-rameter set (cf. Ref.[17]).

For a given experimental excitation energy E∗, we cal-culate the thermal excitation energy available to the sys-

tem, U = E∗−E, at each deformation point of the (β, γ)-plane of the standard quadrupole deformations. The lat-ter energy can be expressed as the difference of the mi-croscopic thermal energies E ≡∑ν nνeν , viz.

U = E({α}, I, T )− E({α}, I, T = 0). (5)

Equations (4) and (5) provide the system of non-linearequations from which T and λ are obtained.

The potential energy surfaces of Fig. 8 were obtainedusing the LSD approach; they were used to calculate theBoltzmann factors of Eq. (1); the resulting distributionsdescribing the probabilities of finding the nucleus at agiven deformation are shown in Fig. 9 of the followingsection. The corresponding results were used as input togenerate the comparison with the experimental results interms of the usual observables as discussed in the follow-ing section.

V. COMPARISON WITH THEORY ANDDISCUSSION

To begin let us compare the experimental results; allthe data for the Giant Dipole Resonance width obtainedfrom the analysis of various spectra are shown in Table Ifor the hot rotating 216Rn. They correspond to an aver-age temperature of 〈T 〉 ≈ 1 MeV and to several valuesof spin which are also reported in the Table. It shouldbe noted that the GDR studies performed so far in thisheavy mass region were focused on the fission channel.The present study is the first investigation of the GDR

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FIG. 9: The probability distributions calculated using the model Boltzmann factors as discussed in the text. Observe thestability of the position of the maximum of the displayed distributions. However, the distributions spread out as function ofspin, their maxima get lower and, at I ∼ 40 ~ the traces of the fission path become visible.

width in connection with the fusion-evaporation decaychannel.

TABLE I: The GDR parameters for 216Rn obtained from thefit to the experimental data.

Data Fold 5-8 Fold 5-30 Fold 9-30 Isomer gated〈I〉[~] 23 26 29 35

Γ [MeV] 7.0±0.3 7.0±0.3 7.0±0.4 7.3±1.0E [MeV] 13.2±0.1 13.2±0.1 13.2±0.3 13.2±0.5

The measured values of the resonance width are foundnot to depend strongly on spin at least in that partic-ular nucleus. They are larger as compared to the zero-temperature widths (Γ0 ≈ 4 MeV) known from othernuclei in the mass range of interest. This difference isof the same magnitude as that found in lighter nuclei atsimilar temperature; it has been interpreted in terms ofthe thermal shape fluctuations. In addition, the fact thatthe width is not changing in any important way with spinindicates that the effective deformation is not changingsignificantly with the spin either.

According to the theoretical prediction in Fig. 8 the216Rn nucleus remains nearly spherical in its groundstate; however, the fission barrier decreases from ∼9 MeV at I = 10 ~ to ∼ 3 MeV at I = 40 ~. Conse-quently, one should not expect any significant presence ofthe large deformations in this nucleus at the temperaturesufficiently high to wash-out the shell-effects. Our exper-

imental results seem to confirm this picture, although, apriori one cannot exclude the possibility that the sensitiv-ity of this particular experiment is not adequate to sam-ple the presence of very large deformations which couldbe weakly populated.

The width of the Boltzmann distribution reflects thethermal shape fluctuation amplitude, and therefore,shows the contribution of the shape fluctuations to theoverall GDR width. The deformation value correspond-ing to the maximum of the distribution is the equilibriumdeformation. Both the equilibrium deformation and thevariance of the distribution play important role in de-scribing the GDR width. From Fig. 9 one expects thatboth of these contributions to the GDR width are notchanging very much with spin. In fact, only a slight in-crease of the GDR width can be expected, as the proba-bility distributions become slightly broader, i.e. samplinglarger deformation but with very small probability.

The calculation of the GDR line-shape obtained withinthe thermal shape fluctuation model as a sum of all pos-sible line-shapes (for all deformation values) weightedwith the Boltzmann factor (cf. Ref. [16]) gives the GDRwidth values plotted as the dashed line in the top panelof Fig. 10. The calculations assume at T = 0 a spher-ical shape with a GDR width of 4 MeV. The obtainedpredictions show that the result of the average over theshape ensemble at T = 1 leads to an increase of the GDRwidth which in the present case is about 7 MeV.

In the top panel of Fig. 10 calculation results are com-pared to the present experimental results and to the pre-

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FIG. 10: Top Panel: The GDR width as a function of spinobtained for experimental GDR strength functions is shownwith points in the top panel in comparison with theoreticalpredictions (the horizontal error bars correspond to the half-width of the spin distribution selected by the gating condi-tions). The dotted line corresponds to results of calculationsperformed according to the Kusnezov formula. The GDRwidth calculated using thermal shape fluctuations based onpotential energies obtained with LSD model is drawn withthe dashed line.Bottom panel: The LSD model predictions for equilibriumdeformation βeq is presented in the bottom panel with dash-dotted line. The dashed line 〈∆β〉 (square root of the varianceof the β distribution) indicates the extent of the possible de-formations and 〈β〉 (solid line) is the average deformation.

dictions based on the same mechanism but using the pa-rameterization reported in [18, 19]. Both approaches givethe results consistent with the experimental values. Inaddition, in the bottom part of the figure we display themain features of the calculated liquid drop shape dis-tribution by showing the static equilibrium quadrupoledeformation (βeq), average dynamical deformation (〈β〉)and the standard deviation of the β distribution (〈∆β〉).The static equilibrium deformation is ∼0 up to spin 30 ~,and then slightly increases, while the standard deviationis almost constant with angular momentum. Both staticand dynamic deformations determine the shape evolutionof investigated nucleus. It is basically constant with spin- so is the behavior of the calculated GDR width.

VI. SUMMARY AND CONCLUSIONS

The present work demonstrates the possibility to studythe nuclear shape around the fission limit through the

GDR γ-decay in exclusive measurements selecting spinsand using the isomer tagging technique. The presentstudy is the first investigation of the Giant Dipole Res-onance width from the fusion-evaporation decay channelin this nuclear mass range.

Theoretical calculations at zero temperature, Ref. [20],predict a super-deformation in the 216Rn nucleus at thequadrupole deformation ∼ 0.5, with the potential barrierseparating the first and the second minima of a moderatesize of about 2 - 3 MeV. At an average temperature of〈T 〉 ∼ 1 MeV, the shell effects are about to disappear asdiscussed in the text. To which extent these two factors,viz. the shell effects and the temperature compete, is acrucial question to examine, since the reliable theoreticalpredictions about the population (or not) of super- andor hyper-deformed minima must be able to control thesetwo factors reliably. The fact that no sign of the super-deformed (more generally large deformation) shapes hasbeen obtained in the present study can be interpreted asanother confirmation of the earlier expectation that thetemperatures 〈T 〉 ∼ 1 MeV wash out the potential barriersof at least Vb ≤ 3 MeV.

According to the present study, the 〈T 〉 ∼ 1 MeV hot216Rn nucleus has nearly spherical (static equilibrium)shapes up to spin I = 30 and weakly oblate deformationswhen approaching the fission limit at I = 40. The stan-dard deviation of the thermal shape fluctuation ensemble〈∆β〉 obtained from our calculations is nearly constantin spin. The average dynamical deformation obtainedwith the Boltzmann factor described in the text is rathersmall (∼ 0.15) and does not change either with angularmomentum within the studied range.

The fact that the 216Rn nucleus investigated here wasfound to be almost spherical up to the fission limit isperhaps not surprising given the fact that the resultsof the calculations were known prior to the experiment;however, the extent to which this result agrees with thecalculations suggests that the progress achieved withinthe LSD approach is indeed encouraging. This has fur-ther implications: since the LSD-model parameters areindependent of the nucleus studied here (they were fittedto all the experimental masses known contemporarily),the parameter-free description of the Jacobi transition in46Ti of Ref. [15] and simultaneously of no-Jacobi transi-tion in 216Rn of this study provides important test cases.However, obviously more such test cases in various massranges will be needed to claim the full success of the newdescription of the nuclear macroscopic energies.

Acknowledgments

This work was supported by the Polish StateCommittee for Scientific Research (KBN GrantNo. 2 P03B 118 22), the Italian INFN, LNL Legnaro andthe Danish Sciences Foundation. A contribution fromthe exchange programme between the Institut Nationalde Physique Nucleaire et de Physique des Particules,

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IN2P3, and Polish Nuclear Physics Laboratories is acknowledged.

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