Using an improved scission-point model, the mass distributions are calculated for induced fission of even Hg isotopes with mass numbers A=174 to 196. With increasing A of a fissioning AHg nucleus the mass distribution evolves from symmetric for 174Hg, to asymmetric for isotopes close to 180Hg, and back to more symmetric for 192,194,196Hg. In the fissioning Hg isotopes their excitation energy weakly influences the shape of the mass distribution. In 180,184Hg, the mass distributions of fission fragments remain asymmetric even at high excitation energies

Adamian, Gurgen

Andreev, Alexander

Andreyev, A.N.

Antonenko, Nikolay

English

White Rose Research Online

PHYSICAL REVIEW C 88, 047604 (2013)Isospin dependence of mass-distribution shape of fission fragments of Hg isotopesA. V. Andreev,1 G. G. Adamian,1 N. V. Antonenko,1 and A. N. Andreyev2,3,41Joint Institute for Nuclear Research, 141980 Dubna, Russia2Instituut voor Kern- en Stralingsfysica, KU Leuven, B-3001 Leuven, Belgium3Department of Physics, University of York, York YO10 5DD, United Kingdom4Advanced Science Research Center, Japan Atomic Energy Agency, Tokai-mura, Ibaraki 319-1195, Japan(Received 18 September 2013; published 25 October 2013)Using an improved scission-point model, the mass distributions are calculated for induced fission of even Hgisotopes with mass numbers A = 174 to 196. With increasing A of a fissioning AHg nucleus the mass distributionevolves from symmetric for 174Hg, to asymmetric for isotopes close to 180Hg, and back to more symmetric for192,194,196Hg. In the fissioning Hg isotopes their excitation energy weakly influences the shape of the massdistribution. In 180,184Hg, the mass distributions of fission fragments remain asymmetric even at high excitationenergies.DOI: 10.1103/PhysRevC.88.047604 PACS number(s): 24.75.+i, 25.85.−w, 24.10.PaAn interest in the theoretical exploration of fission processwas restimulated several years ago after the experiment onβ-delayed fission of 180Tl [1,2], where the mass distributionof fission fragments of post-β-decay daughter nucleus 180Hgwas found to be asymmetric. This was surprising since froma native point of view the mass distributions were expectedto be symmetric for nuclei lighter than thorium. In the earlierexperiments on fission of Au-Po nuclei [3], a weak asymmetryof the mass distributions was observed as well, but thedeviations of these distributions from the Gaussian were foundto be rather small. The experiment [1] was followed by severaltheoretical works describing the observed asymmetric shapeof the fission-fragment mass distribution [4–7]. The existingexperimental data on the fission of 180Hg and 198Hg were welldescribed with our improved scission-point model [4]. In theprevious study we found that the asymmetry of the shape ofmass distribution decreases with increasing mass number A ofthe fissioning even AHg nucleus from A = 180 to A = 198.However, the fissions of the Hg isotopes lighter than 180Hgwere not covered in our previous treatment. In the studyof low-energy fission of neutron-deficient mercury isotopeswith the Brownian Metropolis shape-motion treatment [5],the mass distribution is found to be symmetric for 174Hgand becomes asymmetric with growing mass number of thefissioning mercury isotope to 180Hg. The present work is anextension of our previous calculations [4] to a wider range ofHg isotopes in order to reveal the dependence of the shapeof fission-fragment mass distribution on the mass numberA and excitation energy E∗CN (l = 0) of the fissioning AHgisotope and to compare the results obtained with those ofother theoretical models. Here, l is the angular momentum ofthe fissioning nucleus.We describe the fission observables within the improvedscission-point model [4,8] which operates with the relativeprobabilities of formation of different scission configurationsobtained with the statistical approach by calculating thepotential energy of scission configurations. At the scissionpoint the fissioning nucleus is modeled by a dinuclear systemconsisting of two nearly touching prolate coaxial spheroidswith deformation parameters defined as the ratios of themajor and minor semiaxes of the spheroids βi = ci/ai , wherei = L,H denotes light and heavy fragments of the dinuclearsystem, respectively. The mass and charge numbers of thefragments are Ai and Zi , respectively. A = AL + AH andZ = ZL + ZH are the mass and charge numbers of thefissioning nucleus, respectively. The potential energyU (Ai, Zi, βi, R, l)= ULDL (AL,ZL, βL)+δU shellL (AL,ZL, βL)+UzpvL (AL,ZL) + ULDH (AH,ZH , βH )+ δU shellH (AH,ZH , βH ) + UzpvH (AH,ZH )+V C(Ai, Zi, βi, R)+V N (Ai, Zi, βi, R)+V rot(Ai, Zi, βi, R, l) (1)of the dinuclear system consists of the energies of thefragments, energy of their interaction, and rotational energyV rot in the case of induced fission. The energy of eachfragment consists of the liquid-drop energy ULDi , deformationdependent shell-correction term δU shelli , and the energy of thezero-point vibrations Uzpvi = E2+i coth[E2+i /T (l)], associatedwith the energy E2+i of the first 2+ excited state. The dampingof the shell corrections with temperature is introduced in orderto take into account the influence of the excitation energy.The interaction potential consists of the Coulomb interactionpotential V C of the two uniformly charged spheroids andnuclear interaction potential V N in the form of a doublefolding of Woods-Saxon nuclear densities of the fragments andSkyrme-type density-dependent nucleon-nucleon interaction.The interaction potential has a pocket at a distance betweenthe tips of the fragments of about 0.5 to 1 fm (in the consideredregion of fissioning nuclei) depending on deformations of thefragments. The internuclear distance R in (1) correspondsto the position of this pocket (minimum) R = Rm(βi). Theexcitation energy E∗(Ai, Zi, βi, Rm, l) at the scission iscalculated as the initial excitation energy of the fissioningnucleus plus the difference between the potential energies ofthe fissioning nucleus and the dinuclear system at the scissionpoint. The details of the calculations of the potential energyare given in Ref. [4].047604-10556-2813/2013/88(4)/047604(4) ©2013 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW C 88, 047604 (2013)FIG. 1. Calculated (solid line) and experimental [2] (pointsconnected by dashed line) mass distributions of fission fragmentsfor β-delayed fission of 180Tl (fissioning nucleus is 180Hg).The relative formation probability of the dinuclear systemwith particular masses, charges, and deformations of thefragments is calculated within the statistical approach asfollows:Y (Ai, Zi, βi, l) ∼ exp{−U (Ai, Zi, βi, l)T (l)}. (2)The temperature is T (l) = √E∗l /a, where E∗l =max[{E∗(Ai, Zi, βi, Rm, l)}] and a = A/12 is the leveldensity parameter. In order to obtain the mass distributionin the fission of a particular nucleus with the mass numberA and charge number Z, one should integrate (2) over ZL,βL, and βH , sum over l, and take into account thatAL + AH = A and ZL + ZH = Z. Finally, for the calculationof the mass distribution the following expression is obtained:Y (AL)=∑lmaxl=0(2l + 1)∫exp{−U (Ai,Zi ,βi ,l)T (l)}dZLdβLdβH∑lmaxl=0(2l + 1)∫exp{−U (Ai,Zi ,βi ,l)T (l)}dALdZLdβLdβH.(3)This distribution is normalized to unity. The value of theangular momentum l is limited by either the kinetic angularmomentum lkin = Rb√2μ(Ec.m. − Vb) [R = Rb is the positionof the Coulomb barrier with the height Vb = V C(Ai, Zi, βi =0, R = Rb) + V N (Ai, Zi, βi = 0, R = Rb) in the entrancechannel, μ = m0ALAH/(AL + AH ) is the reduced mass pa-rameter, and m0 is the nucleon mass] or by the calculated crit-ical angular momentum lcr in the entrance channel, dependingon which one is smaller: lmax = min{lkin, lcr}.For a comparison with the recent experimental data [2],we performed the calculations for the fission of the post-β-decay daughter nucleus 180Hg with the excitation energy ofE∗CN (l = 0) = 10.44 MeV (Fig. 1). As in the experiment [2],we obtained an asymmetric fission-fragment mass distribution.In Fig. 1, more experimental points are shown than in thecorresponding figure of Ref. [4]. The calculated average massof the light fragment is AL = 79 and the width of the massdistribution is σAL = 5. The Total Kinetic Energy (TKE) wascalculated by assuming that all the interaction energy at thescission point is transformed into the kinetic energy after alarge separation of the fission fragments. The average valueTKE = 136 MeV is found to be in good agreement with theexperimental observation.To reveal the dependence of the shape of the fission-fragment mass distribution on the mass of the fissioning Hgisotope, we treated the isotopes 174−196Hg with even massnumbers at zero angular momentum l = 0. The calculationswere performed at the excitation energies 2 and 20 MeV abovethe fission barriers. The results are presented in Figs. 2 and 3.For 174Hg, the fission-fragment mass distribution is symmetric.With increasing mass number of Hg the asymmetry appearsand becomes maximal for 180,182Hg. From 182Hg to 190Hg thedip at the symmetric mass division remains, but the asymmetrybecomes smaller due to the decrease of the width of thedistribution. From 190Hg to 196Hg the distribution remainsnarrow and the dip at the symmetric mass division diminishes.This result slightly differs from that of Ref. [5], where themass distribution was found to be asymmetric as well for180−188Hg, but the asymmetry increases with the mass numberof the fissioning Hg isotope. In our calculations (Figs. 2and 3), the asymmetry of the mass distribution decreases withincreasing excitation of the fissioning Hg isotopes. This isin contrast to the results of Ref. [5] where the asymmetryof the fission-fragment mass distribution increases with theexcitation energy in AHg with A < 180.FIG. 2. Calculated mass distributions forthe fission of (a) 174Hg, (b) 176Hg, (c) 178Hg,(d) 180Hg, (e) 182Hg, and (f) 184Hg at l = 0 andexcitation energies of 2 (solid lines) and 20MeV (dashed lines) above the correspondingfission barriers.047604-2BRIEF REPORTS PHYSICAL REVIEW C 88, 047604 (2013)FIG. 3. The same as in Fig. 2, but for thefission of (a) 186Hg, (b) 188Hg, (c) 190Hg, (d)192Hg, (e) 194Hg, and (f) 196Hg.In order to explain the evolution of the shape of thefission-fragment mass distribution calculated, one can analyzethe dependence of the potential energy on the deformationsof the fragments at the scission point. As known, the shelleffects are responsible for the appearance of the asymmetricmass distributions. Because of the strong Coulomb repulsionand, correspondingly, of the large polarization of nuclei, thepotential energy (1) always has a minimum at deformationsof the fission fragments of about βi = 1.5 to 1.8, that islarger than their ground state deformations. Hence, the shelleffects at large deformations mostly influence the shape of thefission-fragment mass distribution. The shell effects in nucleinear the ground state play a minor role if they are not verystrong. The shell effects in nuclei around semimagic 90Zr arecertainly weaker than in the nuclei around doubly magic tinand lead.In the vicinity of the symmetric mass division of 180Hg,the shell corrections in the fragments close to the semimagicnucleus 90Zr are negative at small deformations, but at largerdeformations, around the liquid-drop minimum of the potentialenergy, the shell corrections are positive and comparable withthe shell corrections for the asymmetric split 76Ge + 104Pd [4].As a result, the asymmetric prescission configuration can havea smaller potential energy due to a smaller nucleus-nucleusinteraction. That is why the presence of magic fragmentsdoes not lead to a large yield of symmetric mass split inthe fission of 180Hg. However, for the symmetric scissionconfiguration the rigidity of the 90Zr nucleus leads to a verynarrow minimum on the potential energy surface as a functionof fragment deformations (see [4]). Due to the integrationin (3), this narrow minimum leads to the reduction of theprobability of a symmetric mass split. In contrast, for theasymmetric mass split of 180Hg, the potential energy as afunction of fragment deformations has a wide minimum. Theother factor, which suppresses the symmetric mass split in thefission of 180Hg and nearby isotopes, is rather large valuesof E2+i around 90Zr contributing to the terms UzpvL + UzpvH in(1). All the factors mentioned above result in the asymmetricshape of the mass distribution in the fission of 180Hg and nearbyisotopes.In the isotopes of mercury lighter than 180Hg, the shelleffects become rather weak and the values of E2+i becomecomparable for all mass divisions. Therefore, the shape of thefission-fragment mass distribution is defined by the liquid-drop part of the potential energy that leads to the symmetricmass distributions. For the region of heavier mercury isotopes,the influence of the shell effects at the symmetric mass splitZr + Zr is not so strong as in the 90Zr +90 Zr case. However,the subshell at proton number Z = 40 for Zr results in a dip ata symmetric mass split similar to that in 180Hg [4].In the region of mass numbers A = 180 to 200, one can findnuclei with comparable yields of symmetric and asymmetricfission modes. The mass distributions of fission fragments forthese nuclei are expected to be very wide. In the fission of the196Po nucleus, which is of the present experimental interest,we expect the comparable yields of symmetric and asymmetricfission modes.To study the influence of bombarding energy (excitationenergy and angular momentum) on the shape of the fission-fragment mass distribution, we consider the reactions 36Ar +144Sm → 180Hg at bombarding energies in the center of mass128 (lmax = 18), 136 (lmax = 44), 152 (lmax = 61), 176 (lmax =61), and 200 MeV (lmax = 61) and 40Ar + 144Sm → 184Hg atEc.m. = 125.2 (lmax = 11), 133 (lmax = 42), 148.7 (lmax = 71),172.2 (lmax = 71), and 195.6 MeV (lmax = 71) (Figs. 4 and 5).FIG. 4. Calculated mass distributions for the fission of 180Hg withthe excitation energies of compound nucleus E∗CN (l = 0) = 36 (solidline), 44 (dashed-dotted-dotted line), 60 (dashed line), 84 (dottedline), and 108 MeV (dashed-dotted line).047604-3BRIEF REPORTS PHYSICAL REVIEW C 88, 047604 (2013)FIG. 5. Calculated mass distributions for the fission of 184Hgwith the excitation energies of compound nucleus E∗CN (l = 0) = 34.5(solid line), 42.2 (dashed-dotted-dotted line), 58 (dashed line), 81.5(dotted line), and 105 MeV (dashed-dotted line).For the 36Ar + 144Sm (40Ar + 144Sm) reaction, the bombardingenergy Ec.m. = 128 MeV (125.2 MeV) corresponds to theexcitation energy of the compound nucleus E∗CN (l = 0) =36 MeV (34.5 MeV). Because the zero-point vibration effectsare slowly reduced with increasing excitation energy, the massdistribution of the fission fragments of nuclei 180Hg (184Hg)has a pronounced asymmetric shape even at high excitationenergy E∗CN (l = 0) = 108 MeV or E∗CN (l = lmax) = 101 MeV[E∗CN (l = 0) = 105 MeV or E∗CN (l = lmax) = 96 MeV]. It isinteresting to stress that the experimental mass distributionof the fragments in the fission of post-β-decay daughternucleus 180Hg has a similar shape as the calculated massdistribution of the products in the 36Ar + 144Sm reaction. Athigh angular momenta some dinuclear scission configurationsare energetically more favorable than the compound nucleusconfiguration and the fissionlike fragments in the 36Ar(Ec.m. =152 to 200 MeV) + 144Sm reaction can be mainly produced byquasifission (but not by a fusion-fission process) [9].The improved scission-point model [4,8] provides a gooddescription of the experimental data on fission of 180Hg aswell as of 198Hg (see [4]). In our model, the dependenceof the shell effects on deformations of fission fragments atscission plays the key role in the description of the asymmetryof the fission-fragment mass distribution. With increasingmass number of the fissioning mercury isotope we predictedthe evolution of the fission-fragment mass distribution fromsymmetric for 174Hg, to asymmetric for isotopes around 180Hg,and to more symmetric for 192,194,196Hg. The existence of bothsymmetric and asymmetric fission modes is expected in thefission of 196Po. For all fissioning Hg isotopes, the influenceof the excitation energy on the shape of the mass distributionwas found to be rather weak. In the cases of 180,184Hg the massdistributions have a pronounced asymmetric shape even at thehigh excitation energies of the compound nuclei. All thesepredictions can be checked in the experiments.This work was supported by RFBR (Moscow) and JINRGrant No. 13-302-01. The IN2P3(France)-JINR(Dubna) andPolish-JINR(Dubna) Cooperation Programmes are gratefullyacknowledged.[1] A. N. Andreyev et al., Phys. Rev. Lett. 105, 252502(2010).[2] J. Elseviers et al. (unpublished).[3] M. G. Itkis, N. A. Kondrat’ev, S. I. Mul’gin, V. N. Okolovich,A. Ya. Rusanov, and G. N. Smirenkin, Sov. J. Nucl. Phys. 53,757 (1991).[4] A. V. Andreev, G. G. Adamian, and N. V. Antonenko, Phys. Rev.C 86, 044315 (2012).[5] P. Mo¨ller, J. Randrup, and A. J. Sierk, Phys. Rev. C 85, 024306(2012).[6] M. Warda, A. Staszczak, and W. Nazarewicz, Phys. Rev. C 86,024601 (2012).[7] S. Panebianco, J.-L. Sida, H. Goutte, J.-F. Lemaıˆtre, N. Dubray,and S. Hilaire, Phys. Rev. C 86, 064601 (2012).[8] A. V. Andreev, G. G. Adamian, N. V. Antonenko, S. P. Ivanova,and W. Scheid, Eur. Phys. J. A 22, 51 (2004).[9] Sh. A. Kalandarov, G. G. Adamian, N. V. Antonenko, andW. Scheid, Phys. Rev. C 82, 044603 (2010); 83, 054611 (2011);Sh. A. Kalandarov, G. G. Adamian, N. V. Antonenko, W. Scheid,and J. P. Wieleczko, ibid. 84, 064601 (2011).047604-4

Isospin dependence of mass-distribution shape of fission fragments of Hg isotopes

A. V. Andreev

G. G. Adamian

N. V. Antonenko

A. N. Andreyev

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Physical Review C

https://eprints.whiterose.ac.uk/77073/1/PhysRevC.88.047604.pdf

Isospin dependence of mass-distribution shape of fission fragments of Hg isotopes

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